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Modelling of insulin release from a glucoseresponsive degradable polymeric system
Journal of Membrane Science, 48 (1990) 33-54 Elsevier Science Publishers B.V., Amsterdam - Printed in The Netherlands
MODELLING RESPONSIVE
33
OF INSULIN RELEASE FROM A GLUCOSEDEGRADABLE POLYMERIC SYSTEM
FARIBA FISCHEL-GHODSIAN Department of Engineering Science, Oxford University, Oxford (Great Britain) and J.M. NEWTON The School of Pharmacy, University of London, Brunswick Square, London WClN IAX (Great Britain) (Received July 15,1988; accepted in revised form May 24,1989)
Summary Drug delivery devices which release therapeutic agents at a predetermined constant rate significantly improve drug therapy. However, for diabetic therapy the ideal system is one that delivers insulin in response to blood glucose levels. In this work, a degradable polymeric system containing insulin and glucose-oxidase is described and its performance simulated and compared to that of a normal pancreas. The system considered is a degradable polymer matrix whose erosion is pH dependent, and which is surrounded by a membrane. The feedback mechanism is provided by the enzymatic reaction between glucose and glucose-oxidase. Acid produced from this reaction reduces the pH on the surface of the polymer, and causes an increase in polymer degradation and release rate. Based on simulation results, two possible therapeutic systems are proposed which closely mimic the normal pancreas. One is an implantable system which operates over long periods (6 months), and the other is an injectable device for short periods (one week). The study can therefore provide a useful tool in directing further experimental studies and providing a rational basis for system design.
Introduction Over the last few years polymeric drug delivery systems have shown a great potential in delivering drugs over long periods and maintaining the drug concentration in the blood at a constant level [ 1,2]. For some drugs, however, a constant blood level profile is not optimal. A classic example is insulin, for which the normal physiological concentration varies significantly upon changes in blood glucose level. An ideal delivery of insulin to diabetic patients requires a system which delivers insulin in response to changes in the physiological glucose level. A closed-loop glucose-sensitive system would be an ideal way to deliver insulin, and research in this area includes encapsulation of pancreatic p cells [ 31, synthesis of glycosylated insulin molecules [4,5] and design of glucose-sensi-
0376-7388/90/$03.50
0 1990 Elsevier Science Publishers B.V.
34
tive membranes [ 6,7] and polymeric systems [ 61. One new approach to design a glucose-sensitive system is the use of pH-sensitive degradable polymers [ 9,101. In these polymers, the degradation rate can change by few orders of magnitude over a narrow pH range, which depends on the chemical properties of the polymer molecule [ 111. Since the drug release rate from degradable polymers is proportional to the erosion rate, the high sensitivity of the erosion rate to pH suggests that these polymers may provide a self-regulating drug delivery system. This concept has been successfully shown using area as the drug model
[=I.
In this work, the performance of a glucose-sensitive degradable polymeric system, and the effect of different parameters on its performance, are assessed by mathematical modelling and computer simulations. The findings will provide a guide for further experimental work in designing a system as a replacement for pancreas, which would obviate the need for repeated injections or the problems associated with an indwelling pump system. Mathematical modelling i. Polymeric system In this section, a mathematical model of insulin release dispersed in an erodible polymeric device is formulated. The basic components of this system are a pH-sensitive surface-erodible polymer, surrounded by a membrane containing immobilised glucose-oxidase. As the system is exposed to a glucose solution, glucose diffuses through the membrane and is converted to acid by the enzymatic reaction. The acid produced will reduce the pH on the surface of the polymer. This will increase the surface erosion rate, and more insulin will be released. The release kinetics are modelled by diffusion and surface erosion, and the effect of glucose on the pH on the surface of the polymer, and the effect of pH on polymer erosion. A schematic diagram of the system is shown in Fig. 1. The effect of glucose on the pH at the surface of the polymer is modelled by considering the following: a. Enzymatic reaction between glucose and glucose-oxidase [13,14]: E,, + glucose 2
Eox_o-
k,
Ered+ glucono-lactone
k,. Eredf 0, -
E,, + H,O,
C&3lySe HzOz
9
H,O+$O,
Hydrolysis of glucono-lactone is given by [ 151:
35
glucono-lactone + Hz0 -
khyd
gluconic acid a
kz
k-z
gluconate ion- + H+
and the rate of production of glucono-lactone will be:
&WI [%I [Gl
r=
[%I [Gl+;
[%I +;
tG1 ox
where, k=
lzJG k_,+k,
b. Bicarbonate body buffer reaction [16]: COz+H20
+
HCO, +H+
c. Diffusion of solutes through the polymer and membrane:
$D$
(Fick’s law )
The diffusion coefficient of solutes inside the polymer and membrane is taken
~eleose media
0
U(t)
\ Polymer
s(t)
s(t)+hm
x
, Membiane
Fig. 1. Schematic representation of the model for effect of degradable polymer device.
on insulin release rate from
36
The diffusion coefficient of solutes inside the polymer and membrane is taken as the diffusion coefficient in water divided by a tortuosity factor T. This factor accounts for the tortuous path of diffusion in a porous medium [ 17 ] and is different for the polymer and membrane. A one-dimensional model of the system was constructed based on the following assumptions: 1. No concentration gradient exists in the radial direction. 2. Particles of enzyme and insulin are distributed homogeneously inside the polymer and the enzymatic reaction is homogeneous throughout the pores. 3. Glucose concentration is rate limiting in the production of gluconic acid, due to the lower diffusivity of glucose and the excess amount of the enzyme catalyst. The concentration of oxygen inside the pore is thus assumed to be equal to its concentration in the extracellular fluid. 4. Diffusion coefficients of solutes are independent of the concentration inside the pores (dilute solution). 5. No transport by convection and ion exchange is considered. Only passive diffusion is considered. 6. Effect of electrical field is neglected since the reactions are electrically balanced. Although some gradients in charge density exist at the boundary of the pores and polymer, a uniform charge density and thus a small field can be assumed overall, especially at steady state [ 181. This assumption has also been used by other researchers in similar systems [ 191. 7. Degradation products do not influence the pH, and pH changes are mainly due to changes in glucose concentration. To model this effect, equations of conservation of mass for glucose (G), glucono-lactone (GL), gluconic acid (GA), gluconate ion (GA-), hydrogen ion (H+ ), carbonate ion (HCO; ), and carbon dioxide (CO,) were written in 2 domains: membrane and polymer. For each solute the left hand side of the equation shows the changes in concentration with time, and the right hand side describes the appropriate reaction and diffusion terms. The initial conditions, and the boundary conditions at x = s ( t) + I, are the physiological concentrations of glucose, hydrogen ion, carbonate ion, and carbon dioxide in the body [ 181, and are taken as zero for other solutes (perfect sink condition). In the polymer domain, no flux boundary condition (X/&r= 0) is used at x = u(t), for all the solutes. The unknown concentrations at the boundary between polymer and membrane (x = s (t) ) are related by the flux continuity and partitioning condition. ep D, (dC/dx) polymer = em% ( dC/dr ) membrane
(1)
C, = KC,
(2)
where subscripts p and m represent polymer and membrane, respectively, e is porosity, and K is partition coefficient.
37
The following differential equations, representing the effect of glucose on the pH on the surface of the polymer were generated: d [cl/at=
-k[El
LO21VW{ [021 [Gl +WWO,l
+k,[Wk,x}+ KW~W[W~~2)
(3)
8 [GLll~~=~[El[O21 WI/{ 1021 [Gl +WWO,l
+MGlllz,,)-FZhydW a[GA]/dt=lz,,,[GL]
+k_,[H+]
[GA-‘]
+&JW2[Wl~x2)
(4)
-k,[GA]
+DGA/T(d2[GA]/c3x2)
(5)
a[GA-]/dt=-k_,[H+][GA-]+$[GA]+D,,-/T(d2[GA-]/ax2)
(6)
d[H+]/c3t=-k_2[H+][GA-]+k2[GA]+ka[C02] -k_3[H+] r3[HCO,]/dt=lz,[CO,]
-k_,[H+]
[HC0,]+D,+/T(8’2[H+]/~x2)
(7)
[HCO,] +D
HCO~-/~(~“[H~~,I/~~~) (8)
~[CO,]/~t=-k,[CO,]+k_,[H+][HCO,]+Dco,/T(~”[CO2]/dx2)
(9)
and the pH on the surface of the polymer is: pH= -log[H+]
at x=s(t)
(19)
The values of the reaction rate constants and diffusion coefficients at 25’ C* and their source used in the model are summarised in Tables 1 and 2. The effect of pH on surface erosion rate constant (B) was modelled as a straight line in the resulting pH range, similar to the effects which have been observed experimentally [ 111. The erosion rate was assumed to be independent of pH outside this region, and B was assumed to increase by a maximum of 25 fold**. This is expressed as: B=c(pH)+d
PH,
B=B,,
PH>PH,
B=25B,
PH
(11)
where c and d are the slope and intercept of the straight line, and B. is the *Simulations at 37°C show very similar results [21]. This is due to the fact that higher temperature increases the rate of the enzymatic reaction, the buffer reaction and also the rate of the diffusion. These effects cancel each other and so the overall effect is negligible. **This is chosen in order to achieve a performance similar to the pancreas.
38 TABLE 1 Reported values of the rate constants
for the enzymatic
Rate constant
buffer reaction at 25’ C
Reference
k,=400
set-’
k=4.3x103
M set
k,.=1.2x106 k,,,.,=3.4x
and bicarbonate
13 13 13 14 15 22
M-‘-set-’ see-’ (at pH=7.4)
10-3”
M M-‘-set-’ see-’ see-’ N-‘-see-’
k,,,=2.0x10-4 k_a=5.0x106 kz = 1000b k, = 0.04 k_,=5.5x104
16 16
“The rate constant k,,, strongly depends on pH. Its change with pH in phosphate is given by [ 141 k hyd=1.27x10-1[H+]+1.08x104[OH-]+6.75X10-4(sec-1)
buffer solutions (11).
bk, = k_zkzeq
TABLE 2 Reported
values of the diffusion coefficient
in water at 25” C
Other parameters were taken as [E] = 10-s M, equal to the enzyme concentration used in a similar system [ 81, and [ 0,] = 4.5 x lo- M, oxygen concentration in the extracellular fluid a (25 ) Solute
Diffusion coefficient (cm”/sec)
H+ CO, HCO,H&G, Glucose Glucono-lactone Gluconic acid Gluconate ion Insulin
9.3
1.72 1.17 1.17 0.67 0.67b 0.67b 0.67b 0.2
X 10e5
Reference
17 23 23 23 17
24
“This value is taken equal to the concentration of dissolved insulin in venous blood. bThe diffusion coefficients for glucose, glucono-lactone, gluconic acid, and gluconate ion are assumed to be equal, since they have similar molecular weights.
baseline erosion rate constant when system is exposed to the physiological glucose concentration. The drug release rate from a degradable polymer surrounded by a membrane has been previously modelled [ 26,271 by considering drug diffusion and polymer degradation, and was shown to be: R=D~C,M/(a+M1,)
(12)
39
where D, is insulin diffusion coefficient in the polymer, A is the surface area of the device from which insulin is released, C, is insulin solubility, a is membrane thickness, 1, is diffusion length in the polymer, and M is defined as: M= aTA,/l,T,,,K
(13)
Changes of the diffusional length with time, and the time needed for the system to reach steady state, are given by: dZJdt= -B+D,
Mu/[ (a+M&)C,,/C,-l]
t, S= 1.4D,/B 2(C,/C, - 1) + O.Sa/MB
(14) (15)
and at steady state the release rate is directly proportional to B:
4 ,=BAG{ (G/G) -4
(16)
where C, is insulin loading.
ii. Glucose metabolism The performance of the polymeric system in uiuo was simulated and compared to that of the pancreas, using a simplified version of the model of Hosker et al. [ 281, which describes the physiological regulation between glucose and insulin and correlates excellently with experimental data of intravenous glucose tolerance tests. The effect of glucose on insulin release rate from the pancreas was modelled by assuming an immediate response to changes in glucose level based on in uiuo experiments which have demonstrated insulin response within 30 seconds after a stimulating concentration of glucose reaches the pancreas [ 291. A sigmoidal, steadily increasing curve describes the relationship between insulin secretion and glucose concentration [ 30,311:
R={0.4+0.35(tanh(7.93Xl@
(G,,-9.3x10-3))+0.73)}6~10-3
(17b)
G,>8.13~10-~ Distribution of insulin in the body was modelled by assuming a single volume of distribution (Vi) and first-order disappearance kinetics. Plasma insulin concentration (4) changes as insulin is produced by the pancreas and is cleared through the body:
40 TABLE 3 Steady state values for the glucose metabolism model [ 281 Variable
Basal steady state values for a normal 70-kg man 4X 10m3mol/l 2 X 10m7mg/ml 8 X 10m4mol/min 4 X 10e4 mol/min 4X lo-l4 mol/min 151 13000 ml 0.173 min-’ 4X 10m4mg/min
dl,/dt = R/ Vi- k,I,
(18)
The overall time delay in the action of insulin was accounted for, by introducing an “effective insulin concentration”, I,. I, represents the amount of insulin being effective through processes such as binding to receptors, and enzyme activation. Because of insufficient quantitative data, the same value of clearance rate constant ki was assumed in and out of the effector compartment [ 281: dZ,/dt= ki (Ip-I,)
(19)
The metabolism of glucose in the body was modelled by assuming a single volume of distribution for glucose (V,) with separate turnover for brain, liver and insulin-sensitive periphery including muscle and fat. Blood glucose concentration (G,) changes as glucose is added by food uptake ( GF) or liver ( GL) and is taken up by periphery (GM) and brain (Gn ) : dG,/dt=
(GF+GL--GB-GM)VG
(20)
Brain and muscle glucose uptake and the net production or uptake of glucose in liver has been shown previously [ 301 and are described by the following equations fitted to the experimental curves [ 311: Gn=O.4~10-~
G,/(O.~X~O-~+G~)
(21)
G,={(~.~~G,)/~.OX~O-~+G,} x{(1.0x10-“1,)/@.0x10-7+1,)+0.2x10-3}
(22)
41
GL= [2.131-1.05
(In (ln(25x1061,)}-0.6{ln(1000G,)}]/1000
(23)
Values of the parameters and variables at steady state in the units used in the model are given in Table 3. Simulation studies The performance of the polymeric system and the pancreas were simulated on VAX 11-780, using SIMNON, a simulation program for solving non-linear ordinary differential equations [ 321. The partial differential equations were converted into ordinary differential equations by approximating the diffusion terms using an explicit finite difference method [ 331, and the diffusional length was divided into four equal sections. This provided adequate accuracy and smooth curves, since concentration gradients were small throughout. Division of the length into more sections did not affect the simulation results [ 211, and was not used since it requires more computing time. Simulation studies of the pancreas were carried out by using the Runge-Kutta algorithm [ 341. However, simulating the performance of the polymeric system required an algorithm which deals with stiff equations, as the time constants of the equations were highly non-uniform. This system was simulated using algorithm DAS,3 in SIMNON [ 351, which deals with systems with only few equations causing “stiffness” by partitioning the system into “stiff” and “non-stiff” sections and interacting between them. The correct partitioning of the equations into slow and fast was achieved by placing the equations describing the rate of change of [GA] and [H+ ] (eqns. 5 and 7) in both domains into the fast section. Other equations were placed into the slow section. To investigate the effect of glucose on insulin release rate, both the open loop and closed loop performances of the polymeric system were simulated. i. Open loop performance The effect of glucose concentration on the pH of the surface of the polymer was studied by simulating the pH after one hour exposure to a glucose level of 500 mg/dl. The effect of membrane thickness on pH was simulated for different values of parameters M ’ and K. Parameter M ’ was defined as: M’=l,T&,T,
(24)
and was taken equal for all the solutes, assuming they all have the same tortuosity factor. The partitioning coefficient K was also assumed to be the same for all the solutes. ii. Closed loop performance The closed loop performance of the system was studied by simulating the blood glucose level in a diabetic patient, in whom the device was implanted as
42
a replacement for the normal pancreas. The parameters of the system were chosen to achieve a performance similar to the normal pancreas, considering both the short-term response to glucose and the long-term supply of insulin. The acceptable range for each requirement and the practical value of the parameters, defined a range of values which could be used for each parameter. The requirements specified for the system considering its short and long term performance and size are: 1.As the system is exposed to glucose concentrations higher than 0.01 M, the insulin release rate should increase by 25 fold within 5 minutes. 2. The system should provide a baseline release rate of 4 x 10M4mg/min and a 6-month supply of insulin. 3. The total volume of the system should not exceed 2 cm3, and each side of the system should not exceed 5 cm. The complicated relationship between the specified requirements of the system and its parameters are shown in Fig. 2. Based on the open loop performance and previous analysis [ 261, the following values of the parameters were used (see Appendix). The baseline erosion rate constant was taken as B,, = 1 x lop5 cm/min and the erosion rate was assumed to increase by 25 times over 0.1 pH units. These values are well within the range for reported data for degradable polymers, as erosion rate constants in the range of 1 x 10m6-1 x low3 cm/min, and changes
Sufficient increase in B over the desired pH range
Sufficient baseline insulin
Fig. 2. Representation parameters.
of the relation
between
the specified
properties
of the system
and its
of as high as 1000 times in erosion rate over 2 pH units have been reported [ll]. In addition, synthesis of a degradable polymer whose erosion rate increases as pH decreases around pH 7.4 is currently under development [ 91. The retardation factor 2’ was taken as 10 and 4000 for the membrane and polymer, respectively. The high diffusion coefficient (low T) in the membrane can be achieved by using a hydrogel, since in highly swollen hydrogels the drug diffusivity may approach the value observed in pure water [ 361, and diffusion coefficients corresponding to T equal to 14 and 2.5 have been reported for insulin through porous membranes [ 37,381. The low diffusivisity in the polymer can be achieved by using hydrophobic polymers which show diffusion coefficients of few orders of magnitude lower than in water [ 171. The solubility of insulin was taken as 120 mg/ml, which is the reported value for sodium-insulin [ 241. This high insulin solubility was required to reduce the size of the system. The insulin loading was taken at 480 mg/cm3, within the range of 200 to 500 mg/cm3 used in polymeric systems [ 241. The geometry of the system was taken as slab or disk, and all the surfaces except one (with surface area of 0.11 cm’) were coated to provide a constant surface area and release rate with time. A membrane containing 10m5 M of
. ,.: ,: : :. .:::..: ..; _,_ .:..i ..,: ..,. :_..; a /
0.2
J
_‘:
~‘~~‘.-.:.~::(‘j:‘..l:j:h
0 33
j
: .::;:;i:;: :,
(b)
..__ ,____ / S:$
-0
33
I 0.013
Fig. 3. Schematic able device.
pictures of the proposed polymeric systems:
(a) implantable
device; (b) inject-
44
immobilised glucose-oxidase, with a thickness of 0.013 cm, was surrounding the polymer matrix. Two alternative systems which only differ in their length were proposed: (a) an implantable system of 5 cm length which provides a 6month supply of insulin (Fig. 3a), and (b) an injectable device of 0.2 cm length which provides a one week supply of insulin (Fig. 3b). The size of this device is that of a grain of rice, similar to a commercially available injectable degradable device for the treatment of cancer [ 391. The specified parameters listed above are not unique, since it is not possible to choose one set of parameters as optimal. While in one case size limitation may be the most important factor, in other cases the properties of the polymer or membrane may be the limiting factor. In addition, the specified requirements of the system can be different for different situations, and there can be a trade off between the short-term response to glucose and the long-term supply of insulin. The blood glucose regulation in the proposed systems were simulated and compared with that of the normal pancreas. This was undertaken by simulating the daily plasma glucose profile for a normal diet. A 30 kcal/kg body weight diet, consisting of 45% carbohydrates, was assumed. The total calories were distributed as 20,30, 35, and 15 percent for breakfast, lunch, dinner, and late snack [ 401, which were given at 8am, lpm, 6.30pm, and 10pm. The duration of absorption for each meal was assumed to be 60 min, 120 min, 90 min, and 45 min, respectively. Results and discussion i. Open loop performance The open loop response of the system to glucose is shown in Figs. 4 and 5. Figure 4 shows the pH on the surface of the polymer, after one hour exposure to a glucose level of 500 mg/dl. The result is plotted as a function of the square of the effective thickness of the membrane (JZ’, x 1,)) for different values of PH
M'=O.Ol
Fig. 4. Simulation results for the effect of effective membrane thickness on the pH on the surface of the polymer. Simulation results are shown for different values of the parameter M.
45
M. Other parameters of the system such as the effective diffusion length in the polymer, partition coefficient, and enzyme concentration are taken to be constant as JTpx l,,= 1 cm, K= 1, and [E] = 1 x 10W5M.The figure shows that as membrane thickness increases, the pH decreases to a minimum and then increases. This is the result of two factors: as membrane thickness increases (i) the steady-state pH decreases more since the produced H+ is better protected from the presence of buffer, (ii) the response time increases, since it is proportional to T,&,‘. Combination of these two factors results in an optimal membrane thickness, at which maximum changes in pH occur. The results also show that a larger change in pH occurs as parameter M increases. This is because as the value of MK increases, the concentration of H+ at interface approaches its value inside the membrane, and this is higher than its value inside the polymer, since enzymatic reaction occurs in the membrane. The effect of the partition coefficient, K, on pH is shown in Fig. 5, for PH
T,l,*(cm2)
Fig. 5. Simulation results for the effect of effective membrane thickness on the pH on the surface of the polymer. Simulation results are shown for different values of the parameter k.
0
’0
I
4
8
1
12
I
16
I
20
I
2b
Time (hour)
Fig. 6. Simulation results of the daily blood glucose profile regulated by the pancreas. fast, L - lunch, D -dinner, S -snack.
B-
break-
8
12
16
20
24
Time (hour)
(W
12
16
20
24
Time (hour)
Fig. 7. Simulation results of the daily blood glucose profile regulated by the implantable degradable polymeric system, at t=O (a), and t=6 months (b). B-breakfast, L-lunch, D -dinner, Ssnack.
,/T, x 1,= 1 cm, iW= 100, and [E] = 1 x 1O-5 M. As expected, the value of pH decreases to a greater extent as K increases due to the H+ concentration on the polymer side of the interface becoming higher than the concentration on the membrane side of the interface. Equation (16) indicates that the response time of the system is inversely proportional to B or to the square of B (for low and high values of M, respectively). This suggests that as B increases upon exposure to glucose, the response time decreases. Thus, the system will always show a faster response to an increase in glucose level than to a decrease. This agrees with reported ex-
5
0.025
-
0.02
-
3
c.4
z 2 0.015
A
4 -
s 2 0.01 c?
-
g :: c 0.005
-
o0
4
8
12
16
20
25
0.025
y -
0.02
0
b)
t
’
I
0
4
I
8
12
16
1
20
24
Time (hour)
Fig. 8. Simulation results of the daily blood glucose profile regulated by the injectable degradable polymeric system, at t= 0 (a), and t = 1 week (b). B - breakfast, L - lunch, D - dinner, S snack.
perimental results of a similar system which responds to urea. In that system, the release rate increased immediately after exposure to urea, but decreased slowly (within few hours) to its original value after urea was removed [ 121. ii. Closed loop performance The simulated daily profile of blood glucose level of a normal man (a), and a diabetic patient using the implantable (b) or injectable (c) degradable polymeric system, are shown in Figs. 6-8, for the initial time ( t = 0) and the final time (t=6 months and t= 1 week, for the implantable and injectable system,
48
respectively). Both systems b and c regulate the plasma glucose level in an almost identical manner to that of the normal pancreas. In addition, their performance is constant with time since the diffusion length remains constant. The daily performance of the two systems is the same, since they only differ in their long term supply of insulin. Conclusion The above simulation study shows the considerable potential of pH-sensitive degradable polymers in providing a glucose-responsive insulin delivery system as a replacement for the pancreas. Based on the simulation results, a set of parameters in the range of their practicality is proposed for achieving a performance which almost exactly mimics the pancreas. This was assessed by simulating the daily blood glucose profile of a normal man and a diabetic patient using the polymeric systems. The simulations propose two possible degradable devices for insulin delivery, with an identical response to glucose: (a) an implantable device which operates over long periods (6 months); (b) an injectable device which operates over short periods (1 week ). In order to remove the need for surgery after the device is depleted, both the coating and the membrane of the system can be chosen from degradable polymers whose degradation rate is much slower than the polymer matrix. Thus, their erosion will not be important during insulin release, but they will erode away after complete depletion of the device. An optimal treatment of diabetes may then be possible, by minor surgery every half year, or by a weekly injection. In summary it should be mentioned that this work attempts to quantify a very complicated process, and to provide a comprehensive picture of the system and its important parameters. This seems to be impossible without some simplifications, and we hope this work will provide a basis for more sophisticated models. As the relation between the parameters of the system is very complex, an experimental approach to identify the effect of each parameter on the performance of the system would be extremely difficult, time consuming and costly. The simulation studies undertaken here can therefore be a useful tool in directing and interpreting the current experiments and in providing a rational basis for the design of a degradable, glucose-dependent, insulin delivery system. List of symbols length or thickness surface area erosion rate constant baseline erosion rate
49
ci
G CO
[CO21
D* e
VI [Gl [GA1 [GA- 1 G3 GF
[GLI GL GM
GP tH+l
WO,
[II L
I* K
ki h, lz, kc,,,hrd k, k--2 k 2eq
ks,k--3 ?
i&l R S t T U VG vi
x
1
concentration of solute i insulin solubility insulin loading carbon dioxide concentration diffusion coefficient, subscript indicating solute porosity enzyme concentration glucose concentration gluconic acid concentration gluconate ion concentration brain glucose uptake food glucose uptake glucono-lactone concentration liver glucose uptake or efflux periphery glucose uptake plasma glucose concentration hydrogen ion concentration carbonate ion concentration insulin concentration effective plasma insulin concentration plasma insulin concentration partition coefficient insulin “clearance” rate constant reaction rate constants for enzymatic reaction equilibrium rate constant for gluconic acid reaction rate constants for buffer reaction equilibrium rate constant for bicarbonate diffusional length in polymer membrane thickness oxygen concentration insulin release rate erosion front time tortuosity factor diffusion front distribution volume for glucose distribution volume for insulin spatial distance
50
Subscripts and superscripts
m p ss
membrane polymer steady state
References 1 2 3
R.S. Langer, New drug delivery systems: What the clinician can expect, Drug Ther., 13 (1983 ) 217-231. R.S. Langer and N. Peppas, Chemical and physical structure of polymers as carriers for controlled release of bioactive agents: A review, J. Macromol. Sci., 23 (1983) 61-126. F. Lim and A.M. Sun, Microencapsulated islets as bioartificial endocrine pancreas, Science, 210 (1980)908-910.
4
M. Brownlee and A. Cerami, A glucose controlled insulin delivery system: Semisynthetic insulin bound to lectin, Science, 206 (1979) 1190-1191.
5
S. Sato, S.Y. Jeong, J.C. McRea and SW. Kim, Self-regulating insulin delivery systems. II. In vitro studies. J. Controlled Release 1 (1984) 67-77. K. Ishihara, M. Kobayashi, N. Ishimaru and I. Shinohara, Glucose induced permeation con-
6
trol of insulin through a complex membrane consisting poly(amine), Polym. J., 16 (1984) 625-631. 7
8 9 10 11
12
of immobilized
glucose oxidase and a
J. Kost, T.A. Horbett, B.D. Ratner and M. Singh, Glucose-sensitive membranes containing glucose-oxidase: Activity, swelling and permeability studies, J. Biomed., 19 (1985) 11171133. F. Fischel-Ghodsian, L. Brown, E. Mathiowitz, D. Brandenberg and R. Langer, Enzymatitally controlled drug delivery, Proc. Natl. Acad. Sci. U.S.A., 85 (1988) 2403-2406. J. Heller, Use of bioerodible polymers in self-regulating drug delivery devices, Proc. Int. Symp. Controlled Release Bioactive Mater., 12 (1985) 45-46. J. Heller and S.H. Pangburn, Use of bioerodible polymers in self-regulating drug delivery devices, presented at 191st Amer. Chem. Sot. National Meeting, New York, 1986. J. Heller, R.W. Baker, R.M. Gale and J.O. Rodin, Controlled drug release by polymer dissolution. 1. Partial esters of maleic anhydride copolymers - Properties and theory, J. Appl. Polym. Sci., 22 (1978) 1991-2009. J. Heller and P.V. Trescony, Controlled drug release by polymer dissolution. mediated delivery device, J. Pharm. Sci., 68 (1979) 919-921.
II. Enzyme-
13
S. Nakamura and Y. Ogura, Action Biochem., 63 (1968) 308-316.
14
R.E. Mitchell and F.R. Duke, Kinetics and equilibrium constants conolactone system, Ann. N.Y. Acad. Sci., 172 (1970) 131-138.
15
D.T. Sawyer and J.B. Bagger, The lactone-acid-salt equilibria for D-glucono-&lactone the hydrolysis kinetics for this lactone, J. Amer. Chem. Sot., 81 (1959) 5302-5306.
16
B.H. Gibbons and J.T. Edsall, Rate of hydration of carbon dioxide and dehydration of carbonic acid at 25”C, J. Biol. Chem., 238 (1963) 3502-3507. E.L. Cussler, Diffusion. Mass Transfer in Fluid Systems, Cambridge University Press, Cambridge, MA, 1984.
17
mechanism
of glucose oxidase of Aspergillus
niger, J.
of the gluconic acid-gluand
51 18 19 20 21 22 23 24 25 26 27 28
29
30
31 32 33 34
J. Nussbaum, Electric field control of mechanical and electrochemical properties of poly electrolyte gel membranes, Ph.D. thesis, Massachusetts Institute of Technology, 1986. G.W. Albin, T.A. Horbett, S.R. Miller and N.L. Ricker, Theoretical and experimental studies of glucose sensitive membranes, J. Controlled Release, 6 (1987) 267-291. G.H. Bell, J.N. Davidson and H. Scarborough, Text Book of Physiology and Biochemistry, 6th edn., Churchill Livingstone, New York, NY, 1965, Chap. 33. F. Fischel-Ghodsian, Modelling and simulation in the development of a polymeric glucosedependent insulin delivery system, D.Phil. thesis, Oxford University, 1987. R.P. Bell, The Proton in Chemistry, 2nd edn., Chapman and Hall, London, 1973, Chap. 7. K.W. Wang and W.M. Deen, Chemical kinetic and diffusional limitations on bicarbonate reabsorption by the proximal tubule, Biophys. J., 31 (1980) 161-182. L. Brown, Controlled release polymers: In vivo studies with insulin and other macromolecules, D.Sc. thesis, Massachusetts Institute of Technology, 1983. D.O. Cooney, Biomedical Engineering Principles. An Introduction to Fluid, Heat, and Mass Transport Processes, Marcel Dekker, New York, NY, 1976, Chap. 10. F. Fischel-Ghodsian and J.M. Newton, Analysis of drug release kinetics from degradable polymeric devices, J. Controlled Release, submitted. A.G. Thombre and K.J. Himmelstein, Modelling of drug release kinetics from a laminated device having an erodible drug reservoir, Biomaterials, 5 (1984) 250-254. J.P. Hosker, D.R. Matthews, A.S. Rudenski, M.A. Burnett, P. Darling, E.G. Bown and R.C. Turner, Continuous infusion of glucose with model assessment: Measurement of insulin resistance and/%cell function in man, Diabetologia, 28 (1985) 401-411. D.L. Curry, L.L. Bennett and G.M. Grodsky, Dynamics of insulin secretion by the perfused rat pancreas, Endocrinology, 83 (1968) 572-584. R.C. Turner, R.R. Holman, D. Matthews, T.D.R. Hockaday and J. Peto, Insulin deficiency and insulin resistance interaction in diabetes: Estimation of their relative contribution by feedback analysis from basal plasma insulin and glucose concentrations, Metabolism, 18 ( 1978) 1086-1096. A.S. Rudenski, Personal communications, Oxford University, 1986. K.J. Astrom, A SIMNON tutorial, Department of Automatic Control, Lund Institute of Technology, 1985. G.D. Smith, Numerical Solution of Partial Differential Equations: Finite Difference Methods, 3rd edn., Clarendon Press, Oxford, 1985, Chap. 2. E. Kreyzig, AdvancedEngineering Mathematics, 5th edn., Wiley, New York, NY, 1983, Chap. Zl.
35
36 37
38 39 40
G. Siiderlind, DAS,3 - A program for the numerical integration of partitioned stiff ODES and differential-algebraic systems, Department of Numerical Analysis and Computing Science, The Royal Institute of Technology, Stockholm, 1985. R.W. Korsmeyer and N.A. Peppas, Modelling drug release from swellable systems, Proc. Int. Symp. Controlled Release Bioactive Mater., 10 (1983) 141-148. S.Y. Jeong, S. Sato, J.C. McRea and S.W. Kim, Self-regulating insulin delivery system, presented at Second World Congress on Biomaterials, 10th Annual Meeting of the Society for Biomaterials, Washington, DC, 1984. K. Ishihara, M. Kobayashi and I. Shinohara, Insulin permeation through amphiphilic polymer membranes having 2-hydroxyethyl methacrylate moiety, Polym. J., 16 (1984) 647-651. Editorial, ICI launches once-a-month treatment for prostate cancer, Pharm. J., 238 (1987) 238-357. R.A. Rizza, J.E. Gerich, M.W. Haymond, R.E. Westland, L.D. Hall, A.H. Clemens and F.J. Service, Control of blood sugar in insulin-dependent diabetes: Comparison of an artificial endocrine pancreas, continuous subcutaneous insulin infusion, and intensified conventional insulin therapy, New Engl. J. Med., 303 (1980) 1313-1318.
52
Appendix
Based on the specified requirements, the parameters of the system were chosen as follows: i. Since steady-state release rate is proportional to erosion rate constant, B should increase by 25 fold over the resulting pH range to give a 25fold increase in release rate upon exposure to glucose. To obtain a decrease in pH, as the system is exposed to glucose, the properties of the membranes should be chosen accordingly. Using Fig. 4, the following requirements were taken: T,Zm23 1 X 10 -’ cm2
(A.1)
M’30.1
(A.2)
Figure 5 shows that higher values of K are more desirable, but due to lack of information about this parameter, a value of k= 1 was taken. ii. In order to have a rapid response to glucose: (a) The system should provide a decrease in pH, within 5 minutes’ exposure to glucose. This time constant (tn) is related to the diffusion of glucose inside the membrane, and is proportional to the thickness of the membrane: tD = T,l,=/D
G4.3)
Using the value of diffusion coefficient for glucose: T,Z,,,2<2~10-3
cm2
(A.4)
and using equation (25) and substituting for M’by: M ’ = M&K/a parameter TJ,
(A.5 1 can be written as:
TJ, = T,,a/MK Defining M=cN, N=aB{ (WC,)
(A.6) where N has been defined to be (24): -1)/D,
(A.7)
and substituting in eqn. (A.6) gives: T,Z,=1.2~10-*/{kcB(C,,/C,-1)) The term TJ,’
(A.6)
becomes:
T&,2=1.44~10-8/{T,K2c2B2(C,,/Cs-1)2}
(A.9)
Combination of relationships (A.1 ) and (A.4 ) gives: 7.2~10-‘%T,K~c~B~(C&~--1)~~1.44X10-~
(A.lO)
(b) The release rate of the system should increase within 5 minutes after
53
the erosion rate increases. The time needed for the system to reach steady state defined as the steady-state time of the system is [26]: tss =a/B(1.4/N+0.9/M)
(A.ll)
It can be seen that for each value of N, t, s has its lowest value for M z+ IV. Having M = cN, parameter c should then be: c>l
A.12)
and t, s in dimensional form can be approximated to: t,s=1.40/[T~2(c,/c,-1)]
(A.13)
As B increases by 25 fold, the steady state time decreases by 625 fold. To have a response to glucose within 5 minutes, and also a rapid switching off when glucose is removed, this requirement is applied to the case that B is 6.25 times higher than the baseline erosion rate B, (chosen as 25% of the total increase in B) . Substituting for the value of insulin diffusion coefficient following relation is achieved: T&,2(Co/C,-1)~8.6x10-7
(A.14)
iii. For M > 1 and iV> 10, zero order time of the system is given by (24): t, = a/B
(A.15)
To provide a one-year baseline release rate of 4 x 10m4mg/min of insulin (or 6 month total supply of insulin), we should have: a/& > 5 X 10 -’ min BAC,(C,/C,-
1) =4x 10d4 mg/min
(A.16) (A.17)
iv. Assuming a slab geometry, size requirements of the system give: a<5 cm
(A.18)
Aa<2cm3
(A.19)
A set of parameters was then chosen as follows: Using eqn. (A.18), a value of a=5 was taken. Substituting this value in eqn. (A.16), a value of I?,,= 1 x 10M5 cm/min was taken. Relation (A.14) then becomes: T&&-1)~8.6x103
(A.20)
Using eqn. (A.6)) parameter c was taken as 30. Inequality (A.lO) then becomes: 80~TT,(c,/c,-1)2~160
(A.21)
A low value of T,= 10 was chosen. This gives: 3.8
(A.22)
54
and a value of C-,/C, = 4 was taken. Inequality (A.20) indicates that T, should be larger than 2.9 x 103, so a value of T,=4000 was taken. Equation (A.17) gives: AC,=
13.3 cm2-mg/ml
(A.23)
A value of C,= 120 mg/ml (reported for Na-insulin [ 221) was taken, and the required surface area becomes A = 0.11 cm2. Using eqn. (A.9)) the value of 1, is calculated to be Z,= 0.013 cm, and parameters N, M and M’ will become equal to 5 x 10W3,1.5 X 10e5 and 30, respectively. The resulting change in pH, as the system is exposed to glucose was then computed. The pH was calculated to be 7.3 for [G] = 0.004 M (baseline glucose level), 7.21 for [G] =O.Ol M, and 7.05 for [G] =0.028 M. The 25 times increase of B was then specified to be over the pH range of 7.3 to 7.2, such that: B = 1 x 10W5cm/min
pHa7.3
B = 2.5 x 10 -’ cm/min
pH d 7.2
B=1.5~10-~(pH)+1.105~10-~
7.2 < pH d 7.3
(A24)
MODELLING RESPONSIVE
33
OF INSULIN RELEASE FROM A GLUCOSEDEGRADABLE POLYMERIC SYSTEM
FARIBA FISCHEL-GHODSIAN Department of Engineering Science, Oxford University, Oxford (Great Britain) and J.M. NEWTON The School of Pharmacy, University of London, Brunswick Square, London WClN IAX (Great Britain) (Received July 15,1988; accepted in revised form May 24,1989)
Summary Drug delivery devices which release therapeutic agents at a predetermined constant rate significantly improve drug therapy. However, for diabetic therapy the ideal system is one that delivers insulin in response to blood glucose levels. In this work, a degradable polymeric system containing insulin and glucose-oxidase is described and its performance simulated and compared to that of a normal pancreas. The system considered is a degradable polymer matrix whose erosion is pH dependent, and which is surrounded by a membrane. The feedback mechanism is provided by the enzymatic reaction between glucose and glucose-oxidase. Acid produced from this reaction reduces the pH on the surface of the polymer, and causes an increase in polymer degradation and release rate. Based on simulation results, two possible therapeutic systems are proposed which closely mimic the normal pancreas. One is an implantable system which operates over long periods (6 months), and the other is an injectable device for short periods (one week). The study can therefore provide a useful tool in directing further experimental studies and providing a rational basis for system design.
Introduction Over the last few years polymeric drug delivery systems have shown a great potential in delivering drugs over long periods and maintaining the drug concentration in the blood at a constant level [ 1,2]. For some drugs, however, a constant blood level profile is not optimal. A classic example is insulin, for which the normal physiological concentration varies significantly upon changes in blood glucose level. An ideal delivery of insulin to diabetic patients requires a system which delivers insulin in response to changes in the physiological glucose level. A closed-loop glucose-sensitive system would be an ideal way to deliver insulin, and research in this area includes encapsulation of pancreatic p cells [ 31, synthesis of glycosylated insulin molecules [4,5] and design of glucose-sensi-
0376-7388/90/$03.50
0 1990 Elsevier Science Publishers B.V.
34
tive membranes [ 6,7] and polymeric systems [ 61. One new approach to design a glucose-sensitive system is the use of pH-sensitive degradable polymers [ 9,101. In these polymers, the degradation rate can change by few orders of magnitude over a narrow pH range, which depends on the chemical properties of the polymer molecule [ 111. Since the drug release rate from degradable polymers is proportional to the erosion rate, the high sensitivity of the erosion rate to pH suggests that these polymers may provide a self-regulating drug delivery system. This concept has been successfully shown using area as the drug model
[=I.
In this work, the performance of a glucose-sensitive degradable polymeric system, and the effect of different parameters on its performance, are assessed by mathematical modelling and computer simulations. The findings will provide a guide for further experimental work in designing a system as a replacement for pancreas, which would obviate the need for repeated injections or the problems associated with an indwelling pump system. Mathematical modelling i. Polymeric system In this section, a mathematical model of insulin release dispersed in an erodible polymeric device is formulated. The basic components of this system are a pH-sensitive surface-erodible polymer, surrounded by a membrane containing immobilised glucose-oxidase. As the system is exposed to a glucose solution, glucose diffuses through the membrane and is converted to acid by the enzymatic reaction. The acid produced will reduce the pH on the surface of the polymer. This will increase the surface erosion rate, and more insulin will be released. The release kinetics are modelled by diffusion and surface erosion, and the effect of glucose on the pH on the surface of the polymer, and the effect of pH on polymer erosion. A schematic diagram of the system is shown in Fig. 1. The effect of glucose on the pH at the surface of the polymer is modelled by considering the following: a. Enzymatic reaction between glucose and glucose-oxidase [13,14]: E,, + glucose 2
Eox_o-
k,
Ered+ glucono-lactone
k,. Eredf 0, -
E,, + H,O,
C&3lySe HzOz
9
H,O+$O,
Hydrolysis of glucono-lactone is given by [ 151:
35
glucono-lactone + Hz0 -
khyd
gluconic acid a
kz
k-z
gluconate ion- + H+
and the rate of production of glucono-lactone will be:
&WI [%I [Gl
r=
[%I [Gl+;
[%I +;
tG1 ox
where, k=
lzJG k_,+k,
b. Bicarbonate body buffer reaction [16]: COz+H20
+
HCO, +H+
c. Diffusion of solutes through the polymer and membrane:
$D$
(Fick’s law )
The diffusion coefficient of solutes inside the polymer and membrane is taken
~eleose media
0
U(t)
\ Polymer
s(t)
s(t)+hm
x
, Membiane
Fig. 1. Schematic representation of the model for effect of degradable polymer device.
on insulin release rate from
36
The diffusion coefficient of solutes inside the polymer and membrane is taken as the diffusion coefficient in water divided by a tortuosity factor T. This factor accounts for the tortuous path of diffusion in a porous medium [ 17 ] and is different for the polymer and membrane. A one-dimensional model of the system was constructed based on the following assumptions: 1. No concentration gradient exists in the radial direction. 2. Particles of enzyme and insulin are distributed homogeneously inside the polymer and the enzymatic reaction is homogeneous throughout the pores. 3. Glucose concentration is rate limiting in the production of gluconic acid, due to the lower diffusivity of glucose and the excess amount of the enzyme catalyst. The concentration of oxygen inside the pore is thus assumed to be equal to its concentration in the extracellular fluid. 4. Diffusion coefficients of solutes are independent of the concentration inside the pores (dilute solution). 5. No transport by convection and ion exchange is considered. Only passive diffusion is considered. 6. Effect of electrical field is neglected since the reactions are electrically balanced. Although some gradients in charge density exist at the boundary of the pores and polymer, a uniform charge density and thus a small field can be assumed overall, especially at steady state [ 181. This assumption has also been used by other researchers in similar systems [ 191. 7. Degradation products do not influence the pH, and pH changes are mainly due to changes in glucose concentration. To model this effect, equations of conservation of mass for glucose (G), glucono-lactone (GL), gluconic acid (GA), gluconate ion (GA-), hydrogen ion (H+ ), carbonate ion (HCO; ), and carbon dioxide (CO,) were written in 2 domains: membrane and polymer. For each solute the left hand side of the equation shows the changes in concentration with time, and the right hand side describes the appropriate reaction and diffusion terms. The initial conditions, and the boundary conditions at x = s ( t) + I, are the physiological concentrations of glucose, hydrogen ion, carbonate ion, and carbon dioxide in the body [ 181, and are taken as zero for other solutes (perfect sink condition). In the polymer domain, no flux boundary condition (X/&r= 0) is used at x = u(t), for all the solutes. The unknown concentrations at the boundary between polymer and membrane (x = s (t) ) are related by the flux continuity and partitioning condition. ep D, (dC/dx) polymer = em% ( dC/dr ) membrane
(1)
C, = KC,
(2)
where subscripts p and m represent polymer and membrane, respectively, e is porosity, and K is partition coefficient.
37
The following differential equations, representing the effect of glucose on the pH on the surface of the polymer were generated: d [cl/at=
-k[El
LO21VW{ [021 [Gl +WWO,l
+k,[Wk,x}+ KW~W[W~~2)
(3)
8 [GLll~~=~[El[O21 WI/{ 1021 [Gl +WWO,l
+MGlllz,,)-FZhydW a[GA]/dt=lz,,,[GL]
+k_,[H+]
[GA-‘]
+&JW2[Wl~x2)
(4)
-k,[GA]
+DGA/T(d2[GA]/c3x2)
(5)
a[GA-]/dt=-k_,[H+][GA-]+$[GA]+D,,-/T(d2[GA-]/ax2)
(6)
d[H+]/c3t=-k_2[H+][GA-]+k2[GA]+ka[C02] -k_3[H+] r3[HCO,]/dt=lz,[CO,]
-k_,[H+]
[HC0,]+D,+/T(8’2[H+]/~x2)
(7)
[HCO,] +D
HCO~-/~(~“[H~~,I/~~~) (8)
~[CO,]/~t=-k,[CO,]+k_,[H+][HCO,]+Dco,/T(~”[CO2]/dx2)
(9)
and the pH on the surface of the polymer is: pH= -log[H+]
at x=s(t)
(19)
The values of the reaction rate constants and diffusion coefficients at 25’ C* and their source used in the model are summarised in Tables 1 and 2. The effect of pH on surface erosion rate constant (B) was modelled as a straight line in the resulting pH range, similar to the effects which have been observed experimentally [ 111. The erosion rate was assumed to be independent of pH outside this region, and B was assumed to increase by a maximum of 25 fold**. This is expressed as: B=c(pH)+d
PH,
B=B,,
PH>PH,
B=25B,
PH
(11)
where c and d are the slope and intercept of the straight line, and B. is the *Simulations at 37°C show very similar results [21]. This is due to the fact that higher temperature increases the rate of the enzymatic reaction, the buffer reaction and also the rate of the diffusion. These effects cancel each other and so the overall effect is negligible. **This is chosen in order to achieve a performance similar to the pancreas.
38 TABLE 1 Reported values of the rate constants
for the enzymatic
Rate constant
buffer reaction at 25’ C
Reference
k,=400
set-’
k=4.3x103
M set
k,.=1.2x106 k,,,.,=3.4x
and bicarbonate
13 13 13 14 15 22
M-‘-set-’ see-’ (at pH=7.4)
10-3”
M M-‘-set-’ see-’ see-’ N-‘-see-’
k,,,=2.0x10-4 k_a=5.0x106 kz = 1000b k, = 0.04 k_,=5.5x104
16 16
“The rate constant k,,, strongly depends on pH. Its change with pH in phosphate is given by [ 141 k hyd=1.27x10-1[H+]+1.08x104[OH-]+6.75X10-4(sec-1)
buffer solutions (11).
bk, = k_zkzeq
TABLE 2 Reported
values of the diffusion coefficient
in water at 25” C
Other parameters were taken as [E] = 10-s M, equal to the enzyme concentration used in a similar system [ 81, and [ 0,] = 4.5 x lo- M, oxygen concentration in the extracellular fluid a (25 ) Solute
Diffusion coefficient (cm”/sec)
H+ CO, HCO,H&G, Glucose Glucono-lactone Gluconic acid Gluconate ion Insulin
9.3
1.72 1.17 1.17 0.67 0.67b 0.67b 0.67b 0.2
X 10e5
Reference
17 23 23 23 17
24
“This value is taken equal to the concentration of dissolved insulin in venous blood. bThe diffusion coefficients for glucose, glucono-lactone, gluconic acid, and gluconate ion are assumed to be equal, since they have similar molecular weights.
baseline erosion rate constant when system is exposed to the physiological glucose concentration. The drug release rate from a degradable polymer surrounded by a membrane has been previously modelled [ 26,271 by considering drug diffusion and polymer degradation, and was shown to be: R=D~C,M/(a+M1,)
(12)
39
where D, is insulin diffusion coefficient in the polymer, A is the surface area of the device from which insulin is released, C, is insulin solubility, a is membrane thickness, 1, is diffusion length in the polymer, and M is defined as: M= aTA,/l,T,,,K
(13)
Changes of the diffusional length with time, and the time needed for the system to reach steady state, are given by: dZJdt= -B+D,
Mu/[ (a+M&)C,,/C,-l]
t, S= 1.4D,/B 2(C,/C, - 1) + O.Sa/MB
(14) (15)
and at steady state the release rate is directly proportional to B:
4 ,=BAG{ (G/G) -4
(16)
where C, is insulin loading.
ii. Glucose metabolism The performance of the polymeric system in uiuo was simulated and compared to that of the pancreas, using a simplified version of the model of Hosker et al. [ 281, which describes the physiological regulation between glucose and insulin and correlates excellently with experimental data of intravenous glucose tolerance tests. The effect of glucose on insulin release rate from the pancreas was modelled by assuming an immediate response to changes in glucose level based on in uiuo experiments which have demonstrated insulin response within 30 seconds after a stimulating concentration of glucose reaches the pancreas [ 291. A sigmoidal, steadily increasing curve describes the relationship between insulin secretion and glucose concentration [ 30,311:
R={0.4+0.35(tanh(7.93Xl@
(G,,-9.3x10-3))+0.73)}6~10-3
(17b)
G,>8.13~10-~ Distribution of insulin in the body was modelled by assuming a single volume of distribution (Vi) and first-order disappearance kinetics. Plasma insulin concentration (4) changes as insulin is produced by the pancreas and is cleared through the body:
40 TABLE 3 Steady state values for the glucose metabolism model [ 281 Variable
Basal steady state values for a normal 70-kg man 4X 10m3mol/l 2 X 10m7mg/ml 8 X 10m4mol/min 4 X 10e4 mol/min 4X lo-l4 mol/min 151 13000 ml 0.173 min-’ 4X 10m4mg/min
dl,/dt = R/ Vi- k,I,
(18)
The overall time delay in the action of insulin was accounted for, by introducing an “effective insulin concentration”, I,. I, represents the amount of insulin being effective through processes such as binding to receptors, and enzyme activation. Because of insufficient quantitative data, the same value of clearance rate constant ki was assumed in and out of the effector compartment [ 281: dZ,/dt= ki (Ip-I,)
(19)
The metabolism of glucose in the body was modelled by assuming a single volume of distribution for glucose (V,) with separate turnover for brain, liver and insulin-sensitive periphery including muscle and fat. Blood glucose concentration (G,) changes as glucose is added by food uptake ( GF) or liver ( GL) and is taken up by periphery (GM) and brain (Gn ) : dG,/dt=
(GF+GL--GB-GM)VG
(20)
Brain and muscle glucose uptake and the net production or uptake of glucose in liver has been shown previously [ 301 and are described by the following equations fitted to the experimental curves [ 311: Gn=O.4~10-~
G,/(O.~X~O-~+G~)
(21)
G,={(~.~~G,)/~.OX~O-~+G,} x{(1.0x10-“1,)/@.0x10-7+1,)+0.2x10-3}
(22)
41
GL= [2.131-1.05
(In (ln(25x1061,)}-0.6{ln(1000G,)}]/1000
(23)
Values of the parameters and variables at steady state in the units used in the model are given in Table 3. Simulation studies The performance of the polymeric system and the pancreas were simulated on VAX 11-780, using SIMNON, a simulation program for solving non-linear ordinary differential equations [ 321. The partial differential equations were converted into ordinary differential equations by approximating the diffusion terms using an explicit finite difference method [ 331, and the diffusional length was divided into four equal sections. This provided adequate accuracy and smooth curves, since concentration gradients were small throughout. Division of the length into more sections did not affect the simulation results [ 211, and was not used since it requires more computing time. Simulation studies of the pancreas were carried out by using the Runge-Kutta algorithm [ 341. However, simulating the performance of the polymeric system required an algorithm which deals with stiff equations, as the time constants of the equations were highly non-uniform. This system was simulated using algorithm DAS,3 in SIMNON [ 351, which deals with systems with only few equations causing “stiffness” by partitioning the system into “stiff” and “non-stiff” sections and interacting between them. The correct partitioning of the equations into slow and fast was achieved by placing the equations describing the rate of change of [GA] and [H+ ] (eqns. 5 and 7) in both domains into the fast section. Other equations were placed into the slow section. To investigate the effect of glucose on insulin release rate, both the open loop and closed loop performances of the polymeric system were simulated. i. Open loop performance The effect of glucose concentration on the pH of the surface of the polymer was studied by simulating the pH after one hour exposure to a glucose level of 500 mg/dl. The effect of membrane thickness on pH was simulated for different values of parameters M ’ and K. Parameter M ’ was defined as: M’=l,T&,T,
(24)
and was taken equal for all the solutes, assuming they all have the same tortuosity factor. The partitioning coefficient K was also assumed to be the same for all the solutes. ii. Closed loop performance The closed loop performance of the system was studied by simulating the blood glucose level in a diabetic patient, in whom the device was implanted as
42
a replacement for the normal pancreas. The parameters of the system were chosen to achieve a performance similar to the normal pancreas, considering both the short-term response to glucose and the long-term supply of insulin. The acceptable range for each requirement and the practical value of the parameters, defined a range of values which could be used for each parameter. The requirements specified for the system considering its short and long term performance and size are: 1.As the system is exposed to glucose concentrations higher than 0.01 M, the insulin release rate should increase by 25 fold within 5 minutes. 2. The system should provide a baseline release rate of 4 x 10M4mg/min and a 6-month supply of insulin. 3. The total volume of the system should not exceed 2 cm3, and each side of the system should not exceed 5 cm. The complicated relationship between the specified requirements of the system and its parameters are shown in Fig. 2. Based on the open loop performance and previous analysis [ 261, the following values of the parameters were used (see Appendix). The baseline erosion rate constant was taken as B,, = 1 x lop5 cm/min and the erosion rate was assumed to increase by 25 times over 0.1 pH units. These values are well within the range for reported data for degradable polymers, as erosion rate constants in the range of 1 x 10m6-1 x low3 cm/min, and changes
Sufficient increase in B over the desired pH range
Sufficient baseline insulin
Fig. 2. Representation parameters.
of the relation
between
the specified
properties
of the system
and its
of as high as 1000 times in erosion rate over 2 pH units have been reported [ll]. In addition, synthesis of a degradable polymer whose erosion rate increases as pH decreases around pH 7.4 is currently under development [ 91. The retardation factor 2’ was taken as 10 and 4000 for the membrane and polymer, respectively. The high diffusion coefficient (low T) in the membrane can be achieved by using a hydrogel, since in highly swollen hydrogels the drug diffusivity may approach the value observed in pure water [ 361, and diffusion coefficients corresponding to T equal to 14 and 2.5 have been reported for insulin through porous membranes [ 37,381. The low diffusivisity in the polymer can be achieved by using hydrophobic polymers which show diffusion coefficients of few orders of magnitude lower than in water [ 171. The solubility of insulin was taken as 120 mg/ml, which is the reported value for sodium-insulin [ 241. This high insulin solubility was required to reduce the size of the system. The insulin loading was taken at 480 mg/cm3, within the range of 200 to 500 mg/cm3 used in polymeric systems [ 241. The geometry of the system was taken as slab or disk, and all the surfaces except one (with surface area of 0.11 cm’) were coated to provide a constant surface area and release rate with time. A membrane containing 10m5 M of
. ,.: ,: : :. .:::..: ..; _,_ .:..i ..,: ..,. :_..; a /
0.2
J
_‘:
~‘~~‘.-.:.~::(‘j:‘..l:j:h
0 33
j
: .::;:;i:;: :,
(b)
..__ ,____ / S:$
-0
33
I 0.013
Fig. 3. Schematic able device.
pictures of the proposed polymeric systems:
(a) implantable
device; (b) inject-
44
immobilised glucose-oxidase, with a thickness of 0.013 cm, was surrounding the polymer matrix. Two alternative systems which only differ in their length were proposed: (a) an implantable system of 5 cm length which provides a 6month supply of insulin (Fig. 3a), and (b) an injectable device of 0.2 cm length which provides a one week supply of insulin (Fig. 3b). The size of this device is that of a grain of rice, similar to a commercially available injectable degradable device for the treatment of cancer [ 391. The specified parameters listed above are not unique, since it is not possible to choose one set of parameters as optimal. While in one case size limitation may be the most important factor, in other cases the properties of the polymer or membrane may be the limiting factor. In addition, the specified requirements of the system can be different for different situations, and there can be a trade off between the short-term response to glucose and the long-term supply of insulin. The blood glucose regulation in the proposed systems were simulated and compared with that of the normal pancreas. This was undertaken by simulating the daily plasma glucose profile for a normal diet. A 30 kcal/kg body weight diet, consisting of 45% carbohydrates, was assumed. The total calories were distributed as 20,30, 35, and 15 percent for breakfast, lunch, dinner, and late snack [ 401, which were given at 8am, lpm, 6.30pm, and 10pm. The duration of absorption for each meal was assumed to be 60 min, 120 min, 90 min, and 45 min, respectively. Results and discussion i. Open loop performance The open loop response of the system to glucose is shown in Figs. 4 and 5. Figure 4 shows the pH on the surface of the polymer, after one hour exposure to a glucose level of 500 mg/dl. The result is plotted as a function of the square of the effective thickness of the membrane (JZ’, x 1,)) for different values of PH
M'=O.Ol
Fig. 4. Simulation results for the effect of effective membrane thickness on the pH on the surface of the polymer. Simulation results are shown for different values of the parameter M.
45
M. Other parameters of the system such as the effective diffusion length in the polymer, partition coefficient, and enzyme concentration are taken to be constant as JTpx l,,= 1 cm, K= 1, and [E] = 1 x 10W5M.The figure shows that as membrane thickness increases, the pH decreases to a minimum and then increases. This is the result of two factors: as membrane thickness increases (i) the steady-state pH decreases more since the produced H+ is better protected from the presence of buffer, (ii) the response time increases, since it is proportional to T,&,‘. Combination of these two factors results in an optimal membrane thickness, at which maximum changes in pH occur. The results also show that a larger change in pH occurs as parameter M increases. This is because as the value of MK increases, the concentration of H+ at interface approaches its value inside the membrane, and this is higher than its value inside the polymer, since enzymatic reaction occurs in the membrane. The effect of the partition coefficient, K, on pH is shown in Fig. 5, for PH
T,l,*(cm2)
Fig. 5. Simulation results for the effect of effective membrane thickness on the pH on the surface of the polymer. Simulation results are shown for different values of the parameter k.
0
’0
I
4
8
1
12
I
16
I
20
I
2b
Time (hour)
Fig. 6. Simulation results of the daily blood glucose profile regulated by the pancreas. fast, L - lunch, D -dinner, S -snack.
B-
break-
8
12
16
20
24
Time (hour)
(W
12
16
20
24
Time (hour)
Fig. 7. Simulation results of the daily blood glucose profile regulated by the implantable degradable polymeric system, at t=O (a), and t=6 months (b). B-breakfast, L-lunch, D -dinner, Ssnack.
,/T, x 1,= 1 cm, iW= 100, and [E] = 1 x 1O-5 M. As expected, the value of pH decreases to a greater extent as K increases due to the H+ concentration on the polymer side of the interface becoming higher than the concentration on the membrane side of the interface. Equation (16) indicates that the response time of the system is inversely proportional to B or to the square of B (for low and high values of M, respectively). This suggests that as B increases upon exposure to glucose, the response time decreases. Thus, the system will always show a faster response to an increase in glucose level than to a decrease. This agrees with reported ex-
5
0.025
-
0.02
-
3
c.4
z 2 0.015
A
4 -
s 2 0.01 c?
-
g :: c 0.005
-
o0
4
8
12
16
20
25
0.025
y -
0.02
0
b)
t
’
I
0
4
I
8
12
16
1
20
24
Time (hour)
Fig. 8. Simulation results of the daily blood glucose profile regulated by the injectable degradable polymeric system, at t= 0 (a), and t = 1 week (b). B - breakfast, L - lunch, D - dinner, S snack.
perimental results of a similar system which responds to urea. In that system, the release rate increased immediately after exposure to urea, but decreased slowly (within few hours) to its original value after urea was removed [ 121. ii. Closed loop performance The simulated daily profile of blood glucose level of a normal man (a), and a diabetic patient using the implantable (b) or injectable (c) degradable polymeric system, are shown in Figs. 6-8, for the initial time ( t = 0) and the final time (t=6 months and t= 1 week, for the implantable and injectable system,
48
respectively). Both systems b and c regulate the plasma glucose level in an almost identical manner to that of the normal pancreas. In addition, their performance is constant with time since the diffusion length remains constant. The daily performance of the two systems is the same, since they only differ in their long term supply of insulin. Conclusion The above simulation study shows the considerable potential of pH-sensitive degradable polymers in providing a glucose-responsive insulin delivery system as a replacement for the pancreas. Based on the simulation results, a set of parameters in the range of their practicality is proposed for achieving a performance which almost exactly mimics the pancreas. This was assessed by simulating the daily blood glucose profile of a normal man and a diabetic patient using the polymeric systems. The simulations propose two possible degradable devices for insulin delivery, with an identical response to glucose: (a) an implantable device which operates over long periods (6 months); (b) an injectable device which operates over short periods (1 week ). In order to remove the need for surgery after the device is depleted, both the coating and the membrane of the system can be chosen from degradable polymers whose degradation rate is much slower than the polymer matrix. Thus, their erosion will not be important during insulin release, but they will erode away after complete depletion of the device. An optimal treatment of diabetes may then be possible, by minor surgery every half year, or by a weekly injection. In summary it should be mentioned that this work attempts to quantify a very complicated process, and to provide a comprehensive picture of the system and its important parameters. This seems to be impossible without some simplifications, and we hope this work will provide a basis for more sophisticated models. As the relation between the parameters of the system is very complex, an experimental approach to identify the effect of each parameter on the performance of the system would be extremely difficult, time consuming and costly. The simulation studies undertaken here can therefore be a useful tool in directing and interpreting the current experiments and in providing a rational basis for the design of a degradable, glucose-dependent, insulin delivery system. List of symbols length or thickness surface area erosion rate constant baseline erosion rate
49
ci
G CO
[CO21
D* e
VI [Gl [GA1 [GA- 1 G3 GF
[GLI GL GM
GP tH+l
WO,
[II L
I* K
ki h, lz, kc,,,hrd k, k--2 k 2eq
ks,k--3 ?
i&l R S t T U VG vi
x
1
concentration of solute i insulin solubility insulin loading carbon dioxide concentration diffusion coefficient, subscript indicating solute porosity enzyme concentration glucose concentration gluconic acid concentration gluconate ion concentration brain glucose uptake food glucose uptake glucono-lactone concentration liver glucose uptake or efflux periphery glucose uptake plasma glucose concentration hydrogen ion concentration carbonate ion concentration insulin concentration effective plasma insulin concentration plasma insulin concentration partition coefficient insulin “clearance” rate constant reaction rate constants for enzymatic reaction equilibrium rate constant for gluconic acid reaction rate constants for buffer reaction equilibrium rate constant for bicarbonate diffusional length in polymer membrane thickness oxygen concentration insulin release rate erosion front time tortuosity factor diffusion front distribution volume for glucose distribution volume for insulin spatial distance
50
Subscripts and superscripts
m p ss
membrane polymer steady state
References 1 2 3
R.S. Langer, New drug delivery systems: What the clinician can expect, Drug Ther., 13 (1983 ) 217-231. R.S. Langer and N. Peppas, Chemical and physical structure of polymers as carriers for controlled release of bioactive agents: A review, J. Macromol. Sci., 23 (1983) 61-126. F. Lim and A.M. Sun, Microencapsulated islets as bioartificial endocrine pancreas, Science, 210 (1980)908-910.
4
M. Brownlee and A. Cerami, A glucose controlled insulin delivery system: Semisynthetic insulin bound to lectin, Science, 206 (1979) 1190-1191.
5
S. Sato, S.Y. Jeong, J.C. McRea and SW. Kim, Self-regulating insulin delivery systems. II. In vitro studies. J. Controlled Release 1 (1984) 67-77. K. Ishihara, M. Kobayashi, N. Ishimaru and I. Shinohara, Glucose induced permeation con-
6
trol of insulin through a complex membrane consisting poly(amine), Polym. J., 16 (1984) 625-631. 7
8 9 10 11
12
of immobilized
glucose oxidase and a
J. Kost, T.A. Horbett, B.D. Ratner and M. Singh, Glucose-sensitive membranes containing glucose-oxidase: Activity, swelling and permeability studies, J. Biomed., 19 (1985) 11171133. F. Fischel-Ghodsian, L. Brown, E. Mathiowitz, D. Brandenberg and R. Langer, Enzymatitally controlled drug delivery, Proc. Natl. Acad. Sci. U.S.A., 85 (1988) 2403-2406. J. Heller, Use of bioerodible polymers in self-regulating drug delivery devices, Proc. Int. Symp. Controlled Release Bioactive Mater., 12 (1985) 45-46. J. Heller and S.H. Pangburn, Use of bioerodible polymers in self-regulating drug delivery devices, presented at 191st Amer. Chem. Sot. National Meeting, New York, 1986. J. Heller, R.W. Baker, R.M. Gale and J.O. Rodin, Controlled drug release by polymer dissolution. 1. Partial esters of maleic anhydride copolymers - Properties and theory, J. Appl. Polym. Sci., 22 (1978) 1991-2009. J. Heller and P.V. Trescony, Controlled drug release by polymer dissolution. mediated delivery device, J. Pharm. Sci., 68 (1979) 919-921.
II. Enzyme-
13
S. Nakamura and Y. Ogura, Action Biochem., 63 (1968) 308-316.
14
R.E. Mitchell and F.R. Duke, Kinetics and equilibrium constants conolactone system, Ann. N.Y. Acad. Sci., 172 (1970) 131-138.
15
D.T. Sawyer and J.B. Bagger, The lactone-acid-salt equilibria for D-glucono-&lactone the hydrolysis kinetics for this lactone, J. Amer. Chem. Sot., 81 (1959) 5302-5306.
16
B.H. Gibbons and J.T. Edsall, Rate of hydration of carbon dioxide and dehydration of carbonic acid at 25”C, J. Biol. Chem., 238 (1963) 3502-3507. E.L. Cussler, Diffusion. Mass Transfer in Fluid Systems, Cambridge University Press, Cambridge, MA, 1984.
17
mechanism
of glucose oxidase of Aspergillus
niger, J.
of the gluconic acid-gluand
51 18 19 20 21 22 23 24 25 26 27 28
29
30
31 32 33 34
J. Nussbaum, Electric field control of mechanical and electrochemical properties of poly electrolyte gel membranes, Ph.D. thesis, Massachusetts Institute of Technology, 1986. G.W. Albin, T.A. Horbett, S.R. Miller and N.L. Ricker, Theoretical and experimental studies of glucose sensitive membranes, J. Controlled Release, 6 (1987) 267-291. G.H. Bell, J.N. Davidson and H. Scarborough, Text Book of Physiology and Biochemistry, 6th edn., Churchill Livingstone, New York, NY, 1965, Chap. 33. F. Fischel-Ghodsian, Modelling and simulation in the development of a polymeric glucosedependent insulin delivery system, D.Phil. thesis, Oxford University, 1987. R.P. Bell, The Proton in Chemistry, 2nd edn., Chapman and Hall, London, 1973, Chap. 7. K.W. Wang and W.M. Deen, Chemical kinetic and diffusional limitations on bicarbonate reabsorption by the proximal tubule, Biophys. J., 31 (1980) 161-182. L. Brown, Controlled release polymers: In vivo studies with insulin and other macromolecules, D.Sc. thesis, Massachusetts Institute of Technology, 1983. D.O. Cooney, Biomedical Engineering Principles. An Introduction to Fluid, Heat, and Mass Transport Processes, Marcel Dekker, New York, NY, 1976, Chap. 10. F. Fischel-Ghodsian and J.M. Newton, Analysis of drug release kinetics from degradable polymeric devices, J. Controlled Release, submitted. A.G. Thombre and K.J. Himmelstein, Modelling of drug release kinetics from a laminated device having an erodible drug reservoir, Biomaterials, 5 (1984) 250-254. J.P. Hosker, D.R. Matthews, A.S. Rudenski, M.A. Burnett, P. Darling, E.G. Bown and R.C. Turner, Continuous infusion of glucose with model assessment: Measurement of insulin resistance and/%cell function in man, Diabetologia, 28 (1985) 401-411. D.L. Curry, L.L. Bennett and G.M. Grodsky, Dynamics of insulin secretion by the perfused rat pancreas, Endocrinology, 83 (1968) 572-584. R.C. Turner, R.R. Holman, D. Matthews, T.D.R. Hockaday and J. Peto, Insulin deficiency and insulin resistance interaction in diabetes: Estimation of their relative contribution by feedback analysis from basal plasma insulin and glucose concentrations, Metabolism, 18 ( 1978) 1086-1096. A.S. Rudenski, Personal communications, Oxford University, 1986. K.J. Astrom, A SIMNON tutorial, Department of Automatic Control, Lund Institute of Technology, 1985. G.D. Smith, Numerical Solution of Partial Differential Equations: Finite Difference Methods, 3rd edn., Clarendon Press, Oxford, 1985, Chap. 2. E. Kreyzig, AdvancedEngineering Mathematics, 5th edn., Wiley, New York, NY, 1983, Chap. Zl.
35
36 37
38 39 40
G. Siiderlind, DAS,3 - A program for the numerical integration of partitioned stiff ODES and differential-algebraic systems, Department of Numerical Analysis and Computing Science, The Royal Institute of Technology, Stockholm, 1985. R.W. Korsmeyer and N.A. Peppas, Modelling drug release from swellable systems, Proc. Int. Symp. Controlled Release Bioactive Mater., 10 (1983) 141-148. S.Y. Jeong, S. Sato, J.C. McRea and S.W. Kim, Self-regulating insulin delivery system, presented at Second World Congress on Biomaterials, 10th Annual Meeting of the Society for Biomaterials, Washington, DC, 1984. K. Ishihara, M. Kobayashi and I. Shinohara, Insulin permeation through amphiphilic polymer membranes having 2-hydroxyethyl methacrylate moiety, Polym. J., 16 (1984) 647-651. Editorial, ICI launches once-a-month treatment for prostate cancer, Pharm. J., 238 (1987) 238-357. R.A. Rizza, J.E. Gerich, M.W. Haymond, R.E. Westland, L.D. Hall, A.H. Clemens and F.J. Service, Control of blood sugar in insulin-dependent diabetes: Comparison of an artificial endocrine pancreas, continuous subcutaneous insulin infusion, and intensified conventional insulin therapy, New Engl. J. Med., 303 (1980) 1313-1318.
52
Appendix
Based on the specified requirements, the parameters of the system were chosen as follows: i. Since steady-state release rate is proportional to erosion rate constant, B should increase by 25 fold over the resulting pH range to give a 25fold increase in release rate upon exposure to glucose. To obtain a decrease in pH, as the system is exposed to glucose, the properties of the membranes should be chosen accordingly. Using Fig. 4, the following requirements were taken: T,Zm23 1 X 10 -’ cm2
(A.1)
M’30.1
(A.2)
Figure 5 shows that higher values of K are more desirable, but due to lack of information about this parameter, a value of k= 1 was taken. ii. In order to have a rapid response to glucose: (a) The system should provide a decrease in pH, within 5 minutes’ exposure to glucose. This time constant (tn) is related to the diffusion of glucose inside the membrane, and is proportional to the thickness of the membrane: tD = T,l,=/D
G4.3)
Using the value of diffusion coefficient for glucose: T,Z,,,2<2~10-3
cm2
(A.4)
and using equation (25) and substituting for M’by: M ’ = M&K/a parameter TJ,
(A.5 1 can be written as:
TJ, = T,,a/MK Defining M=cN, N=aB{ (WC,)
(A.6) where N has been defined to be (24): -1)/D,
(A.7)
and substituting in eqn. (A.6) gives: T,Z,=1.2~10-*/{kcB(C,,/C,-1)) The term TJ,’
(A.6)
becomes:
T&,2=1.44~10-8/{T,K2c2B2(C,,/Cs-1)2}
(A.9)
Combination of relationships (A.1 ) and (A.4 ) gives: 7.2~10-‘%T,K~c~B~(C&~--1)~~1.44X10-~
(A.lO)
(b) The release rate of the system should increase within 5 minutes after
53
the erosion rate increases. The time needed for the system to reach steady state defined as the steady-state time of the system is [26]: tss =a/B(1.4/N+0.9/M)
(A.ll)
It can be seen that for each value of N, t, s has its lowest value for M z+ IV. Having M = cN, parameter c should then be: c>l
A.12)
and t, s in dimensional form can be approximated to: t,s=1.40/[T~2(c,/c,-1)]
(A.13)
As B increases by 25 fold, the steady state time decreases by 625 fold. To have a response to glucose within 5 minutes, and also a rapid switching off when glucose is removed, this requirement is applied to the case that B is 6.25 times higher than the baseline erosion rate B, (chosen as 25% of the total increase in B) . Substituting for the value of insulin diffusion coefficient following relation is achieved: T&,2(Co/C,-1)~8.6x10-7
(A.14)
iii. For M > 1 and iV> 10, zero order time of the system is given by (24): t, = a/B
(A.15)
To provide a one-year baseline release rate of 4 x 10m4mg/min of insulin (or 6 month total supply of insulin), we should have: a/& > 5 X 10 -’ min BAC,(C,/C,-
1) =4x 10d4 mg/min
(A.16) (A.17)
iv. Assuming a slab geometry, size requirements of the system give: a<5 cm
(A.18)
Aa<2cm3
(A.19)
A set of parameters was then chosen as follows: Using eqn. (A.18), a value of a=5 was taken. Substituting this value in eqn. (A.16), a value of I?,,= 1 x 10M5 cm/min was taken. Relation (A.14) then becomes: T&&-1)~8.6x103
(A.20)
Using eqn. (A.6)) parameter c was taken as 30. Inequality (A.lO) then becomes: 80~TT,(c,/c,-1)2~160
(A.21)
A low value of T,= 10 was chosen. This gives: 3.8
(A.22)
54
and a value of C-,/C, = 4 was taken. Inequality (A.20) indicates that T, should be larger than 2.9 x 103, so a value of T,=4000 was taken. Equation (A.17) gives: AC,=
13.3 cm2-mg/ml
(A.23)
A value of C,= 120 mg/ml (reported for Na-insulin [ 221) was taken, and the required surface area becomes A = 0.11 cm2. Using eqn. (A.9)) the value of 1, is calculated to be Z,= 0.013 cm, and parameters N, M and M’ will become equal to 5 x 10W3,1.5 X 10e5 and 30, respectively. The resulting change in pH, as the system is exposed to glucose was then computed. The pH was calculated to be 7.3 for [G] = 0.004 M (baseline glucose level), 7.21 for [G] =O.Ol M, and 7.05 for [G] =0.028 M. The 25 times increase of B was then specified to be over the pH range of 7.3 to 7.2, such that: B = 1 x 10W5cm/min
pHa7.3
B = 2.5 x 10 -’ cm/min
pH d 7.2
B=1.5~10-~(pH)+1.105~10-~
7.2 < pH d 7.3
(A24)