Part-2: Analytical Expressions of Concentrations of Glucose, Oxygen, and Gluconic Acid in a Composite Membrane for Closed-Loop Insulin Delivery for the Non-steady State Conditions

Abstract

A mathematical model developed by Abdekhodaie and Wu (J Membr Sci 335:21–31, 2009), which describes a dynamic process involving an enzymatic reaction and diffusion of reactants and product inside glucose-sensitive composite membrane has been discussed. This theoretical model depicts a system of non-linear non-steady state reaction diffusion equations. These equations have been solved using new approach of homotopy perturbation method and analytical solutions pertaining to the concentrations of glucose, oxygen, and gluconic acid are derived. These analytical results are compared with the numerical results, and limiting case results for steady state conditions and a good agreement is observed. The influence of various kinetic parameters involved in the model has been presented graphically. Theoretical evaluation of the kinetic parameters like the maximal reaction velocity (V _max) and Michaelis–Menten constants for glucose and oxygen (K _g and K _ox) is also reported. This predicted model is very much useful for designing the glucose-responsive composite membranes for closed-loop insulin delivery.

Appendix: Approximate analytical solution of Eqs. (8, 9) from the relation between the concentrations u, v, and w

From the Eqs. (7, 8), we can obtain the following equation.

$$\frac{{\partial^{2} }}{{\partial X^{2} }}\left( {\frac{{u\gamma_{\text{s1}} }}{2} - \gamma_{{{\text{E}}_{ 1} }} v} \right) = \frac{\partial }{\partial \tau }\left( {\frac{{u\gamma_{\text{s1}} }}{2} - \gamma_{{{\text{E}}_{ 1} }} v} \right)$$

(26)

$${\text{Let}}\;M = \left( {\frac{{u\gamma_{\text{s1}} }}{2} - \gamma_{{{\text{E}}_{ 1} }} v} \right)$$

(27)

Then (26) becomes

$$\frac{{\partial^{2} M}}{{\partial X^{2} }} = \frac{\partial \,M}{\partial \tau }$$

(28)

Using the boundary conditions in Eqs. (11, 12), the boundary conditions for M will be

$$N = 0{\text{ when }}\tau = 0$$

(29)

$$\frac{\partial M}{\partial X} = 0\;{\text{when}}\;X = 0$$

(30)

$$M = \frac{{\gamma_{S1} }}{2} - \gamma_{E1} \; {\text{when}}\;X = 1$$

(31)

Now by applying Laplace transform in (28) and using complex inversion formula and proceeding as in Appendix A, the solution of (28) will be

$$M = \left( {{\frac{{\gamma_{s1} }}{2} - \gamma_{{E_{1} }} v}} \right)\;\left[ {1 + 4\sum\limits_{n = 0}^{\infty } {\frac{{( - 1)^{ - (n + 1)} }}{(2n + 1)\pi }} \cos \left( {\frac{(2n + 1)\pi X}{2}} \right){\text{e}}^{{\left( {\frac{{ - (2n + 1)^{2} \pi^{2} }}{4}} \right)\tau }} } \right]$$

(32)

From (27), we have

$$v = \left( {\frac{{u\gamma_{s1} }}{{2\gamma_{{E_{1} }} }} - \frac{M}{{\gamma_{{E_{1} }} }}} \right)$$

(33)

The analytical expression for the concentration v can be obtained by substituting (A9) and (32) in (33). Similarly, the relation between u and w from the Eqs. (7, 9) is obtained by

$$\frac{{\partial^{2} }}{{\partial X^{2} }}\left( {u\gamma_{E} + w\gamma_{{E_{1} }} } \right) = \frac{{\partial^{{}} }}{\partial \tau }\left( {u\gamma_{E} + w\gamma_{{E_{1} }} } \right)$$

(34)

Let

$$N = \left( {u\gamma_{E} + w\gamma_{{E_{1} }} } \right)$$

(35)

Then (34) becomes

$$\frac{{\partial \,^{2} N}}{{\partial X^{2} }} = \frac{{\partial^{{}} N}}{\partial \tau }$$

(36)

Using the boundary conditions in (10) to (12), the boundary conditions for N will be

$$N = 0{\text{ when}}\;\tau = 0$$

(37)

$$\frac{\partial N}{\partial X} = 0\;{\text{when}}\;X = 0$$

(38)

$$M = \gamma_{E} \;{\text{when}}\;X = 1$$

(39)

Now by applying Laplace transform in (36, 37, 38 and 39) and using complex inversion formula (Appendix A, Online resource), the solution of (36) becomes as

$$N = \gamma_{E} \left[ {1 + \sum\limits_{n = 0}^{\infty } {\frac{{4( - 1)^{ - (n + 1)} }}{(2n + 1)\pi }} \cos \left( {\frac{(2n + 1)\pi X}{2}} \right)\text{e}^{{\left( {\frac{{ - (2n + 1)^{2} \pi^{2} }}{4}} \right)\tau }} } \right]$$

(40)

From (35), we have

$$w = \frac{N}{{\gamma_{E1} }} - \frac{{u\gamma_{E} }}{{\gamma_{E1} }}.$$

(41)

Using the above equation, we can obtain the Eq. (15) (the analytical expression for the concentration of w) in the text.