Skip to main content
SpringerLink
Log in

Numerical simulation of a glucose sensitive composite membrane closed-loop insulin delivery system

  • Research Paper
  • Published:
Bioprocess and Biosystems Engineering Aims and scope Submit manuscript

Abstract

Closed-loop insulin delivery system works on pH modulation by gluconic acid production from glucose, which in turn allows regulation of insulin release across membrane. Typically, the concentration variation of gluconic acid can be numerically modeled by a set of non-linear, non-steady state reaction diffusion equations. Here, we report a simpler numerical approach to time and position dependent diffusivity of species using finite difference and differential quadrature (DQ) method. The results are comparable to that obtained by analytical method. The membrane thickness directly determines the concentrations of the glucose and oxygen in the system, and inversely to the gluconic acid. The advantage with the DQ method is that its parameter values need not be altered throughout the analysis to obtain the concentration profiles of the glucose, oxygen and gluconic acid. Our work would be useful for modeling diabetes and other systems governed by such non-linear and non-steady state reaction diffusion equations.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4

Similar content being viewed by others

References

  1. Abdekhodie MJ, Wu XY (2009) Modeling of a glucose sensitive composite membrane for closed-loop insulin delivery. J Membr Sci 335:21–31

    Article  Google Scholar 

  2. Rajendran L, Bieniasz L (2013) Analytical expressions for the steady-state concentrations of glucose, oxygen and gluconic acid in a composite membrane for closed-loop insulin delivery. J Membr Biol 246(2):121–129

    Article  CAS  Google Scholar 

  3. Yu J, Zhang Y, Ye Y, DiSanto R, Sun W, Ranson D, Ligler FS, Buse JB, Gu Z (2015) Microneedle-array patches loaded with hypoxia-sensitive vesicles provide fast glucose-responsive insulin delivery. Proc Natl Acad Sci 112(27):8260–8265

    Article  CAS  Google Scholar 

  4. Bae YH, Kim SW (1993) Hydrogel delivery systems based on polymer blends, block co-polymers or interpenetrating networks. Adv Drug Deliv Rev 11(1):109–135. doi:10.1016/0169-409X(93)90029-4

    Article  CAS  Google Scholar 

  5. Wu X, Zhang Q, Arshady R (2003) Stimuli sensitive hydrogels. Polymer structure and phase transition. Polymeric biomaterials Citus Books, London, pp 157–194

    Google Scholar 

  6. Wu X, Zhang Q, Arshady R (2003) Stimuli sensitive hydrogels. Response and release modulation. Polymeric biomaterials. Citus Books, London, pp 195–231

    Google Scholar 

  7. He J-H (1999) Homotopy perturbation technique. Comput Methods Appl Mech Eng 178(3):257–262

    Article  Google Scholar 

  8. Adomian G (1984) Convergent series solution of nonlinear equations. J Comput Appl Math 11(2):225–230

    Article  Google Scholar 

  9. Adomian G, Witten M (1994) Computation of solutions to the generalized Michaelis–Menton equation. Appl Math Lett 7(4):45–48

    Article  Google Scholar 

  10. Rida S (2010) Approximate analytical solutions of generalized linear and nonlinear reaction–diffusion equations in an infinite domain. Phys Lett A 374(6):829–835

    Article  CAS  Google Scholar 

  11. He J-H (2002) Modified Lindstedt–Poincare methods for some strongly non-linear oscillations: Part I: expansion of a constant. Int J Non-Linear Mech 37(2):309–314

    Article  Google Scholar 

  12. He J-H (2006) Some asymptotic methods for strongly nonlinear equations. Int J Mod Phys B 20(10):1141–1199

    Article  Google Scholar 

  13. Shou D-H, He J-H (2007) Application of parameter-expanding method to strongly nonlinear oscillators. Int J Nonlinear Sci Numer Simul 8(1):121–124

    Article  Google Scholar 

  14. Bellman RE, Roth RS (1979) A scanning technique for systems identification. J Math Anal Appl 71(2):403–411. doi:10.1016/0022-247X(79)90200-2

    Article  Google Scholar 

  15. Bellman R, Kashef B, Vasudevan R (1974) The inverse problem of estimating heart parameters from cardiograms. Math Biosci 19(3):221–230. doi:10.1016/0025-5564(74)90040-6

    Article  Google Scholar 

  16. Hu LC, Hu C-R (1974) Identification of rate constants by differential quadrature in partly measurable compartmental models. Math Biosci 21(1):71–76

    Article  Google Scholar 

  17. Mingle JO (1977) The method of differential quadrature for transient nonlinear diffusion. J Math Anal Appl 60(3):559–569

    Article  Google Scholar 

  18. Civan F, Sliepcevich C (1983) Solution of the Poisson equation by differential quadrature. Int J Numer Meth Eng 19(5):711–724

    Article  Google Scholar 

  19. Civan F, Sliepcevich CM (1983) Application of differential quadrature to transport processes. J Math Anal Appl 93(1):206–221. doi:10.1016/0022-247X(83)90226-3

    Article  Google Scholar 

  20. Civan F (1993) Comments on application of generalized differential quadrature to solve two-dimensional incompressible Navier–Stokes equation by C. Shu and B. E. Richards. Int J Numer Methods Fluids 17:921–922

    Article  Google Scholar 

  21. Shu C, Richard B (1992) Parallel simulation of incompressible viscous flows by generalized differential quadrature. Comp Syst Eng 3(1–4):271–281

    Article  Google Scholar 

  22. Shu C, Richards BE (1992) Application of generalized differential quadrature to solve two dimensional incompressible Navier–Stokes equations. Int J Numer Meth Fluids 15(7):791–798

    Article  Google Scholar 

  23. Quan J, Chang C (1989) New insights in solving distributed system equations by the quadrature method—I. Analysis. Comput Chem Eng 13(7):779–788

    Article  CAS  Google Scholar 

  24. Quan J, Chang C-T (1989) New insights in solving distributed system equations by the quadrature method—II. Numerical experiments. Comput Chem Eng 13(9):1017–1024

    Article  CAS  Google Scholar 

  25. Malik M, Civan F (1995) A comparative study of differential quadrature and cubature methods vis-à-vis some conventional techniques in context of convection–diffusion–reaction problems. Chem Eng Sci 50(3):531–547

    Article  CAS  Google Scholar 

  26. Gutierrez R, Laura P (1994) Solution of the Helmholtz equation in a parallelogrammic domain with mixed boundary conditions using the differential quadrature method. J Sound Vib 178(2):269–271

    Article  Google Scholar 

  27. Civan F, Sliepcevich C (1984) On the solution of the Thomas–Fermi equation by differential quadrature. J Comput Phys 56(2):343–348

    Article  Google Scholar 

  28. Civan F, Sliepcevich C (1984) Differential quadrature for multi-dimensional problems. J Math Anal Appl 101(2):423–443

    Article  Google Scholar 

  29. Malik M, Bert C, Kukreti A (1970) Differential quadrature solution of uniformly loaded circular plate resting on elastic half-space. WIT Trans Eng Sci 1:385–396

    Google Scholar 

  30. Chen W (1996) Differential quadrature method and its applications in engineering. Department of Mechanical Engineering, Shanghai Jiao Tong University, Shanghai

    Google Scholar 

  31. Joy RA, Rajendran L (2012) Mathematical modelling and transient analytical solution of a glucose sensitive composite membrane for closed-loop insulin delivery using he’s variational iteration method. Int Rev Chem Eng 4:516–523

    Google Scholar 

  32. Bert CW, Malik M (1996) Differential quadrature method in computational mechanics: a review. Appl Mech Rev 49(1):1–28

    Article  Google Scholar 

  33. van Stroe-Biezen S, Everaerts F, Janssen L, Tacken R (1993) Diffusion coefficients of oxygen, hydrogen peroxide and glucose in a hydrogel. Anal Chim Acta 273(1–2):553–560

    Article  Google Scholar 

Download references

Acknowledgements

The work is supported by a research Grant the Science and Engineering Board, Department of Science and Technology (DST) under the DST Centre for Mathematical Biology.

Author information

Authors and Affiliations

  1. Department of Computational and Data Sciences, Indian Institute of Science, C. V. Raman Avenue, Bangalore, Karnataka, 560012, India

    Shashi Bajaj Mukherjee, Soumyendu Raha & Debnath Pal

  2. Radiological Physics and Advisory Division, Bhabha Atomic Research Centre, Mumbai, Maharashtra, 400085, India

    Debabrata Datta

Authors
  1. Shashi Bajaj Mukherjee
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Debabrata Datta
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Soumyendu Raha
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Debnath Pal
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Debnath Pal.

Ethics declarations

Conflict of interest

The authors have no competing interests.

Electronic supplementary material

Below is the link to the electronic supplementary material.

Supplementary material 1 (DOC 51 kb)

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Mukherjee, S.B., Datta, D., Raha, S. et al. Numerical simulation of a glucose sensitive composite membrane closed-loop insulin delivery system. Bioprocess Biosyst Eng 40, 1453–1462 (2017). https://doi.org/10.1007/s00449-017-1803-1

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00449-017-1803-1

Keywords

Search

Navigation