Skip to main content
SpringerLink
Log in

A Geometry Model for the Simulation of Drug Diffusion through the Stratum Corneum

  • Regular Article
  • Published:
Computing and Visualization in Science

Abstract

We present a three-dimensional geometry model with tetrakaidecahedra for the biphasic model stratum corneum (SC) membrane ΩSC consisting of corneocytes embedded in a lipid matrix. Two practical domains for ΩSC are realized: the simple model SC-membrane ΩsSC and a realistic model SC-membrane ΩrSC with dimensions for abdominal human SC. The new geometry model uses tetrakaidecahedra as basic units. It is possible to assemble the tetrakaidecahedra one upon the other and side by side without gaps in a densest packing and with minimal area for all required interfaces. Geometric characteristics such as length, depth, height and angles of the corneocytes as well as the thickness of the lipid channels can be chosen arbitrarily. Furthermore, we are able to control the shift of the corneocytes and our concept allows to assemble many corneocytes in rows, columns and layers all embedded in a lipid matrix. With the aid of this concept the non-steady-state problem of drug diffusion within a three-dimensional, biphasic model SC-membrane, such as ΩsSC or ΩrSC, having homogeneous lipid and corneocyte phases is solved numerically with a multigrid method. The numerical computations are done with our simulation system UG. Our method for solving the diffusion problem is validated with homogeneous model SC-membranes with varying size of corneocytes and lipid channels, different numbers of corneocytes, and corneocyte alignment. Several time-dependent drug concentration profiles within the heterogeneous model SC-membranes are calculated and graphically shown for different values of relative corneocyte permeability ɛ  = D cor/D lip.

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

Similar content being viewed by others

References

  1. Albery W., Hadgraft J. (1978) Percutaneous absorption: in vivo experiments. J. Pharm. Pharmacol. 31, 140–147

    Google Scholar 

  2. Bastian P., Wittum G. On robust and adaptive multi-grid methods. In: Hemker, P., Wesseling, P. (eds.) Multigrid methods. Proceedings of the 4th European Multigrid Conference, Amsterdam, July 1993, Birkhäuser, Basel (1994)

  3. Bastian P., Birken K., Johannsen K., Lang S., Neuss N., Rentz-Reichert H., Wieners C. (1997) UG—a flexible software toolbox for solving partial differential equations. Comput. Vis. Sci. 1, 27–40

    Article  MATH  Google Scholar 

  4. Bastian P., Birken K., Johannsen K., Lang S., Reichenberger V., Wieners C., Wittum G., Wrobel C. A parallel software-platform for solving problems of partial differential equations. In: Krause, E., Jäger, W. (eds.) High performance computing in science and engineering, Springer, pp. 326–339, Berlin Heidelberg New York (1999)

  5. Boderke P., Schittkowski K., Wolf M., Merkle H.-P. (2000) Modeling of diffusion and concurrent metabolism in cutaneous tissue. J. Theor. Biol. 204, 393–407

    Article  Google Scholar 

  6. Brakke K.: http://www.susqu.edu/facstaff/b/brakke/kelvin/ kelvin.html

  7. Christophers E., Kligman A.M. (1964) Visualization of the cell layers of the stratum corneum. J. Invest. Dermatol. 42, 407–409

    Google Scholar 

  8. Crank J. The mathematics of diffusion, 2nd edn. Oxford: Oxford University Press, pp. 49–53 (1975)

  9. Elias P. M. (1983) Epidermal lipids, barrier function, and desquamation. J. Invest. Dermatol. 80, 44–49

    Article  Google Scholar 

  10. Feuchter D., Stemmermann U., Wittum G. Description and generation of geometries and grids for layered domains. 17th GAMM Seminar Leipzig on construction of grid generation algorithms http://www.mis.mpg.de//gamm/2001/, ISBN 3-00-007753-7, pp. 29–54(2001)

  11. Feuchter D., Heisig M., Liu Y., Wittum G.: Simulation der Arzneimitteldiffusion durch das Stratum Corneum. I. Geometrie- und Gittererzeugung mit Tetrakaidekaedern, WiR Preprint 04/2004, Universität Heidelberg (in German)

  12. Feuchter D.: Suitable grids for anisotropic layered domains. In: WiR-BaWü workshop on modeling and computation in environmental science MCES 04, Hohenwart Forum, Germany, 10–13 October (2004)

  13. Feuchter D., Wäcker L., Wahner R. Eval_Volume_and_ Area, a software tool to evaluate the volume and the outer surface of UG grids, Simulation in Technology, University of Heidelberg (2005)

  14. Frasch H.F. (2002) A random walk model of skin permeation. Risk. Anal. 22, 265–276

    Article  Google Scholar 

  15. Frasch H.F., Barbero A.M. (2003) Steady-state flux and lag time in the stratum corneum lipid pathway: results from finite element models. J. Pharm. Sci. 92, 2196–2207

    Article  Google Scholar 

  16. Fritsch P. Dermatologie Venerologie. S. 6. Springer, Berlin Heidelberg NewYork (2004)

  17. Fuchs A.: Optimierte Delaunay-Triangulierungen zur Vernetzung getrimmter NURBS-Körper. ISBN 3–8265-6757-9, pp. 55–58. Shaker Verlag Aachen (1999) (in German)

  18. Gienger G., Knoch A., Merkle H.P. (1986) Modelling and numerical computation of drug transport in laminates: model case evaluation of transdermal delivery system. J. Pharm. Sci. 75, 9–15

    Article  Google Scholar 

  19. Guy R.H., Hadgraft J. (1984) Prediction of drug disposition kinetics in skin and plasma following topical administration. J. Pharm. Sci. 73, 883–887

    Article  Google Scholar 

  20. Guy R.H., Hadgraft J. (1988) Physicochemical aspects of percutaneous penetration and its enhancement. Pharm. Res. 5, 753–758

    Article  Google Scholar 

  21. Hackbusch W. (1985) Multi-grid methods and applications. Springer, Berlin Heidelberg NewYork

    MATH  Google Scholar 

  22. Hackbusch W. (1989) On first and second order box schemes. Computing 41, 277–296

    Article  MathSciNet  MATH  Google Scholar 

  23. Hackbusch W. Iterative Lösung groβer schwachbesetzter Gleichungssysteme. Teubner, Stuttgart (1991) (in German)

  24. Hales T. http://www.math.pitt.edu/˜thales/flyspeck, http://www.math.lsa.umich.edu/˜hales/kepler.html (in German)

  25. Hebisch U. http://www.mathe.tu-freiberg.de/˜hebisch/cafe/ platonische.html (in German)

  26. Heisig M., Lieckfeldt, R., Wittum, G., Mazurkevich, G., Lee, G. (1995) A non steady-state, biphasic model for solute through stratum corneum. ICA-Berichte 95/3, Universität Stuttgart

  27. Heisig M., Lieckfeldt R., Wittum G., Mazurkevich G., Lee G. (1996) Non steady-state descriptions of drug permeation through stratum corneum. I. The biphasic brick-and-mortar model. Pharm. Res.13, 421–426

    Article  Google Scholar 

  28. Heisig M., Feuchter D., Liu Y., Wagner C., Wittum G. A three-dimensional non-steady-state, biphasic model of drug permeation through stratum corneum. AAPS Annual Meeting and Exposition, Baltimore, USA, 7–11 November, 2004

  29. Kepler, J.: Strena seu de nive sexangula (in Latin). http:// www.thelatinlibrary.com/kepler/strena.html (1611)

  30. Johnson M.E., Blankschtein D., Langer R. (1997) Evaluation of solute permeation through the stratum corneum: lateral bilayer diffusion as the primary transport mechanism. J. Pharm. Sci. 86, 1162– 1172

    Article  Google Scholar 

  31. Lang S., Wittum G. (2005) Large-scale density-driven flow simulation using parallel unstructured grid adaption and local multigrid methods. Concurrency Comput. 17, 1415–1440

    Article  Google Scholar 

  32. Lee A.J., King J.R., Barrett D.A. (1997) Percutaneous absorption: a multiple pathway model. J. Control. Release 45, 141–151

    Article  Google Scholar 

  33. Leppmaier M. Kugelpackungen von Kepler bis heute. Vieweg (1997) (in German)

  34. Lieckfeldt R., Lee G. W. J., Wittum G., Heisig M.: Diffusant concentration profiles within corneocytes and lipid phase of stratum corneum. In: Proceedings of the International Symposium Control. Rel. Bioact. Mater. 20, 1993, Washington, USA

  35. Lieckfeldt R., Villalaín J., Gómes-Fernándes J., Lee G. (1993) Diffusivity and structural polymorphism in some model stratum corneum lipid systems. Biochim. Biophys. Acta. 1151, 182–188

    Google Scholar 

  36. Manitz R., Lucht W. Strehmel K., Weiner R., Neubert R. (1998) On mathematical modeling of dermal and transdermal drug delivery. J. Pharm. Sci. 87, 873–879

    Article  Google Scholar 

  37. Menton D.N., Eisen A.Z. (1971) Structure and organization of mammalian stratum corneum. J. Ultrastruct. Res. 35, 247–264

    Article  Google Scholar 

  38. Peck K.D., Higuchi W.I. Characterization of the passive transdermal diffusional route of polar/ionic permeants. In: Potts, R.O., Guy, R.H. (eds.) Mechanisms of Transdermal Drug Delivery. Marcel Dekker, New York (1997)

  39. Richter T., Müller J.H., Schwarz U.D., Wepf R., Wiesendanger R. (2001) Investigation of the swelling of human skin cells in liquid media by tapping mode scanning force microscopy. Appl. Phys. A. 72, 125–128

    Article  Google Scholar 

  40. Schmauder M. (1995) Diffusion through the human skin – an unexpected application of a PDE-black-box-solver. ZAMM. 75, 707–708

    MathSciNet  Google Scholar 

  41. Schätzlein A., Cevc G. (1998) Non-uniform cellular packing of the stratum corneum and permeability barrier function of intact skin: a high-resolution confocal laser scanning microscopy study using highly deformable vesicles (Transfersomes). Br. J. Dermatol. 138, 583–592

    Article  Google Scholar 

  42. Springer M.: Nachgehakt: Ein erschöpfender Beweis. Spektrum der Wissenschaft, September 2003, p. 13 (2003) (in German)

  43. Talreja P.S., Kasting G.B., Kleene N.K., Pickens W.L., Wang T.-F. (2001) Visualization of the lipid barrier and measurement of lipid pathlength in human stratum corneum. AAPS Pharm. Sci. 3(2): 1–9

    Article  Google Scholar 

  44. Thomson W., Lord Kelvin (1887) On the division of space with minimum partitional area. Phil. Mag. 24, 503

    Google Scholar 

  45. Weaire D., Phelan R. (1994) A counterexample to Kelvin’s conjecture on minimal surfaces. Phil. Mag. Lett. 69, 107–110

    Article  Google Scholar 

  46. Weaire D. (1997) The Kelvin problem: foam structures of minimal surface area. Taylor & Francis, London

    Google Scholar 

  47. Wesseling P. (1991) An introduction to multigrid methods. Wiley, Chichester

    Google Scholar 

  48. Wilschut A., ten Berge W.F., Robinson P.J., McKone T.E. (1995) Estimating skin permeation. The validation of five mathematical skin permeation models. Chemosphere 30, 1275–1296

    Google Scholar 

  49. Wittum G. (1989) On the robustness of ILU-smoothing. SIAM J. Sci. Stat. Comput. 10, 699–717

    Article  MathSciNet  MATH  Google Scholar 

  50. Wittum G. Mehrgitterverfahren. Spektrum der Wissenschaft, pp 78–90 (1990) (in German)

  51. Wittum G. Filternde Zerlegungen–Schnelle Löser für groβe Gleichungssysteme. Teubner Skripten zur Numerik Band 1, Teubner, Stuttgart (1992) (in German)

  52. Wolf M. Mathematisch-physikalische Berechnungs- und Simulationsmodelle zur Beschreibung und Entwicklung therapeutischer Systeme. Habilitationsschrift, Rheinische Friedrich-Schiller-Universität zu Bonn, Bonn (1993) (in German)

  53. Yu B., Dong C.-Y., So P.T.C., Blankschtein D., Langer R. (2001) In vitro visualization and quantification of oleic acid-induced changes in transdermal transport using two-photon fluorescence microscopy. J. Invest. Dermatol. 117, 16–25

    Article  Google Scholar 

  54. Yu B., Kim K.H., So P.T.C., Blankschtein D., Langer R. (2002) Topographic heterogeneity in transdermal transport revealed by high-speed two-photon fluorescence microscopy. Determination of representative skin sample sizes. J. Invest. Dermatol. 118, 1085–1088

    Google Scholar 

  55. Yu B., Kim K.H., So P.T.C., Blankschtein D., Langer R. (2003) Visualization of oleic acid induced transdermal diffusion pathways using two-photon fluorescence microscopy. J. Invest. Dermatol. 120, 448–455

    Article  Google Scholar 

  56. Ziherl P., Kamien D. (2001) Maximizing entropy by minimizing area: towards a new principle of self-organization. Department of Physics and Astronomy, University of Pennsylvania, Philadelphia

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. IWR, Simulation in Technology (SiT), University of Heidelberg, Im Neuenheimer Feld 368, 69120, Heidelberg, Germany

    D. Feuchter, M. Heisig & G. Wittum

Authors
  1. D. Feuchter
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. M. Heisig
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. G. Wittum
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to D. Feuchter.

Additional information

Communicated by R.H.W. Hoppe.

Dedicated to Peter Deuflhard on the occasion of his 60th birthday.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Feuchter, D., Heisig, M. & Wittum, G. A Geometry Model for the Simulation of Drug Diffusion through the Stratum Corneum. Comput. Visual Sci. 9, 117–130 (2006). https://doi.org/10.1007/s00791-006-0017-x

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00791-006-0017-x

Keywords

Search

Navigation