Journal of Mathematical Biology

, Volume 57, Issue 5, pp 675–695 | Cite as

Modeling capsule tissue growth around disk-shaped implants: a numerical and in vivo study

Article
First Online:
Received:
Revised:

Abstract

We propose a new mathematical model that describes the growth of fibrous tissue around rigid, disk-shaped implants. A solution methodology based on an efficient regularized iterative method is presented to calibrate the model from some measurements of the capsule tissue concentration. Numerical results obtained with synthetic data are presented to demonstrate the ability of the proposed solution methodology to determine the model parameters corresponding to a given implant. In addition, numerical results obtained with experimental data are presented to illustrate the validity of the proposed model.

Keywords

Biomedical implants Fibrous capsule formation Finite difference method Implicit scheme Newton method Tikhonov regularization Inverse problem Design problem 

Mathematics Subject Classification (2000)

35 39 65 92 
This is a preview of subscription content, log in to check access.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Li D.J., Ohsaki K., Li K., Cui P.C., Ye Q., Baba K., Wang Q.C., Tenshin S., Yamamoto T.T.: Thickness of fibrous capsule after implantation of hydroxyapatite in subcutaneous tissue in rats. J. Biomed. Mater. Res. 45, 322–326 (1999)CrossRefGoogle Scholar
  2. 2.
    Zhu C.S., Ohsaki K., Li K., Ye Q., Trn Y.H., Ohba Y., Moriyama K.: Long-term observation of subcutaneous tissue reaction to synthetic auditory ossicle (Bioceram©) in rats. J. Med. Invest. 46, 97–103 (1999)Google Scholar
  3. 3.
    Anderson J.M.: Biological responses to materials. Annu. Rev. Mater. Res. 31, 81–110 (2001)CrossRefGoogle Scholar
  4. 4.
    Ravin A.G., Olbrich K.C., Levin L.S., Usala A-l., Klizman B.: Long and short-term effects of biological hydrogels on capsule microvascular density around implants in rats. J. Biomed. Mater. Res. 58, 313–318 (2001)CrossRefGoogle Scholar
  5. 5.
    Sanders J.E., Bale S.D., Neumann T.: Tissue response to microfibers of different polymers: polyster, polyethylene, polylactic acid, and polyurethane. J. Biomed. Mater. Res. 62, 222–227 (2002)CrossRefGoogle Scholar
  6. 6.
    Lehle K., Lohn S., Reinerth G., Schubert T., Preuner J.G., Birnbaum D.E.: Cytological evaluation of the tissue-implant reaction associated with subcutaneous implantation of polymers coated with titaniumcarboxonitride in vivo. Biomaterials 25, 5457–5466 (2004)CrossRefGoogle Scholar
  7. 7.
    Kontio R., Ruuttila P., Lindroos L., Suuronen R., Salo A., Lindqvist C., Virtanen I., Konttinen Y.T.: Biodegradable polydioxanone and poly(L/D)lactide implants: an experimental study on peri-implant tissue response. Int. J. Oral Maxillofac. Surg. 34, 766–776 (2005)CrossRefGoogle Scholar
  8. 8.
    Sackenheim M.M.: Radio frequency ablation: the key to cancer treatment. J. Diagnost. Med. Sonogr. 19(2), 88–92 (2003)CrossRefGoogle Scholar
  9. 9.
    Yamada M., Tanemura K., Okada S., Iwanami A., Nakamura M., Mizuno H., Ozawa M., Ohyama-Goto R., Kitamura N., Kawano M., Tan-Takeuchi K., Ohtsuka C., Miyawaki A., Takashima A., Ogawa M., Toyama Y., Okano H., Kondo T.: Electrical stimulation modulates fate determination of differentiating embryonic stem cells. Stem Cells 25, 562–570 (2007)CrossRefGoogle Scholar
  10. 10.
    Agren M.S., Engel M.A., Mertz P.M.: Collagenase during burn wound healing: influence of a hydrogel dressing and pulsed electrical stimulation. Plast. Reconstruct. Surg. 94(3), 518–524 (1994)CrossRefGoogle Scholar
  11. 11.
    Kopf A.W., Bart R.S., Schrager D., Lazar M., Popkin G.L.: Curettage-electrodesiccation treatment of basal cell carcinomas. Arch. Dermatol. 113(4), 439–443 (1977)CrossRefGoogle Scholar
  12. 12.
    Perumpanani A.J., Sherratt J.A., Norbury J.: Mathematical modeling of capsule formation and multinodulatority in benign tumor growth. Nonlinearity 10, 1599–1614 (1997)MATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    Sherratt J.A.: Traveling wave solutions of a mathematical model for tumor encapsulation. SIAM J. Appl. Math. 60, 392–407 (1999)CrossRefMathSciNetGoogle Scholar
  14. 14.
    Pozrikidis C.: Modeling and Simulation of Capsules and Biological Cells. Chapman & Hall/CRC, London (2003)MATHGoogle Scholar
  15. 15.
    Ryan, P.: A mathematical and laboratory investigation of an electrical current-based method for disrupting the growth of fibrous tissue around biomaterial implants. Master’s Thesis, California State University Northridge (2007)Google Scholar
  16. 16.
    Dautray R., Lions J.: Mathematical Analysis and Numerical Methods for Science and Technology, vol.~5, Evolution Problems I. Springer, Heidelberg (1992)MATHGoogle Scholar
  17. 17.
    Isakov V.: Inverse Problems for Partial Differential Equations. Springer, Heidelberg (1997Google Scholar
  18. 18.
    Knabner P., Angermann L.: Numerical Methods for Elliptic and Parabolic Partial Differential Equations. Springer, Heidelberg (2003)MATHGoogle Scholar
  19. 19.
    Strikwerda, J.C.: Finite Difference Schemes and Partial Differential Equations, 2nd edn. SIAM (2004)Google Scholar
  20. 20.
    Hadamard J.: Lectures on Cauchy’s Problem in Linear Partial Differential Equations. Yale University Press, New Haven (1923)MATHGoogle Scholar
  21. 21.
    Engl H.W., Hanke M., Nebauer A.: Regularization of Inverse Problems. Kluwer, Dordrecht (1996)MATHGoogle Scholar
  22. 22.
    Gilyazov S.F., Goldman N.L.: Regularization of Ill-posed Problems by Iteration Methods. Kluwer, Dordrecht (2000)MATHGoogle Scholar
  23. 23.
    Tikhonov A.N.: Regularization of incorrectly posed problems. Soviet Math. Doklady 4, 1624–1627 (1963)MATHGoogle Scholar
  24. 24.
    Tikhonov A.N., Arsenin V.Y.: Solutions of Ill-posed Problems. Winston and Sons, Washington (1977)MATHGoogle Scholar
  25. 25.
    Morozov V.A.: Choice of parameter for the solution of functional equations by the regularization method. Dokl. Akad. Nauk. SSSR 175, 1000–1003 (1967)MathSciNetGoogle Scholar
  26. 26.
    Morozov V.A.: On the solution of functional equations by the method of the regularization. Soviet Math. Doklady 167, 414–417 (1966)Google Scholar
  27. 27.
    Wahba, G.: Spline Models for Observations Data. SIAM (1990)Google Scholar
  28. 28.
    Hansen P.C., O’Leary D.P.: The use of the L-curve in the regularization of discrete ill-posed problems. SIAN 14, 1487–1503 (1993)MATHMathSciNetGoogle Scholar
  29. 29.
    Haberman R.: Applied Partial Differential Equations. Prentice Hall, Englewood Cliffs (1983)MATHGoogle Scholar
  30. 30.
    Golub G.H., Van Loan C.F.: Matrix Computations. The Johns Hopkins University Press, Baltimore (1983)MATHGoogle Scholar
  31. 31.
    Davis, T.A.: Direct Methods for Sparse Linear Systems. SIAM (2006)Google Scholar
  32. 32.
    Colton D., Kress R.: Inverse Acoustic and Electromagnetic Scattering Theory. Springer, Heidelberg (1992)MATHGoogle Scholar

Copyright information

© Springer-Verlag 2008

Authors and Affiliations

  1. 1.Department of MathematicsCalifornia State University NorthridgeLos AngelesUSA
  2. 2.Department of BiologyCalifornia State University NorthridgeLos AngelesUSA

Personalised recommendations