Evolution and memory effects in the homogenization limit for electrical conduction in biological tissues

Mathematical Models and Methods in Applied Sciences

Volumn 14, Issue 9, 2004, Pages 1261-1295

  Amar, Micol ^a   Andreucci, Daniele ^a   Gianni, Roberto ^a   Bisegna, Paolo ^b  

^a SAPIENZA UNIVERSITY OF ROME   (Italy)

^b UNIVERSITY OF ROME TOR VERGATA   (Italy)

Author keywords

Dynamical condition; Electrical conduction in biological tissues; Evolution equation with memory; Homogenization

Indexed keywords

EID: 1842512759     PISSN: 02182025     EISSN: None     Source Type: Journal    
DOI: 10.1142/S0218202504003623     Document Type: Article

References (36)

1. S. Agmon, A. Douglis and L. Nirenberg, Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary value conditions I, Commun. Pure Appl. Math. 12 (1959) 623-727.
2. R. Alexandre, Some results in homogenization tackling memory effects, Asympt. Anal. 15 (1997) 229-259.
3. M. Amar, D. Andreucci, P. Bisegna and R. Gianni, Electrical conduction in finely mixed media, in preparation.
4. M. Amar, D. Andreucci, P. Bisegna and R. Gianni, An elliptic equation with history, C. R. Acad. Sci. Paris, Ser. I 338 (2004) 595-598.
5. M. Amar, D. Andreucci, P. Bisegna and R. Gianni, Existence and uniqueness for an elliptic problem with evolution arising in electrodynamics, to appear.
6. M. Amar, D. Andreucci, P. Bisegna and R. Gianni, Homogenization limit for electrical conduction in biological tissues in the radio-frequency range, Comptes Rend. Méc. 331 (2003) 503-508.
7. L. Ambrosio, N. Fusco and D. Pallara, Functions of Bounded Variation and Free Discontinuity Problems, Oxford Mathematical Monographs (Oxford Univ. Press, 2000).
8. Y. Amirat, K. Hamdache and A. Ziani, Homogenisation of parametrised families of hyperbolic problems, Proc. Roy. Soc. Edinburgh A120 (1992) 199-221.
9. D. Andreucci and R. Gianni, Global existence and blow up in a problem with nonlocal dynamical boundary conditions, Adv. Differential Equations 1 (1996) 729-752.
10. H. Attouch, Variational Convergence for Functionals and Operators (Pitman, 1984).
11. N. Bakhvalov and G. Panasenko, Homogenization: Averaging Processes in Periodic Media. Mathematical Problems in the Mechanics of Composite Materials (Kluwer, 1989).
12. A. Bensoussan, J. L. Lions and G. Papanicolaou, Asymptotic Analysis for Periodic Structures (North-Holland, 1978).
13. P. Bisegna, G. Caruso and F. Lebon, Bioelectrical impedance analysis: A matter of homogenization of composites with imperfect interfaces, in Proc. 15th AIMETA Congress of Theoretical and Applied Mechanics, ed. G. Augusti (2001).
14. A. Braides and A. Defranceschi, Homogenization of Multiple Integrals (Oxford Univ. Press, 1998).
15. J. D. Bronzino, The Biomedical Engineering Handbook (CRC Press, 1999).
16. S. Challal, Homogénéisation d'un problème á mémoire instantanée dans un milieu très finement perforé, Comptes Rend. Acad. Sci. Série I Math. 317 (1993) 149-154.
17. D. Cioranescu and P. Donato, An Introduction to Homogenization (Oxford Univ. Press, 1999).
18. P. Colli Franzone, L. Guerri and M. Pennacchio, Mathematical models and problems in electrocardiology, Riv. Mat. dell'Univ. Parma 2 (1999) 123-142.
19. A. De Lorenzo, A. Andreoli, J. Matthie and P. Withers, Predicting body cell mass with bioimpedence by using theoretical methods: A technological review, J. Appl. Physiol. 82 (1997) 1542-1558.
20. J. Escher, Nonlinear elliptic systems with dynamic boundary conditions, Math. Z. 210 (1992) 413-439.
21. J. Escher, Stable equilibria to elliptic equations in unbounded domains with nonlinear dynamic boundary conditions, Anal. 20 (2000) 325-351.
22. K. R. Foster and H. P. Schwan, Dielectric properties tissues and biological materials: A critical review, Critical Rev. Biomedical Eng. 17 (1989) 25-104.
23. E. Frénod and F. Watbled, The Vlasov equation with strong magnetic field and oscillating electric field as a model for isotop resonant separation, Electronic J. Differential Equations 2002 (2002) 1-20.
24. V. V. Jikov, S. M. Kozlov and O. A. Oleinik, Homogenization of Differential Operators and Integral Functionals (Springer-Verlag, 1994).
25. E. R. Kandell, J. H. Schwartz and T. M. Jessell, Principles of Neural Science 4th edn. (McGraw-Hill, 2000).
26. W. Krassowska and J. C. Neu, Homogenization of syncytial tissues, Critical Rev. Biomed. Eng. 21 (1993) 137-199.
27. O. A. Ladyzhenskaja, and N. N. Ural'ceva,, Linear and Quasilinear Elliptic Equations (Academic Press, 1968).
28. F. Lene and D. Leguillon, Eacute;tude de l'influence d'un glissement entre les constituants d'un matériau composite sur ses coefficients de comportement effectifs, J. Méc. 20 (1981) 509-536.
29. J. L. Lions, Some Methods in the Mathematical Analysis of Systems and Their Controls (Science Press, 1981).
30. R. Lipton, Heat conduction in fine scale mixtures with interfacial contact resistance, SIAM J. Appl. Math. 58 (1998) 55-72.
31. O. A. Oleinik, A. S. Shamaev and G. A. Yosifian, Mathematical Problems in Elasticity and Homogenization (North-Holland, 1992).
32. J. F. Rodrigues, V. A. Solonnikov and F. Yi, On a parabolic system with time derivative in the boundary conditions and related free boundary problems, Math. Anna. 315 (1999) 61-95.
33. J. Sanchez-Hubert and E. Sánchez-Palencia, Introduction aux Methods Asymptotiques et à l'Homogénéisation (Masson, 1992).
34. E. Sánchez-Palencia, Non Homogeneous Media and Vibration Theory, Lecture Notes in Physics, Vol. 127 (Springer-Verlag, 1980).
35. L. Tartar, Memory effects and homogenization, Arch. Rat. Mech. Anal. 111 (1990) 121-133.
36. G. Yosifian, On some homogenization problems in perforated domains with nonlinear boundary conditions, Applicable Anal. 65 (1997) 257-288.