Previous |  Up |  Next


periodic homogenization; two-scale convergence; carcinogenesis; reaction-diffusion system; surface diffusion
In the context of periodic homogenization based on two-scale convergence, we homogenize a linear system of four coupled reaction-diffusion equations, two of which are defined on a manifold. The system describes the most important subprocesses modeling the carcinogenesis of a human cell caused by Benzo-[a]-pyrene molecules. These molecules are activated to carcinogens in a series of chemical reactions at the surface of the endoplasmic reticulum, which constitutes a fine structure inside the cell. The diffusion on the endoplasmic reticulum, modeled as a Riemannian manifold, is described by the Laplace-Beltrami operator. For the binding process to the surface of the endoplasmic reticulum, different scalings with powers of the homogenization parameter are considered. This leads to three qualitatively different models in the homogenization limit.
[1] Allaire, G.: Homogenization and two-scale convergence. SIAM J. Math. Anal. 23 1482-1518 (1992). DOI 10.1137/0523084 | MR 1185639 | Zbl 0770.35005
[2] Bakhvalov, N. S., Panasenko, G.: Homogenization: Averaging Processes in Periodic Media: Mathematical Problems in the Mechanics of Composite Materials. Translated from the Russian. Mathematics and Its Applications: Soviet Series 36 Kluwer Academic Publishers, Dordrecht (1989). MR 1112788
[3] Bensoussan, A., Lions, J.-L., Papanicolaou, G.: Asymptotic Analysis for Periodic Structures. Studies in Mathematics and its Applications 5 North-Holland Publ. Company, Amsterdam (1978). MR 0503330 | Zbl 0404.35001
[4] Besong, D. O.: Mathematical Modelling and Numerical Solution of Chemical Reactions and Diffusion of Carcinogenic Compounds in Cells. KTH Numerical Analysis and Computer Science TRITA NA E04152, Stockholm (2004).
[5] Canon, É., Pernin, J.-N.: Homogenization of diffusion in composite media with interfacial barrier. Rev. Roum. Math. Pures Appl. 44 23-36 (1999). MR 1841959 | Zbl 0994.35023
[6] Cioranescu, D., Donato, P.: An Introduction to Homogenization. Oxford Lecture Series in Mathematics and Its Applications 17 Oxford University Press, Oxford (1999). MR 1765047 | Zbl 0939.35001
[7] Davies, E. B.: Heat Kernels and Spectral Theory. Cambridge Tracts in Mathematics 92 Cambridge University Press, Cambridge (1989). MR 0990239 | Zbl 0699.35006
[8] Carmo, M. P. do: Riemannian Geometry. Translated from the Portuguese. Mathematics: Theory & Applications Birkhäuser, Boston (1992). MR 1138207 | Zbl 0752.53001
[9] Evans, L. C.: Partial Differential Equations. Graduate Studies in Mathematics 19 American Mathematical Society, Providence (2010). DOI 10.1090/gsm/019/02 | MR 2597943 | Zbl 1194.35001
[10] Gelboin, H. V.: Benzo[a]pyrene metabolism, activation, and carcinogenesis: Role and regulation of mixed-function oxidases and related enzymes. Physiological Reviews 60 1107-1166 (1980).
[11] Gossauer, A.: Struktur und Reaktivität der Biomoleküle. Willey-VCH-Verlag Weinheim (2003), German.
[12] Graf, I., Peter, M. A.: Diffusion on surfaces and the boundary periodic unfolding operator with an application to carcinogenesis in human cells. SIAM J. Math. Anal. Accepted.
[13] Jiang, H., Gelhaus, S. L., Mangal, D., Harvey, R. G., Blair, I. A., Penning, T. M.: Metabolism of benzo[a]pyrene in human bronchoalveolar H358 cells using liquid chromatography-mass spectrometry. Chem. Res. Toxicol. 20 1331-1341 (2007). DOI 10.1021/tx700107z
[14] Jikov, V. V., Kozlov, S. M., Oleinik, O. A.: Homogenization of Differential Operators and Integral Functionals. Translated from the Russian. Springer, Berlin (1994). MR 1329546
[15] Marchenko, V. A., Khruslov, E. Ya.: Homogenization of Partial Differential Equations. Translated from the Russian. Progress in Mathematical Physics 46 Birkhäuser, Boston (2006). MR 2182441 | Zbl 1113.35003
[16] Neuss-Radu, M.: Some extensions of two-scale convergence. C. R. Acad. Sci., Paris, Sér. I 322 899-904 (1996). MR 1390613 | Zbl 0852.76087
[17] Nguetseng, G.: A general convergence result for a functional related to the theory of homogenization. SIAM J. Math. Anal. 20 608-623 (1989). DOI 10.1137/0520043 | MR 0990867 | Zbl 0688.35007
[18] Pelkonen, O., Nebert, D. W.: Metabolism of polycyclic aromatic hydrocarbons: Etiologic role in carcinogenesis. Pharmacological Reviews 34 189-212 (1982).
[19] Peter, M. A.: Coupled reaction-diffusion processes inducing an evolution of the microstructure: Analysis and homogenization. Nonlinear Anal. 70 806-821 (2009). DOI 10.1016/ | MR 2468421 | Zbl 1151.35308
[20] Peter, M. A., Böhm, M.: Scalings in homogenisation of reaction, diffusion and interfacial exchange in a two-phase medium. Proceedings of Equadiff 11 International Conference on Differential Equations, Czecho-Slovak series Bratislava 369-376 (2005).
[21] Peter, M. A., Böhm, M.: Different choises of scaling in homogenization of diffusion and interfacial exchange in a porous medium. Math. Methods Appl. Sci. 31 1257-1282 (2008). DOI 10.1002/mma.966 | MR 2431426
[22] Peter, M. A., Böhm, M.: Multiscale modelling of chemical degradation mechanisms in porous media with evolving microstructure. Multiscale Model. Simul. 7 1643-1668 (2009). DOI 10.1137/070706410 | MR 2505062 | Zbl 1181.35019
[23] Phillips, D. H.: Fifty years of benzo[a]pyrene. Nature 303 468-472 (1983). DOI 10.1038/303468a0
[24] Sanchez-Palencia, E.: Non-homogeneous Media and Vibration Theory. Lecture Notes in Physics 127 Springer, Berlin (1980). MR 0578345 | Zbl 0432.70002
[25] Showalter, R. E.: Microstructure models of porous media. Homogenization and Porous Media U. Hornung Interdisciplinary Applied Mathematics 6 Springer, New York 183-202 (1997). DOI 10.1007/978-1-4612-1920-0_9 | MR 1434324
[26] Showalter, R. E.: Monotone Operators in Banach Space and Nonlinear Partial Differential Equations. Mathematical Surveys and Monographs 49 American Mathematical Society, Providence (1997). MR 1422252 | Zbl 0870.35004
[27] Sims, P., Grover, P. L., Swaisland, A., Pal, K., Hewer, A.: Metabolic activation of benzo(a)pyrene proceeds by a diol-epoxide. Nature 252 326-328 (1974). DOI 10.1038/252326a0
[28] Wloka, J.: Partial Differential Equations. Translated from the German. Cambridge University Press, Cambridge (1987). MR 0895589 | Zbl 0623.35006
Partner of
EuDML logo