Homogenization of a carcinogenesis model with different scalings with the homogenization parameter

Graf, Isabell ; Peter, Malte A.

Mathematica Bohemica, Tome 139 (2014) no. 2, pp. 163-184

Abstract (VO)

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.

Détail

DOI : 10.21136/MB.2014.143847

Classification : 35B10, 35B27, 35B45, 35K51, 35K58, 92C37
Keywords: periodic homogenization; two-scale convergence; carcinogenesis; reaction-diffusion system; surface diffusion

Bibliographie

[1] Allaire, G.: Homogenization and two-scale convergence. SIAM J. Math. Anal. 23 1482-1518 (1992). | DOI | MR | Zbl

[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

[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 | Zbl

[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 | Zbl

[6] Cioranescu, D., Donato, P.: An Introduction to Homogenization. Oxford Lecture Series in Mathematics and Its Applications 17 Oxford University Press, Oxford (1999). | MR | Zbl

[7] Davies, E. B.: Heat Kernels and Spectral Theory. Cambridge Tracts in Mathematics 92 Cambridge University Press, Cambridge (1989). | MR | Zbl

[8] Carmo, M. P. do: Riemannian Geometry. Translated from the Portuguese. Mathematics: Theory & Applications Birkhäuser, Boston (1992). | MR | Zbl

[9] Evans, L. C.: Partial Differential Equations. Graduate Studies in Mathematics 19 American Mathematical Society, Providence (2010). | DOI | MR | Zbl

[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). | DOI

[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

[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

[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 | Zbl

[16] Neuss-Radu, M.: Some extensions of two-scale convergence. C. R. Acad. Sci., Paris, Sér. I 322 899-904 (1996). | MR | Zbl

[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 | MR | Zbl

[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 | MR | Zbl

[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 | MR

[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 | MR | Zbl

[23] Phillips, D. H.: Fifty years of benzo[a]pyrene. Nature 303 468-472 (1983). | DOI

[24] Sanchez-Palencia, E.: Non-homogeneous Media and Vibration Theory. Lecture Notes in Physics 127 Springer, Berlin (1980). | MR | Zbl

[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 | MR

[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 | Zbl

[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

[28] Wloka, J.: Partial Differential Equations. Translated from the German. Cambridge University Press, Cambridge (1987). | MR | Zbl