×

Mathematical homogenization in the modelling of digestion in the small intestine. (English) Zbl 1350.92011

Summary: Digestion in the small intestine is the result of complex mechanical and biological phenomena which can be modelled at different scales. In a previous article, we introduced a system of ordinary differential equations for describing the transport and degradation-absorption processes during the digestion. The present article sustains this simplified model by showing that it can be seen as a macroscopic version of more realistic models including biological phenomena at lower scales. In other words, our simplified model can be considered as a limit of more realistic ones by averaging-homogenization methods on biological processes representation.

MSC:

92C30 Physiology (general)
35B27 Homogenization in context of PDEs; PDEs in media with periodic structure
34C29 Averaging method for ordinary differential equations
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] Barles, G., Solutions de viscosité des équations de Hamilton-Jacobi, 17 (1994), Springer-Verlag: Springer-Verlag, Paris · Zbl 0819.35002
[2] Barles, G., Nonlinear Neumann boundary conditions for quasilinear degenerate elliptic equations and applications, J. Differential Equations, 154, 1, 191-224 (1999) · Zbl 0924.35051 · doi:10.1006/jdeq.1998.3568
[3] Barles, G.; Da Lio, F.; Lions, P.-L.; Souganidis, P. E., Ergodic problems and periodic homogenization for fully nonlinear equations in half-space type domains with Neumann boundary conditions, Indiana Univ. Math. J., 57, 5, 2355-2375 (2008) · Zbl 1173.35013 · doi:10.1512/iumj.2008.57.3363
[4] Crandall, M. G.; Ishii, H.; Lions, P.-L., User’s guide to viscosity solutions of second order partial differential equations, Bull. Amer. Math. Soc. (N.S.), 27, 1, 1-67 (1992) · Zbl 0755.35015 · doi:10.1090/S0273-0979-1992-00266-5
[5] Evans, L. C., The perturbed test function method for viscosity solutions of nonlinear PDE, Proc. Roy. Soc. Edinburgh Sect. A, 111, 3-4, 359-375 (1989) · Zbl 0679.35001 · doi:10.1017/S0308210500018631
[6] Evans, L. C., Periodic homogenisation of certain fully nonlinear partial differential equations, Proc. Roy. Soc. Edinburgh Sect. A, 120, 3-4, 245-265 (1992) · Zbl 0796.35011 · doi:10.1017/S0308210500032121
[7] Ishii, H., Perron’s method for Hamilton-Jacobi equations, Duke Math. J., 55, 2, 369-384 (1987) · Zbl 0697.35030 · doi:10.1215/S0012-7094-87-05521-9
[8] Keener, J.; Sneyd, J., Mathematical physiology. Vol. II: Systems physiology, 8/ (2009), Springer: Springer, New York · Zbl 1273.92018 · doi:10.1007/978-0-387-79388-7_1
[9] Logan, J. D.; Joern, A.; Wolesensky, W., Location, time, and temperature dependence of digestion in simple animal tracts, J. Theoret. Biol., 216, 1, 5-18 (2002) · doi:10.1006/jtbi.2002.2541
[10] Mernone, A. V.; Mazumdar, J. N.; Lucas, S. K., A mathematical study of peristaltic transport of a Casson fluid, Math. Comput. Modelling, 35, 7-8, 895-912 (2002) · Zbl 1022.76058 · doi:10.1016/S0895-7177(02)00058-4
[11] Miftahof, R.; Akhmadeev, N., Dynamics of intestinal propulsion, J. Theoret. Biol., 246, 2, 377-393 (2007) · Zbl 1120.92022 · doi:10.1016/j.jtbi.2007.01.006
[12] Piccinini, L. C., Homogeneization problems for ordinary differential equations, Rend. Circ. Mat. Palermo (2), 27, 1, 95-112 (1978) · Zbl 0416.34019 · doi:10.1007/BF02843869
[13] Randall, D.; Burggren, W.; French, K.; Eckert, R., Eckert Animal Physiology: Mechanisms and Adaptations (1997), W.H. Freeman & Company
[14] Rivest, J.; Bernier, J. F.; Pomar, C., A dynamic model of protein digestion in the small intestine of pigs, J Anim Sci, 78, 2, 328-340 (2000)
[15] Taghipoor, M.; Lescoat, P.; Licois, J.-R.; Georgelin, Ch.; Barles, G., Mathematical modeling of transport and degradation of feedstuffs in the small intestine, Journal of Theoretical Biology, 294, 114-121 (2012) · Zbl 1397.92149 · doi:10.1016/j.jtbi.2011.10.024
[16] Yamauchi, K. E., Review of a histological intestinal approach to assessing the intestinal function in chickens and pigs, Animal Science Journal, 78, 356-370 (2007)
[17] Zhao, X. T.; McCamish, M. A.; Miller, R. H.; Wang, L.; Lin, H. C., Intestinal transit and absorption of soy protein in dogs depend on load and degree of protein hydrolysis., J Nutr, 127, 12, 2350-2356 (1997)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.