The study of diffusion equation in the pulmonary

Document Type : Research Paper

Author

Department of Applied Mathematics, Faculty of Mathematics, Khaje Nasiruddin Tusi University of Technology, Tehran

Abstract

 Biomathematics is an important field in applied mathematics which is associated with to simulate living organisms behaviors. In this field, by studying the data and information which are obtained from nature, natural processes can be simulated mathematically. In this study, by introducing the concepts of physiology concern to breathing system, the breathing system in pulmonary circulation is simulated by mathematical equations. The resulting equation includes diffusion term and is called diffusion equation. In addition, the goal of study is oxygenation of blood in the pulmonary capillaries. Further, the simultaneous transport of oxygen and carbon dioxide with effect of convection is considered in the pulmonary circulation. This process, include parabolic partial differential equation.

Keywords

References

[1] C. S. Desai and L. D. Johnson, Evaluation of some numerical schemes for consolidation, Int. J. Numer. Methods Eng., 7 243–254 (1973).
[2] A. Aminataei, M. Sharan and M. P. Singh, A numerical solution for the nonlinear convective-facilitated diffusion reaction problem for the process of blood oxygenation in the lungs, J. Nat. Acad. Math., 3 182-197 (1985).
[3] A. Aminataei, M. Sharan and M. P. Singh, A numerical model for the process of gas exchange in the pulmonary capillaries, Indian J. Pure Appl. Math., 18 (1987) 1040-1060.
[4] M. Sharan, A. Aminataei and M. P. Singh, A numerical steady of non-steady transport of gases in the pulmonary capillaries, J. Math. Biol., 25 (1987) 433–452.
[5] M. Sharan, A. Aminataei and M. P. Singh, The process of gas exchange in the pulmonary circulation incorporating the contribution of axial diffusion, Int. J. BioMedical Comput., 20 (1987) 191-209.
[6] M. Sharan, M. P. Singh and A. Aminataei, A numerical model for the blood oxygenation in the pulmonary capillaries-effec of pulmonary membrane resistance, BioSystems, 20 (1987) 355.
[7] A. Aminataei, M. Sharan and M. P. Singh, Two layer model for the process of blood oxygenation in the pulmonar capillaries-parabolic profiles in the core as well as in the plasma layer., Appl. Math. Modelling, 12 (1988) 601–609.
[8] M. Sharan, M. P. Singh and A. Aminataei, A mathematical model for the computation of the oxygen dissociation curve in human blood, BioSystems, 22 (1989) 249-260.
[9] M. P. Singh, M. Sharan and A. Aminataei, Development of mathematical formulae for oxygen and carbon dioxide dissociation curves in the blood, IMA J. Maths. Appl., in Medicine and Biology, 6 (1989) 25-46.
[10] M. Sharan, M. P. Singh and A. Aminataei, A numerical model for studying the effect of plasma layer on the process on blood oxygenation in the pulmonary capillaries, Transactions of the ASME J. Biomech Eng., 112 (1990) 457-463.
[11] A. Aminataei, A numerical two layer model for blood oxygenation in lungs, Amir-Kabir J. Sci. Tech., 12 no. 45 (2001) 63-85.
[12] A. Aminataei, A mathematical model for oxygen dissociation curve in the blood, Euro. J. Scien. Res., 6 no. 1 (2005) pp. 5.
[13] A. Aminataei and S. Hassani, An efficient numerical method for the solution of initial and boundary values problems, J. of Sci., Al-Zahra Univ., 22 no. 2 (2009) 13-30.
[14] A. Aminataei, Comparison of explicit and implicit approaches to numerical solution of uni-dimensional equation of diffu-sion, J. of Sci., Al-Zahra Univ., 15 no. 2 (2002) 1-20.
[15] A. Aminataei, A numerical simulation of the unsteady convective-diffusion equation, The J. of Damghan Univ. of Basic Scis., 1 no. 2 (2008) 73-87.
[16] A. Aminataei, Numerical simulation of the process of oxygen mass transport in the human pulmonary capillaries incorpo-rating the contribution of axial diffusion, J. of Sci., Kharazmi Univ., 13 no. 3 (2013) 779-796.
[17] W. F. Ganong, Review of medical physiology, 11th ed., New York. Mc Graw Hill, 2001 172-290.
[18] A. C. Guyton, Text-book of medical physiology, W. B. Saunders Company, Philadelphia, 1981.
[19] Y. C. Fung, Biomechanics-Mechanical properties of living tissues, Springer-Verlag., New York, 1981.
[20] L. Dintenfass, Influence of plasma proteins on the in-vivo and in-vitro rheological properties of blood, 5 th Int’l Bio-rheology Congress Abstracts, Bio-rheology, 20 (1983) pp. 384.
[21] W. J. Dorson, R. Yee, K. G. Larsen, R. G. Elgas and M. E. Voorlees, Oxygen diffusion in blood and plasma, Proc. 24 th ACEMB, 13 (1971) pp. 280.
[22] R. M. Navari, J. L. Gainer and K. R. Hall, A predictine theory for diffusion in polymer and protein solutions, AICHE J., 17 (1971) pp. 1028.
[23] T. K. Goldstick, V. T. Ciuryla and L. Zuckerman, Diffusion of oxygen in plasma and blood, Adv. In Expt’l Medi and Biol., 75 (1976) pp. 183.
[24] J. L. Gainer and W. B. Stringfield, Effect of transferring on diffusion of oxygen in plasma, 28 (1982) pp. 836.
[25] M. M. Lih, Transport Phenomena in Medicine and Biology, John Wiley Sons, Inc., New York, 1975.
[26] E. P. Hill, G. Power and L. D. Longo, Kinetics of o2 and co2 exchange, in: Bioengineering aspects of the lung, J. B. West, Marcel Dekker, Inc., New York, 1977 459-514.
[27] A. Aminataei, Blood oxygenation in the pulmonary circulation: a review, Euro. J. Scien. Res., 10 no. 2 (2005) 55-71.
[28] F. P. Chinard, G. J. Vosburgh and T. Enns, Am. J. Physiol., 183 (1955) 221-234.
[29] C. A. Goresky and G. G. Bach, Ann. NY Acad. Sci., 170 (1970) 18-45.
[30] A. Krough, J. Physiol. London, 52 (1918-19) 391-408.
[31] A. Krough, J. Physiol. London, 52 (1918-19) 409-415.
[32] T. Weis-Fogh, J. Exp. Biol., 41 (1964) 229-256.
[33] K. Schmidt-Nielsen, How Animals Work, Cambridge Univ. Press., (1972) pp. 114.
[34] R. R. Tarica, E. Koushanpour and W. F. Stevens, Chem. Eng. Progr. Symp. Ser., 67 no. 114 (1971) 28-34.
[35] W. Moll, In Oxygen Transport in Blood and Tissue, Ed. Lubbers et al., Stuttgart: G. Thieme, (1968) 70-79.
[36] F. C. Romanul, Nature, 201 (1964) 307-308.
[37] R. B. Bird, W. E. Stewart and E. N. Lightfoot, Transport Phenomena, New York: Wiley, 1960 pp. 780.
[38] R. Skalak, In Biomechanics: Its Foundations and Objectives, Y. C. Fung and Eds, Prentice-Hall, Englewood Cliffs, 1972.
[39] Y. C. Fung and B. W. Zweifach, Ann. Rev. Fluid Mech., 3 (1971) 189-210.
[40] Y. C. Fung, Microvasc. Res., 5 (1973) 34-48.
[41] E. D. Crandall and R. W. Flumerfelt, Morphological aspects of the pulmonary circulation and of the airways, NATO Advisory Group for Aerospace Research and Development, Conf. Proc., no. 65 (1970) 447-475.
[42] M. Intaglietta and B. W. Zweifach, Microcirculatory Basis of Fluid Exchange, Adv. Biol. Med. Phys., 15 (1974) 111-159.
[43] B. Aberg, Restricted convection diffusion across capillary walls, Thesis. Uppsala Univ., Sweden, 1973.
[44] D. D. Reneau, D. F. Bruley, H. I. Bicher and M. H. Knisely, Systems analysis of transport processes in human brain-Part I. Presented at First Pacific, Chem. Eng. Congr., Kyoto, Japan, 1972.
[45] K. Winkler, L. Bass, S. Keiding and N. Tygstrup, Presented at the Alfred Benzon Symposium. VI, Regulation of Hepatic Metabolism, Copenhagen, May 1973 20-24.

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History

  • Receive Date: 17 July 2013
  • Revise Date: 31 July 2014
  • Accept Date: 23 August 2014
  • Publish Date: 22 August 2016

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