Skip to main content

Advertisement

SpringerLink
Log in

Numerical simulation of effect of convection-diffusion on oxygen transport in microcirculation

  • Published:
Applied Mathematics and Mechanics Aims and scope Submit manuscript

Abstract

The entire process of oxygen transport in microcirculation by developing a 3D porous media model is calculated numerically with coupled solid deformation-fluid seepage-convection and diffusion. The principal novelty of the model is that it takes into account volumetric deformation of both capillary and tissues resulting from capillary fluctuation. How solid deformation, fluid seepage, and convection-diffusion combine to affect oxygen transport is examined quantitatively: (1) Solid deformation is more significant in the middle of capillary, where the maximum value of volumetric deformation reaches about 0.5%. (2) Solid deformation has positive influence on the tissue fluid so that it flows more uniformly and causes oxygen to be transported more uniformly, and eventually impacts oxygen concentration by 0.1%–0.5%. (3) Convection-diffusion coupled deformation and seepage has a maximum (16%) and average (3%) increase in oxygen concentration, compared with pure molecular diffusion. Its more significant role is to allow oxygen to be transported more evenly.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Similar content being viewed by others

References

  1. Pittman, R. N. Oxygen gradients in the microcirculation. Acta Physiologica, 202(3), 311–322 (2011)

    Article  Google Scholar 

  2. Skalak, R. Mechanics of the Microcirculation, 1st ed., Prentice-Hall, New Jersey, 457–499 (1971)

    Google Scholar 

  3. Den Uil, C. A., Klijn, E., Lagrand, W. K., Brugts, J. J., Ince, C., Spronk, P. E., and Simoons, M. L. The microcirculation in health and critical disease. Progress in Cardiovascular Diseases, 51(2), 161–170 (2008)

    Article  Google Scholar 

  4. Abularrage, C. J., Sidawy, A. N., Aidinian, G., Singh, N., Weiswasser, J. M., and Arora, S. Evaluation of the microcirculation in vascular disease. Journal of Vascular Surgery, 42(5), 574–581 (2005)

    Article  Google Scholar 

  5. Michel, C. C. Handbook of Physiology: the Cardiovascular System, 1st ed., American Physiological Society, Bethesda, 375–410 (1984)

    Google Scholar 

  6. Landis, E. M. and Pappenheimer, J. R. Handbook of Physiology: Circulation, 1st ed., American Physiological Society, Washington D. C., 961–1034 (1963)

    Google Scholar 

  7. Krogh, A. The number and distribution of capillaries in muscles with calculations of the oxygen pressure head necessary for supplying the tissue. The Journal of Physiology, 52(6), 409–415 (1919)

    Article  Google Scholar 

  8. Krogh, A. The supply of oxygen to the tissues and the regulation of the capillary circulation. The Journal of Physiology, 52(6), 457–474 (1919)

    Article  Google Scholar 

  9. Middleman, S. Transport Phenomena in the Cardiovascular System, Wiley-InterScience, New York, 116–140 (1972)

    Google Scholar 

  10. Goldman, D. Theoretical models of microvascular oxygen transport to tissue. Microcirculation 15(8), 795–811 (2008)

    Article  Google Scholar 

  11. Popel, A. S. Theory of oxygen transport to tissue. Critical Reviews in Biomedical Engineering, 17(3), 257–321 (1989)

    Google Scholar 

  12. Pittman, R. N. Regulation of Tissue Oxygenation, 1st ed., Morgan and Claypool Life Sciences, San Rafael, 1–100 (2011)

    Google Scholar 

  13. Greene, A. S., Tonellato, P. J., Zhang, Z., Lombard, J. H., and Cowley, A. J. Effect of microvascular rarefaction on tissue oxygen delivery in hypertension. American Journal of Physiology, 262(5), 1486–1493 (1992)

    Google Scholar 

  14. McGuire, B. J. and Secomb, T. W. A theoretical model for oxygen transport in skeletal muscle under conditions of high oxygen demand. Journal of Applied Physiology, 91(5), 2255–2265 (2001)

    Google Scholar 

  15. Mikelic, A. and Primicerio, M. A diffusion-consumption problem for oxygen in a living tissue perfused by capillaries. Nonlinear Differential Equations and Applications, 13(3), 349–367 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  16. Yulianti, K. and Gunawan, A. Y. An asymptotic study for the steady model of oxygen diffusion in tissue region. ITB Journal of Science, 44(2), 164–178 (2012)

    MathSciNet  Google Scholar 

  17. Grinberg, O., Novozhilov, B., Grinberg, S., Friedman, B., and Swartz, H. M. Axial oxygen diffusion in the Korgh model: modifications to account for myocardial oxygen tension in isolated perfused rat hearts measured by EPR oximetry. Advances in Experimental Medicine and Biology, 566, 127–134 (2005)

    Article  Google Scholar 

  18. Titcombe, M. S., Titcombe, E. S., and Michael, M. J. An asymptotic study of oxygen transport from multiple capillaries to skeletal muscle tissue. Journal of Applied Mathematics, 60(5), 1767–1788 (2000)

    MATH  Google Scholar 

  19. Salathe, E. P. Mathematical analysis of oxygen concentration in a two dimensional array of capillaries. Journal of Mathematical Biology, 46(4), 287–308 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  20. Sharma, G. C. and Jain, M. A computational solution of mathematical model for oxygen transport in peripheral nerve. Computers in Biology and Medicine, 34(7), 633–645 (2004)

    Article  Google Scholar 

  21. Hellums, J. D., Nair, P. K., Huang, N. S., and Oshima, N. Simulation of intraluminal gas transport in the microcirculation. Annals of Biomedical Engineering, 24(1), 1–24 (1996)

    Article  Google Scholar 

  22. Moschandreou, T. E., Ellis, C. G., and Goldman, D. Influence of tissue metabolism and capillary oxygen supply on arteriolar oxygen transport: a computational model. Mathematical Biosciences, 232(1), 1–10 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  23. Pozrikidis, C. and Farrow, D. A. A model of fluid flow in solid tumors. Annals of Biomedical Engineering, 31(2), 181–194 (2003)

    Article  Google Scholar 

  24. Pozrikidis, C. Numerical simulation of blood and interstitial flow through a solid tumor. Journal of Mathematical Biology, 60(1), 75–94 (2010)

    Article  MathSciNet  Google Scholar 

  25. Chapman, S. J., Shipley, R. J., and Jawad, R. Multiscale modeling of fluid transport in tumors. Bulletin of Mathematical Biology, 70(8), 2334–2357 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  26. Wang, P. and Olbricht, W. L. Retro-convection enhanced drug delivery: a computational study. Annals of Biomedical Engineering, 38(8), 2512–2519 (2010)

    Article  Google Scholar 

  27. Schuff, M. M., Gore, J. P., and Nauman, E. A. A mixture theory model of fluid and solute transport in the microvasculature of normal and malignant tissues, I, theory. Journal of Mathematical Biology, 66(6), 1179–1207 (2013)

    Article  MATH  MathSciNet  Google Scholar 

  28. Schuff, M. M., Gore, J. P., and Nauman, E. A. A mixture theory model of fluid and solute transport in the microvasculature of normal and malignant tissues, II, factor sensitivity analysis, calibration, and validation. Journal of Mathematical Biology, 67(6–7), 1307–1337 (2013)

    Article  MATH  MathSciNet  Google Scholar 

  29. Pittman, R. N. Oxygen transport in the microcirculation and its regulation. Microcirculation, 20(2), 117–137 (2013)

    Article  Google Scholar 

  30. Popel, A. S., Goldman, D., and Vadapalli, A. Modeling of oxygen diffusion from the blood vessels to intracellular organelles. Advances in Experimental Medicine and Biology, 530, 485–495 (2003)

    Article  Google Scholar 

  31. Yao, H. and Gu, W. Y. Convection and diffusion in charged hydrated soft tissue: a mixture theory approach. Biomechanics and Modeling in Mechanobiology, 6(1–2), 63–72 (2007)

    Article  Google Scholar 

  32. Byrne, H. and Preziosi, L. Modelling solid tumour growth using the theory of mixtures. Mathematical Medicine and Biology, 20(4), 341–366 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  33. Zweifach, B. W. Quantitative studies of microcirculatory structure and function, II, direct measurement of capillary pressure in splanchnic mesenteric vessels. Circulation Research, 34(6), 858–866 (1974)

    Article  Google Scholar 

  34. Bear, J. Dynamics of Fluids in Porous Media, Elsevier, New York (1988)

    MATH  Google Scholar 

  35. Hall, J. E. Guyton and Hall Textbook of Medical Physiology, 12th ed., Elsevier Saunders, New York, 495–504 (2010)

    Google Scholar 

  36. Casciari, J. J., Sotirchos, S. V., and Sutherland, R. M. Variation in tumour cell growth rates and metabolism with oxygen-concentration, glucose-concentration and extracellar pH. Journal of Cellular Physiology, 151(2), 386–394 (1992)

    Article  Google Scholar 

  37. Anderson, A. R. A. A hybrid mathematical model of solid tumor invasion: the importance of cell adhesion. Mathematical Medicine and Biology, 22(2), 163–186 (2005)

    Article  MATH  Google Scholar 

  38. Caro, C. G., Pedley, T. J., Schroter, R. C., and Seed, W. A. The Mechanics of the Circulation, 2nd ed., Oxford University Press, Cambridge, 368–372 (2012)

    MATH  Google Scholar 

  39. He, Y. and Himeno, R. Finite element analysis on fluid filtration in system of permeable curved capillary and tissue. Journal of Mechanics in Medicine and Biology, 12(4), 1250077 (2012)

    Article  Google Scholar 

  40. Liu, G., Gabhann, F. M., and Popel, A. S. Effect of fiber type and size on the heterogeneity of oxygen distribution in exercising Skeletal muscle. PLoS ONE, 7(9), 44375 (2012)

    Article  Google Scholar 

  41. Cai, Y., Xu, S., Wu, J., and Long, Q. Coupled modeling of tumor growth and blood perfusion. Journal of Theoretical Biology, 279(1), 90–101 (2011)

    Article  Google Scholar 

  42. Zhu, Q., Zhang, A., Liu, P. X., and Xu, L. Study of tumor growth under hyperthermia condition. Computational and Mathematical Methods in Medicine, 2012, 198145 (2012)

    Article  Google Scholar 

  43. Macklin, P. and Lowengrub, J. Nonlinear simulation of the effect of microenvironment on tumor growth. Journal of Theoretical Biology, 245(4), 677–704 (2007)

    Article  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Graduate School of Systems Life Sciences, Kyushu University, Fukuoka, 819-0395, Japan

    N. Zhao

  2. Department of Informatics, Graduate School of Information Science and Electrical Engineering, Kyushu University, Fukuoka, 819-0395, Japan

    K. Iramina

Authors
  1. N. Zhao
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. K. Iramina
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to N. Zhao.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Zhao, N., Iramina, K. Numerical simulation of effect of convection-diffusion on oxygen transport in microcirculation. Appl. Math. Mech.-Engl. Ed. 36, 179–200 (2015). https://doi.org/10.1007/s10483-015-1908-7

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10483-015-1908-7

Key words

Chinese Library Classification

2010 Mathematics Subject Classification

Search

Navigation