Encyclopedia of Systems Biology

2013 Edition
| Editors: Werner Dubitzky, Olaf Wolkenhauer, Kwang-Hyun Cho, Hiroki Yokota

Parabolic Differential Equations, Diffusion Equation

  • Mahendra KavdiaEmail author
Reference work entry
DOI: https://doi.org/10.1007/978-1-4419-9863-7_273



The diffusion equation describes the spatial and the temporal relationship of concentration (C) over a region and the most general form is
$$ \frac{{\partial C}}{{\partial \text t}} = \rm D{\nabla^2}C $$
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  1. Byrne H, Preziosi L (2003) Modelling solid tumour growth using the theory of mixtures. Math Med Biol 20(4):341–366PubMedGoogle Scholar
  2. Deen WM (1998) Analysis of transport phenomena. Topics in chemical engineering. Oxford University Press, New York, p xix, 597Google Scholar
  3. Deonikar P, Kavdia M (2010) A computational model for nitric oxide, nitrite and nitrate biotransport in the microcirculation: effect of reduced nitric oxide consumption by red blood cells and blood velocity. Microvasc Res 80(3):464–476PubMedGoogle Scholar
  4. Eikenberry SE et al (2009) Virtual glioblastoma: growth, migration and treatment in a three-dimensional mathematical model. Cell Prolif 42(4):511–528PubMedGoogle Scholar
  5. Farlow SJ (1993) Partial differential equations for scientists and engineers, Dover books on advanced mathematics. Dover, New York, p ix, 414Google Scholar
  6. Ferreira SC Jr, Martins ML, Vilela MJ (2002) Reaction–diffusion model for the growth of avascular tumor. Phys Rev E Stat Nonlin Soft Matter Phys 65(2 Pt 1):021907PubMedGoogle Scholar
  7. Fournier RL (2007) Basic transport phenomena in biomedical engineering, 2nd edn. Taylor & Francis, New York, p xxii, 450Google Scholar
  8. Goldman D, Popel AS (1999) Computational modeling of oxygen transport from complex capillary networks. Relation to the microcirculation physiome. Adv Exp Med Biol 471:555–563PubMedGoogle Scholar
  9. Ji JW et al (2006) A computational model of oxygen transport in skeletal muscle for sprouting and splitting modes of angiogenesis. J Theor Biol 241(1):94–108PubMedGoogle Scholar
  10. Krogh A (1919) The number and distribution of capillaries in muscles with calculations of the oxygen pressure head necessary for supplying the tissue. J Physiol 52(6):409–415PubMedGoogle Scholar
  11. McGuire BJ, Secomb TW (2001) A theoretical model for oxygen transport in skeletal muscle under conditions of high oxygen demand. J Appl Physiol 91(5):2255–2265PubMedGoogle Scholar
  12. Popel AS (1989) Theory of oxygen transport to tissue. Crit Rev Biomed Eng 17(3):257–321PubMedGoogle Scholar
  13. Truskey GA, Yuan F, Katz DF (2009) Transport phenomena in biological systems, 2nd edn, Pearson Prentice Hall bioengineering. Pearson Prentice Hall, Upper Saddle River, p xxiii, 860Google Scholar
  14. Tsoukias NM (2008) Nitric oxide bioavailability in the microcirculation: insights from mathematical models. Microcirculation 15(8):813–834PubMedGoogle Scholar

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© Springer Science+Business Media, LLC 2013

Authors and Affiliations

  1. 1.Department of Biomedical EngineeringWayne State UniversityDetroitUSA