Modelling of hydrogen diffusion in the retina


  • Wafaa Mansoor Murdoch University
  • Graeme Hocking
  • Duncan Farrow



A simple mathematical model for diffusion of hydrogen within the retina has been developed. The model consists of three, well-mixed, one dimensional layers that exchange hydrogen via a diffusion process. A Fourier series method is applied to compute the hydrogen concentration. The effect of important parameters is examined and discussed. The results may contribute to an understanding of the hydrogen clearance technique to estimate blood flow. A two dimensional numerical method for the hydrogen diffusion is also presented. It is shown that the predominant features of the process are captured quite well by the simpler model.


  • V. A. Alder, D. Y. Yu, S. J. Cringle and E. N. Su. Experimental approaches to diabetic retinopathy. Asia-Pac. J. Ophthalmol. 4:20–25, 1992.
  • J. C. Arciero, P. Causin and F. Malgoroli. Mathematical methods for modeling the microcirculation. AIMS Biophys. 4:362–399, 2017. doi:10.3934/biophy.2017.3.362
  • D. E. Farrow, G. C. Hocking, S. J. Cringle and D.-Y. Yu. Modeling Hydrogen clearance from the retina. ANZIAM J. 59:281–292, 2018. doi:10.1017/S1446181117000426
  • A. B. Friedland. A mathematical model of transmural transport of oxygen to the retina. Bull. Math. Biol. 40:823–837, 2018; doi:10.1007/BF02460609
  • D. Goldman. Theoretical models of microvascular oxygen transport to tissue. Microcirculation 15:795–811, 2008. doi:10.1080/10739680801938289
  • A. C. Hindmarsh. ODEPACK, A Systematized Collection of ODE Solvers. In Scientific Computing, R. S. Stepleman, et al., Eds., pp. 55-64. North-Holland, Amsterdam, 1983.
  • S. S. Kety. The theory and applications of the exchange of inert gas at the lungs and tissues. Pharmacol. Rev. 3:1–41, 1951.
  • B. P. Leonard. A stable and accurate convective modelling procedure based on quadratic upstream interpolation. Comput. Methods Appl. Mech. Eng. 19:59–98, 1979. doi:10.1016/0045-7825(79) 90034-3
  • S. L. Mitchell. Coupling transport and chemistry: numerics, analysis and applications. PhD thesis, University of Bath, UK, 2003.
  • G. A. Winchell. Mathematical model of inert gas washout from the retina: evaluation of hydrogen washout as a means of determining retinal blood flow in the cat. Master\textquoteright s Thesis, Northwestern University, Evanston, USA, 1983.
  • D. Y. Yu, V. A. Alder and S. J. Cringle. Measurement of blood flow in rat eyes by hydrogen clearance. Am. J. Physiol. (Heart Circ. Physiol.) 261:H960–H968, 1991. doi:10.1152/ajpheart.1991.261.3.H960
  • D. Y. Yu, S. J. Cringle, V. A. Alder, E. N. Su, and P. K. Yu, Intraretinal oxygen distribution and choroidal regulation in the avascular retina of guinea pigs. Am. J. Physiol. (Heart Circ. Physiol.) 270:H965-H973, 1996. doi:10.1152/ajpheart.1996.270.3.H965
  • S. Cringle, D.-Y. Yu, V. Alder, E.-N. Su, and P. Yu. Choroidal regulation of oxygen supply to the guinea pig retina. In A. G. Hudetz, and D. F. Bruley (Eds.), Oxygen Transport to Tissue XX, pp. 385–389. Springer, 1998. doi:10.1007/978-1-4615-4863-8





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