Hostname: page-component-6b989bf9dc-md2j5 Total loading time: 0 Render date: 2024-04-14T03:15:04.941Z Has data issue: false hasContentIssue false

DISPERSAL OF HYDROGEN IN THE RETINA—A THREE-LAYER MODEL

Published online by Cambridge University Press:  10 May 2022

W. F. MANSOOR
Affiliation:
Department of Mathematics, College of Education for Pure Sciences-Ibn Al-Haitham, University of Baghdad, Baghdad, Iraq; e-mail: wafaa.mansoor2@murdoch.edu.au Mathematics & Statistics, Murdoch University, Perth, WA, Australia
G. C. HOCKING*
Affiliation:
Mathematics & Statistics, Murdoch University, Perth, WA, Australia
D. E. FARROW
Affiliation:
School of Physics, Mathematics and Computing, University of Western Australia, Perth, WA, Australia; e-mail: duncan.farrow@uwa.edu.au

Abstract

Two simple mathematical models of advection and diffusion of hydrogen within the retina are discussed. The work is motivated by the hydrogen clearance technique, which is used to estimate blood flow in the retina. The first model assumes that the retina consists of three, well-mixed layers with different thickness, and the second is a two-dimensional model consisting of three regions that represent the layers in the retina. Diffusion between the layers and leakage through the outer edges are considered. Solutions to the governing equations are obtained by employing Fourier series and finite difference methods for the two models, respectively. The effect of important parameters on the hydrogen concentration is examined and discussed. The results contribute to understanding the dispersal of hydrogen in the retina and in particular the effect of flow in the vascular retina. It is shown that the predominant features of the process are captured by the simpler model.

Type
Research Article
Copyright
© The Author(s), 2022. Published by Cambridge University Press on behalf of Australian Mathematical Publishing Association Inc.

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Alder, V. A., Yu, D. Y., Cringle, S. J. and Su, E. N., “Experimental approaches to diabetic retinopathy”, Asia-Pac. J. Ophthalmol. 4 (1992) 2025. https://research-repository.uwa.edu.au/en/publications/experimental-approaches-to-diabetic-retinopathy.Google Scholar
Campbell, M. and Humphries, P., “The blood-retina barrier: tight junctions and barrier modulation”, Adv. Exp. Med. Biol. 763 (2012) 7084. https://www.ncbi.nlm.nih.gov/pubmed/23397619.CrossRefGoogle ScholarPubMed
Cringle, S. J., Yu, D. Y., Alder, V. A. and Su, E. N., “Retinal blood flow by hydrogen clearance polarography in the streptozotocin-induced diabetic rat”, Invest. Ophthalmol. Vis. Sci. 34 (1993) 17161721.Google ScholarPubMed
Farrow, D. E., Hocking, G. C., Cringle, S. J. and Yu, D.-Y., “Modeling hydrogen clearance from the retina”, ANZIAM J. 59 (2018) 281292; doi:10.1017/S1446181117000426.Google Scholar
Goldman, D., “Theoretical models of microvascular oxygen transport to tissue”, Microcirculation 15 (2008) 795811; doi:10.1080/10739680801938289.CrossRefGoogle ScholarPubMed
Kety, S. S., “The theory and applications of the exchange of inert gas at the lungs and tissues”, Pharmacol. Rev. 3 (1951) 141. http://pharmrev.aspetjournals.org/content/3/1/1.Google Scholar
Leonard, B. P., “A stable and accurate convective modelling procedure based on quadratic upstream interpolation”, Comput. Methods Appl. Mech. Engrg. 19 (1979) 5998; doi:10.1016/0045-7825(79) 90034-3.CrossRefGoogle Scholar
Mansoor, W. F., Hocking, G. C. and Farrow, D. E., “Modelling of hydrogen diffusion in the retina”, ANZIAM J. 61 (2020) C119C136; doi:10.21914/anziamj.v61i0.14995.CrossRefGoogle Scholar
Margolis, R. and Spaide, R. F., “A pilot study of enhanced depth imaging optical coherence tomography of the choroid in normal eyes”, Amer. J. Ophthalmol. 147 (2009) 811815; doi:10.1016/j.ajo.2008.12.008.CrossRefGoogle ScholarPubMed
Merkle, C. W., Leahy, C. and Srinivasan, V. J., “Dynamic contrast optical coherence tomography images transit time and quantifies microvascular plasma volume and flow in the retina and choriocapillaris”, Biomed. Opt. Express 7 (2016) 42894312; doi:10.1364/BOE.7.004289.CrossRefGoogle ScholarPubMed
Mitchell, S. J., Morton, K. W. and Spence, A., “Analysis of box schemed for reactive flow problems”, SIAM J. Sci. Comput. 27 (2006) 12021223; doi:10.1137/030601910.CrossRefGoogle Scholar
Nickla, D. L. and Wallman, J., “The multifunctional choroid”, Prog. Retin. Eye Res. 29 (2010) 144168; doi:10.1016/j.preteyeres.2009.12.002.CrossRefGoogle ScholarPubMed
Sohn, E. H., Khanna, A., Tucker, B. A., Abràmoff, M. D., Stone, E. M. and Mullins, R. F., “Structural and biochemical analyses of choroidal thickness in human donor eyes”, Invest. Ophthalmol. Vis. Sci. 55 (2014) 13521360; doi:10.1167/iovs.13-13754.CrossRefGoogle ScholarPubMed
Strang, R., Horton, P. W. and Gillespie, F. C., “Theoretical basis of the measurement of choroidal blood flow using a radioactive inert gas clearance technique”, Phys. Med. Biol. 24 (1979) 964975; doi:10.1088/0031-9155/24/5/009.CrossRefGoogle Scholar
White, B. R., Pierce, M. C., Nassif, N., Cense, B., Hyle-Park, B., Tearney, G. J., Bouma, B. E., Chen, T. C. and de Boer, J. F., “In vivo dynamic human retinal blood flow imaging using ultra-high-speed spectral domain optical Doppler tomography”, Opt Express 11 (2003) 34903497; doi:10.1364/OE.11.003490.CrossRefGoogle Scholar
Winchell, G. A., “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’s Thesis, Northwestern University, Evanston, USA, 1983. https://books.google.com.au/books?id=CAKoGwAACAAJ.Google Scholar
Yu, D. Y., Alder, V. A. and Cringle, S. J., “Measurement of blood flow in rat eyes by hydrogen clearance”, Amer. J. Physiol. 261 (1991) H960H968; doi:10.1152/ajpheart.1991.261.3.H960.Google ScholarPubMed
Yu, D. Y., Cringle, S. J., Alder, V. A., Su, E. N. and Yu, P. K., “Intraretinal oxygen distribution and choroidal regulation in the avascular retina of guinea pigs”, Amer. J. Physiol. 270 (1996) H965H973; doi:10.1152/ajpheart.1996.270.3.H965.Google ScholarPubMed
Zouache, M. A., Eames, I. and Luthert, P. J., “Blood flow in the choriocapillaris”, J. Fluid Mech. 774 (2015) 3766; doi:10.1017/jfm.2015.243.CrossRefGoogle Scholar