Abstract
The ability to characterise functional capillary supply (FCS) plays a key role in developing effective therapeutic interventions for numerous pathological conditions, such as chronic ischaemia in skeletal or cardiac muscle. Detailed tissue geometry, such as muscle fibre size, has been incorporated into indices of FCS by considering the distribution of Voronoi tessellations (‘capillary domains’) generated from vessel locations in a plane perpendicular to muscle fibre orientation, implicitly assuming that each Voronoi polygon represents the area of supply of its enclosed capillary. However, to assess the capacity of FCS in muscle, we are naturally led to use a modelling framework that can account for the local anatomic and metabolic heterogeneities of muscle fibres. Such a framework can be used to explore the validity of the Voronoi polygon representation of FCS regions while also providing a general platform for robust predictions of FCS.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
Krogh, A.: The number and distribution of capillaries in muscles with calculations of the oxygen pressure head necessary for supplying the tissue. J. Physiol. 52, 391–408, 409–415, 457–474 (1919)
Egginton, S., Ross, H.F.: Planar analysis of tissue capillary supply. In: Oxygen Transport in Biological Systems. Society for Experimental Biology Seminar Series, vol. 51, pp. 165–195. Cambridge University Press, Cambridge (1992)
World Health Organization: The top 10 causes of death. http://www.who.int/mediacentre/factsheets/fs310/en/. Accessed 22 Dec 2013
Egginton, S.: Morphometric analysis of tissue capillary supply. In: Boutilier, R.G. (ed.) Vertebrate Gas Exchange from Environment to Cell. Advances in Comparative and Environmental Physiology, vol. 6, pp. 73–141. Springer, Berlin (1990)
Egginton, S., Gaffney, E.A.: Tissue capillary supply—it’s quality not quantity that counts! Exp. Physiol. 95(10), 971–979 (2010)
Degens, H., Deveci, D., Botto-Van Bemden, A., Hoofd, L.J.C, Egginton, S.: Maintenance of heterogeneity of capillary spacing is essential for adequate oxygenation in the soleus muscle of the growing rat. Microcirculation 13, 467–476 (2006)
Wüst, R.C.I., Gibbings, S.L., Degens, H.: Fiber capillary supply related to fiber size and oxidative capacity in human and rat skeletal muscle. In: Liss, P., Hansell, P., Bruley, D.F., Harrison, D.K. (eds.) Oxygen Transport to Tissue XXX. Advances in Experimental Medicine and Biology, vol. 645, pp. 75–80. Springer, New York (2009)
Kreuzer, F.: Oxygen supply to tissues: the krogh model and its assumptions. Experientia 38, 1415–1426 (1982)
Gonzalez-Fernandez, J.M., Atta, S.E.: Concentration of oxygen around capillaries in polygonal regions of supply. Math. Biosci. 13, 55–69 (1972)
Hoofd, L., Turek, Z., Kubat, K., Ringnalda, B.E.M., Kazda, S.: Variability of intercapillary distance estimated on histological sections of rat heart. Adv. Exp. Med. Biol. 191, 239–247 (1985)
Al-Shammari, A.A., Gaffney, E.A., Egginton, S.: Modelling capillary oxygen supply capacity in mixed muscles: Capillary domains revisited. J. Theor. Biol. 356, 47–61 (2014), DOI: 10.1016/j.jtbi.2014.04.016
Hoofd, L., Turek, Z., Olders, J.: Calculation of oxygen pressures and fluxes in a flat plane perpendicular to any capillary distribution. In: Rakusan, K., Biro, G., Goldstick, T.K., Turek, Z. (eds.) Oxygen Transport to Tissue XI, pp. 187–196. Plenum Press, New York (1989)
Wang, C.Y., Bassingthwaitghte, J.B.: Capillary supply regions. Math. Biosci. 173, 103–114 (2001)
Al-Shammari, A.A., Gaffney, E.A., Egginton, S.: Modelling capillary oxygen supply capacity in mixed muscles. Capillary domains revisited. J. Theor. Boil. 356, 47–61 (2014). doi:10.1016/j.jtbi.2014.o4.
The Mathworks, Inc.: Partial differential equations toolbox: user’s guide (R2013b). http://www.mathworks.co.uk/help/pdf_doc/pde/pde.pdf (2013). Accessed 2 Nov 2013
Acknowledgements
AA is supported by a studentship from Kuwait University, Kuwait, and by travel grants from the Mathematical Institute and Linacre College, University of Oxford, UK.
Author information
Authors and Affiliations
Corresponding author
Editor information
Rights and permissions
Copyright information
© 2014 Springer International Publishing Switzerland
About this paper
Cite this paper
Al-Shammari, A.A., Gaffney, E.A., Egginton, S. (2014). Modelling Oxygen Capillary Supply to Striated Muscle Tissues. In: Ansari, A. (eds) Advances in Applied Mathematics. Springer Proceedings in Mathematics & Statistics, vol 87. Springer, Cham. https://doi.org/10.1007/978-3-319-06923-4_2
Download citation
DOI: https://doi.org/10.1007/978-3-319-06923-4_2
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-06922-7
Online ISBN: 978-3-319-06923-4
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)