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A Mathematical and Numerical Investigation of the Hemodynamical Origins of Oscillations in Microvascular Networks

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Abstract

Evidence is presented to show that self-sustained oscillations of purely hemodynamical origin are possible in some arcade-type microvascular networks supplied with steady boundary conditions, but that in others the oscillations disappear with sufficient reduction of the time step Δt, showing them to be numerical artefacts. In an attempt to elucidate the mechanisms involved in the onset of fluctuations, we proceed to perform a linear stability analysis for the convective model of Kiani et al. (Microvasc. Res. 45:219–232, 1993; Am. J. Physiol. 266(35):H1822–H1828, 1994), and show that this leads via a system of delay differential equations to a nonlinear eigenvalue problem. This result generalises the characteristic equation obtained by Carr et al. (Ann. Biomed. Eng. 33:764–771, 2005) and Geddes et al. (SIAM J. Appl. Dyn. Syst. 6(4):694–727, 2007) who solved a special case in a two node network. An implicit numerical method is proposed for the computation of blood flows in networks using the convective model.

In a moderate size subnetwork of one of the networks chosen by Kiani et al. (Am. J. Physiol. 266(35):H1822–H1828, 1994), the topology, vessel lengths, and diameters of which were based on microvascular networks in the rat mesentery, we compare results generated using the original explicit numerical method of Kiani et al. (Am. J. Physiol. 266(35):H1822–H1828, 1994) with those from our implicit scheme. From the linear stability theory, a critical value D RBC,crit of a red blood cell diameter parameter D RBC in the plasma skimming model of Fenton et al. (Pflügers Arch. 403:396–401, 1985b) is identified for the onset of oscillations about steady state and both the explicit and implicit methods are used to calculate the inflow hematocrit solutions in all vessels of the subnetwork at the critical parameter value, subject to perturbed initial conditions. The results of the implicit method are demonstrated to be in excellent and superior agreement with the predictions of the linear analysis in this case. For values of D RBC slightly larger than D RBC,crit the bifurcating periodic solutions calculated using either the explicit or implicit schemes are characteristic of those of a supercritical Hopf bifurcation and the graphs of D RBC vs. oscillation amplitude would seem to converge as Δt→0.

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Acknowledgements

Sylvie Lorthois is thanked for having suggested a study of this problem to the second author a couple of years ago and for proposing Voronoi diagrams as a way of generating large complex arcade-type network topologies. The authors wish to thank Russell T. Carr, Mohammad Kiani, and Axel Pries for supplying them with helpful technical information about the computations performed in some of their articles. This work was partially supported by a Discovery Grant from the Natural Sciences and Engineering Research Council of Canada.

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  1. Département de mathématiques et de statistique, Université de Montréal, C.P. 6128, succ. centre-ville, Montréal, QC H3C 3J7, Canada

    Yasmine Tawfik & Robert G. Owens

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  1. Yasmine Tawfik
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Correspondence to Robert G. Owens.

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Tawfik, Y., Owens, R.G. A Mathematical and Numerical Investigation of the Hemodynamical Origins of Oscillations in Microvascular Networks. Bull Math Biol 75, 676–707 (2013). https://doi.org/10.1007/s11538-013-9825-6

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