A mathematical and numerical investigation of the hemodynamical origins of oscillations in microvascular networks.

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.

Mathematical modelling of the cell-depleted peripheral layer in the steady flow of blood in a tube.

In an earlier paper, Moyers-Gonzalez et al. [J. Fluid. Mech. 617 (2008), 327-354] used kinetic theory to derive a non-homogeneous haemorheological model and applied this to simulate the properties of steady flow of blood in a tube. By adjusting the tube haematocrit to match that of the experimental fitted curve of Pries et al. [Circ. Res. 67 (1990), 826-834] the authors showed that it was possible to quantitatively predict the apparent viscosity values presented in a later paper by Pries et al. [Am. J. Physiol. 263 (1992), 1770-1778]. In the present paper, it is the discharge haematocrit rather than the tube haematocrit that is prescribed. We further develop the predictive capacities of the original model of Moyers-Gonzalez et al. [J. Fluid. Mech. 617 (2008), 327-354] by introducing a cell-free peripheral layer next to the tube wall where, following the ideas of Sharan and Popel [Biorheology 38 (2001), 415-428], dissipation in this layer is accounted for by allowing the viscosity there to exceed that of plasma. Using both the apparent viscosity data of Pries et al. [Am. J. Physiol. 263 (1992), 1770-1778] and the relative tube haematocrit relation proposed by Sharan and Popel [Biorheology 38 (2001), 415-428], we predict the thickness of the cell-free layer and the relative viscosity in this layer. The predicted thickness of the cell-free layer as a function of both a pseudo-shear rate and the tube diameter for 45% haematocrit blood is shown to be in very close conformity with the experimental measurements of Reinke et al. [Am. J. Physiol. 253 (1987), 540-547]. With increasing discharge haematocrit the cell-free layer thickness is shown to decrease, as observed in several experimental papers [Bugliarello and Hayden, Trans. Soc. Rheol. VII (1963), 209-230, Bugliarello and Sevilla, Biorheology 7 (1970), 85-107, Soutani et al., Am. J. Physiol. 268 (1995), 1959-1965]. Our prediction of the relative viscosity in the cell-free layer shows a similar trend to that computed by Sharan and Popel [Biorheology 38 (2001), 415-428]. Finally, for sufficiently large pseudo-shear rates it is shown that the Deborah number (a non-dimensional relaxation time) may be taken to be a constant, thus greatly simplifying our haemorheological model and allowing for a partially analytic solution to the problem of steady non-homogeneous flow of blood in a tube.

Numerical simulations of pulsatile blood flow using a new constitutive model.

In the present paper we use a new constitutive equation for whole human blood [R.G. Owens, A new microstructure-based constitutive model for human blood, J. Non-Newtonian Fluid Mech. (2006), to appear] to investigate the steady, oscillatory and pulsatile flow of blood in a straight, rigid walled tube at modest Womersley numbers. Comparisons are made with the experimental results of Thurston [Elastic effects in pulsatile blood flow, Microvasc. Res. 9 (1975), 145-157] for the pressure drop per unit length against volume flow rate and oscillatory flow rate amplitude. Agreement in all cases is very good. In the presentation of the numerical and experimental results we discuss the microstructural changes in the blood that account for its rheological behaviour in this simple class of flows. In this context, the concept of an apparent complex viscosity proves to be useful.