Biomedical Engineering Reference
In-Depth Information
form is out of sought and numerical simulations are necessary. We should note that
numerical simulations are useful, not only from a mathematical point of view but
also from a biological perspective. In fact, a specific perturbation in the hemostatic
system, which usually cannot be assessed through laboratory tests, can easily be in-
troduced in a phenomenologic mathematical model and predictions on the effects of
that specific perturbation can be evaluated through numerical simulations and their
corresponding results.
In addition to the biochemical reactions, the rheology of blood also plays a crucial
role in the regulation of hemostasis. While plasma can be considered as a Newtonian
fluid, whole blood exhibits non-Newtonian characteristics, particularly at low shear
rates. These are mainly due to erythrocytes ability to aggregate and to form a three-
dimensional (3D) microstructure (
rouleaux
) at low shear rates, their deformability
and their tendency to align with the flow field at high shear rates [80]. The formation
and breakup of this 3D microstructure, as well as the elongation and recovery of ery-
throcytes, contribute to the non-Newtonian behaviour of blood, like shear-thinning
viscosity, thixotropy, viscoelasticity and possibly a yield stress. In particular, the
shear-thinning viscosity of blood is manifested by the fact that at rest or at low shear
rates (below 1
s
−
1
) blood seems to have a high apparent viscosity, while at high
shear rates there is a reduction in its viscosity. Generalized Newtonian models like
the power-law, Cross, Carreau, Carreau-Yasuda, or modifications of these models
have been obtained by fitting experimental data in one dimensional flows and are
used to capture the shear dependence of blood viscosity [75]. Moreover, blood cells
are essentially elastic membranes filled with a fluid and it seems reasonable, at least
under certain flow conditions, to expect blood to behave like a viscoelastic fluid, due
to its ability to store and release elastic energy from its branched 3D microstructures.
Thurston [89] was among the earliest to recognize the viscoelastic nature of blood
at low shear rates. Some viscoelastic constitutive models for describing blood rhe-
ology have been proposed in the recent literature, in particular the empirical three
constant generalized Oldroyd-B model (Yeleswarapu model) studied in [97] and the
model developed by Anand and Rajagopal [1] which includes relaxation times de-
pending on the shear rate and gives good agreement with experimental data in steady
Poiseuille and oscillatory flows. We also refer to the microstructure based model de-
veloped in [20, 61].
In most part of the arterial system of healthy individuals blood can be modelled
as a Newtonian fluid [26]. However, there are flow regimes and clinical situations
where non-Newtonian effects of blood can probably be observed. These include,
for normal blood, regions of stable recirculation like in the venous system and parts
of the arterial vasculature where geometry has been altered and RBC aggregates be-
come more stable, like downstream a stenosis, inside a saccular aneurysm or in some
cerebral anastomoses. In addition, several pathologies are accompanied by signifi-
cant changes in the mechanical properties of blood and this results in alterations in
blood viscosity and viscoelastic properties. For a detailed discussion of the rheolog-
ical properties of blood and corresponding continuum mathematical models, see for
instance [74, 75] and references therein.
