Page 107 - Biomedical Engineering and Design Handbook Volume 1, Fundamentals

84 BIOMECHANICS OF THE HUMAN BODY

The governing equation for the tube deformation may be given by the tube law, which is also an
equation of state that relates the transmural pressure to the local cross-sectional area,
⎛ A ⎞
PP = K p • F ⎜ ⎝ A ⎠ ⎟ (3.41)
−
e
0
where A is the unstressed circular cross section and K is the wall stiffness coefficient. Solution of
0 p
these governing equations for given boundary conditions provides the one-dimensional flow pattern
of the coupled fluid-structure problem of fluid flow through a collapsible elastic tube.
Shapiro (1977) defined the speed index, S = u/c, similar to the Mach number in gas dynamics,
and demonstrated different cases of subcritical (S < 1) and supercritical (S > 1) flows. It has been
shown experimentally in simple experiments with compliant tubes that gradual reduction of the
downstream pressure progressively increases the flow rate until a maximal value is reached (Holt,
1969; Conrad, 1969). The one-dimensional theory demonstrates that for a given tube (specific geo-
metry and wall properties) and boundary conditions, the maximal steady flow that can be conveyed
in a collapsible tube is attained for S = 1 (e.g., when u = c) at some position along the tube (Dawson
and Elliott, 1977; Shapiro, 1977; Elad et al., 1989). In this case, the flow is said to be “choked” and
further reduction in downstream pressure does not affect the flow upstream of the flow-limiting site.
Much of its complexity, however, is still unresolved either experimentally or theoretically (Kamm
et al., 1982; Kamm and Pedley, 1989; Elad et al., 1992).

3.5 BLOOD FLOW IN THE MICROCIRCULATION

The concept of a closed circuit for the circulation was established by Harvey (1578–1657). The
experiments of Hagen (1839) and Poiseuille (1840) were performed in an attempt to elucidate the
flow resistance of the human microcirculation. During the past century, major strides have been
made in understanding the detailed fluid mechanics of the microcirculation and in depicting a con-
crete picture of the flow in capillaries and other small vessels.

3.5.1 The Microvascular Bed
We include in the term “microcirculation” those vessels with lumens (internal diameters) that are
some modest multiple—say 1 to 10—of the major diameter of the unstressed RBC. This definition
includes primarily the arterioles, the capillaries, and the postcapillary venules. The capillaries are of
particular interest because they are generally from 6 to 10 mm in diameter, i.e., about the same size
as the RBC. In the larger vessels, RBC may tumble and interact with one another and move from
streamline to streamline as they course down the vessel. In contrast, in the microcirculation the RBC
must travel in single file through true capillaries (Berman and Fuhro, 1969; Berman et al., 1982).
Clearly, any attempt to adequately describe the behavior of capillary flow must recognize the par-
ticulate nature of the blood.

3.5.2 Capillary Blood Flow
The tortuosity and intermittency of capillary flow argue strongly that the case for an analytic descrip-
tion is lost from the outset. To disprove this, we must return to the Navier-Stokes equations for a
moment and compare the various acceleration and force terms, which apply in the microcirculation.
The momentum equation, which is Newton’s second law for a fluid, can be written as
(A) (B) (C) (D)
∂u
ρ + ρ u( • ∇ u = −∇ + F shear (3.42)
P
)
∂t