CHAPTER 5 • VASCULAR FUNCTION
101
Resistance is directly proportional to vessel
length. Therefore, a vessel that is twice as
long as another vessel with the same radius
will have twice the resistance to flow. In
the body, individual vessel lengths do not
change appreciably; therefore, changes in
vessel length have only a minimal effect on
resistance.
Resistance to flow is directly related to the
viscosity of the blood. Viscosity is related to
friction generated by interactions between
fluid molecules in the plasma and suspended
formed substances (e.g., red blood cells) as
the blood is flowing. Viscosity also takes into
account the friction generated between the
blood and the lining of the vessel. Therefore,
viscosity can be thought of as a force that
opposes blood flow. If viscosity increases two-
fold, the resistance to flow increases twofold,
thereby decreasing flow by one-half at con-
stant AP. At normal body temperatures, the
viscosity of plasma is about 1.8 times the vis-
cosity of water. The viscosity of whole blood
is about three to four times the viscosity of
water owing to the presence of red cells and
proteins. Blood viscosity normally does not
change much; however, it can be significantly
altered by changes in hematocrit and tem-
perature and by low flow states. Hematocrit
is the volume of red blood cells expressed
as a percentage of a given volume of whole
blood. If hematocrit increases from a normal
value of 40% to an elevated value of 60% (this
is termed polycythemia), the blood viscos-
ity approximately doubles. Decreasing blood
temperature
increases
viscosity
by
about
2% per degree centigrade. The flow rate of
blood also affects viscosity. At very low flow
states
in
the
microcirculation— as
occurs
during
circulatory
shock—the
blood
vis-
cosity can increase severalfold. This occurs
because at low flow states, cell-to-cell and
protein-to-cell adhesive interactions increase,
which can cause erythrocytes to adhere to one
another and increase the blood viscosity.
Of the
three
independent variables
in
Equation
5-6,
vessel
radius
is
the
most
important
quantitatively
for
determining
resistance to flow. Because radius and resist-
ance are inversely related, an increase in
radius reduces resistance.
Furthermore,
a
change in radius alters resistance inversely to
the fourth power of the radius.
For example, a
twofold increase in radius decreases resistance
16-fold! Therefore, vessel resistance is exqui-
sitely sensitive to changes in radius. Because
changes in radius and diameter are directly
proportional, diameter can be substituted for
radius in Equation 5-6.
If the expression for resistance (Equation
5-6) is combined with the equation describ-
ing the relationship between flow, pressure,
and resistance (F = AP/R; Equation 5-5), the
following relationship is obtained:
Eq. 5-7
A
P
r4
n
L
This
relationship
(Poiseuille’s
equation)
was first described by the French physician
Poiseuille (1846). The full equation contains n
in the numerator, and the number 8 in the
denominator
(a
constant
of
integration).
Equation 5-7 describes how flow is related to
perfusion pressure, radius, length, and viscos-
ity. In the body, however, flow does not con-
form precisely to this relationship because the
equation assumes the following: (1) the vessels
are long, straight, rigid tubes; (2) the blood
behaves as a Newtonian fluid in which viscos-
ity is constant and independent of flow; and
(3) the blood is flowing under steady laminar
flow (nonturbulent) conditions. Despite these
assumptions, which clearly are not always
achieved in vivo, the relationship is important
because it describes the dominant influence
of vessel radius on resistance and flow, and
therefore provides a conceptual framework to
understand how physiologic and pathologic
changes in blood vessels and blood viscosity
affect pressure and flow.
The relationship between flow and radius
(Equation 5-7) for a single vessel is shown
graphically in Figure 5.7. In this analysis,
laminar flow conditions are assumed, and
driving pressure, viscosity, and vessel length
are held constant. As vessel radius decreases
from a relative value of 1.0, a dramatic fall in
flow occurs because flow is directly related
to radius to the fourth power. For example,
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