6.11 Case Study: Functional Adaptation in Arteries
315
stress zero. Thus, σ zp < 1 and σ za > 1, while σ zP = σ zA = 0, emphasizing
the difference between constituent stresses (lowercase p and a) and muscle stresses
(uppercase P and A).
6.11 Case Study: Functional Adaptation in Arteries
Arteries serve dual functions. In addition to transporting blood, they play an
important role in regulating flow during adaptation to functional demands. Blood
flow is regulated primarily by arterioles, which are small-caliber arteries composed
largely of smooth muscle, just upstream of the capillaries.
To understand functional adaptation in arteries, it is helpful to think in terms of
Laplace’s law
σ =
P a
h
(6.116)
for the average circumferential stress in a tube, as well as a couple of formulas from
fluid dynamics. As a first approximation, the flow of blood through an artery can be
treated as steady flow of viscous fluid through a rigid tube. For a tube of length L
and inner radius a, the classical Poiseuille formula gives the flow resistance (Fung
1997)
R =
P
Q
=
8μL
πa 4 ,
(6.117)
where P = P in − P out is the pressure drop from inlet to outlet, and μ is the fluid
viscosity. In addition, the fluid shear stress (drag) on the wall is
τ =
4μQ
πa 3 ,
(6.118)
where Q is the volumetric flow rate. These last two relations show that flow
resistance and shear stress decrease nonlinearly with increasing radius. In addition,
it is important to realize that flow is driven by a difference in pressure (pressure
gradient) between the ends of the tube. If the pressure is the same at both ends, there
is no flow. Likewise, an equal increase (or decrease) in pressure at the ends changes
the average pressure in the tube, but does not change P or the flow rate.
Arteries respond to changes in both blood pressure and flow. For both perturbations, the response consists of an acute phase (hours to days) and a chronic
phase (several days to weeks). The smooth muscle in an artery is oriented circumferentially on average and maintains a basal tone, i.e., partial contraction, under
homeostatic conditions. During the acute phase the muscle contracts or relaxes to
alter vessel radius. During the chronic phase, growth and remodeling make these
geometric changes “permanent.”
315
stress zero. Thus, σ zp < 1 and σ za > 1, while σ zP = σ zA = 0, emphasizing
the difference between constituent stresses (lowercase p and a) and muscle stresses
(uppercase P and A).
6.11 Case Study: Functional Adaptation in Arteries
Arteries serve dual functions. In addition to transporting blood, they play an
important role in regulating flow during adaptation to functional demands. Blood
flow is regulated primarily by arterioles, which are small-caliber arteries composed
largely of smooth muscle, just upstream of the capillaries.
To understand functional adaptation in arteries, it is helpful to think in terms of
Laplace’s law
σ =
P a
h
(6.116)
for the average circumferential stress in a tube, as well as a couple of formulas from
fluid dynamics. As a first approximation, the flow of blood through an artery can be
treated as steady flow of viscous fluid through a rigid tube. For a tube of length L
and inner radius a, the classical Poiseuille formula gives the flow resistance (Fung
1997)
R =
P
Q
=
8μL
πa 4 ,
(6.117)
where P = P in − P out is the pressure drop from inlet to outlet, and μ is the fluid
viscosity. In addition, the fluid shear stress (drag) on the wall is
τ =
4μQ
πa 3 ,
(6.118)
where Q is the volumetric flow rate. These last two relations show that flow
resistance and shear stress decrease nonlinearly with increasing radius. In addition,
it is important to realize that flow is driven by a difference in pressure (pressure
gradient) between the ends of the tube. If the pressure is the same at both ends, there
is no flow. Likewise, an equal increase (or decrease) in pressure at the ends changes
the average pressure in the tube, but does not change P or the flow rate.
Arteries respond to changes in both blood pressure and flow. For both perturbations, the response consists of an acute phase (hours to days) and a chronic
phase (several days to weeks). The smooth muscle in an artery is oriented circumferentially on average and maintains a basal tone, i.e., partial contraction, under
homeostatic conditions. During the acute phase the muscle contracts or relaxes to
alter vessel radius. During the chronic phase, growth and remodeling make these
geometric changes “permanent.”