318
6 Growth
While functional adaptation after birth is certainly important, the most dramatic
growth occurs during embryonic development, as both the heart and arteries must
adapt to the increasing hemodynamic demands of the rapidly growing embryo.
In the embryo, blood vessels begin as small tubes composed of a single layer of
endothelial cells. Interestingly, a primitive vascular bed forms even before blood
flow begins (Ribatti et al. 2015). After the heart starts to pump blood, the blood
vessels grow and the vascular bed expands. Later, progenitor smooth muscle cells
are recruited into the vessel wall, where they produce elastin and collagen. As an
artery matures, these constituents organize to create the media and adventitia.
In experiments on chick embryos more than a century ago, Thoma (1893) found
that hemodynamic loads play a major role in regulating the growth of blood vessels
during development, and numerous studies have shown that mature arteries exhibit
similar behavior (Humphrey 2002). Thus, we assume that the growth laws (6.119)
for arteries hold throughout the lifetime of an organism.
Contraction Law Under homeostatic conditions, we assume that the timeaveraged fluid shear stress τ is at its homeostatic value τ 0 , and the smooth muscle
in the media maintains a normal tone with contraction ratio K θ = K 0 . If the
fluid shear stress τ increases, the artery relaxes ( ˙
K > 0) to increase radius; if τ
decreases, the artery contracts ( ˙
K < 0). Contraction maintains τ = τ 0 until growth
has a chance to fix the new radius during the chronic phase of functional adaptation,
allowing normal tone to be gradually reestablished. These considerations lead to a
contraction law of the form
˙
K θ = b τ (τ − τ 0 )K θ − b K (K θ − K 0 ),
(6.120)
where b τ and b K are positive constants. The second term reestablishes normal
tone (K θ = K 0 ), while circumferential growth keeps the fluid shear stress at its
homeostatic value (τ = τ 0 ). Note that K θ decreases ( ˙
K < 0) if K θ > K 0 , and vice
versa. In addition, because the wall is fully relaxed when K θ = 1, we stipulate that
K θ must not exceed unity.
6.11.2 Model for an Artery
We now consider a model for growth and adaptation of an artery in response
to changes in loading conditions during development and maturity. The artery is
treated as a tube consisting of two incompressible, pseudoelastic layers representing
the contractile media (inner layer) and passive adventitia (outer layer). Developing
arteries stretch and grow longer with increasing blood pressure. Hence, rather than
specifying the axial strain, the ends of the tube are assumed to be closed but free
to move in response to the axial force caused by pressure on the end caps. Time
t = 0 corresponds to the time at which significant flow begins in the embryo, and
any initial stresses are neglected. The initial geometry is depicted in Fig. 6.24.
6 Growth
While functional adaptation after birth is certainly important, the most dramatic
growth occurs during embryonic development, as both the heart and arteries must
adapt to the increasing hemodynamic demands of the rapidly growing embryo.
In the embryo, blood vessels begin as small tubes composed of a single layer of
endothelial cells. Interestingly, a primitive vascular bed forms even before blood
flow begins (Ribatti et al. 2015). After the heart starts to pump blood, the blood
vessels grow and the vascular bed expands. Later, progenitor smooth muscle cells
are recruited into the vessel wall, where they produce elastin and collagen. As an
artery matures, these constituents organize to create the media and adventitia.
In experiments on chick embryos more than a century ago, Thoma (1893) found
that hemodynamic loads play a major role in regulating the growth of blood vessels
during development, and numerous studies have shown that mature arteries exhibit
similar behavior (Humphrey 2002). Thus, we assume that the growth laws (6.119)
for arteries hold throughout the lifetime of an organism.
Contraction Law Under homeostatic conditions, we assume that the timeaveraged fluid shear stress τ is at its homeostatic value τ 0 , and the smooth muscle
in the media maintains a normal tone with contraction ratio K θ = K 0 . If the
fluid shear stress τ increases, the artery relaxes ( ˙
K > 0) to increase radius; if τ
decreases, the artery contracts ( ˙
K < 0). Contraction maintains τ = τ 0 until growth
has a chance to fix the new radius during the chronic phase of functional adaptation,
allowing normal tone to be gradually reestablished. These considerations lead to a
contraction law of the form
˙
K θ = b τ (τ − τ 0 )K θ − b K (K θ − K 0 ),
(6.120)
where b τ and b K are positive constants. The second term reestablishes normal
tone (K θ = K 0 ), while circumferential growth keeps the fluid shear stress at its
homeostatic value (τ = τ 0 ). Note that K θ decreases ( ˙
K < 0) if K θ > K 0 , and vice
versa. In addition, because the wall is fully relaxed when K θ = 1, we stipulate that
K θ must not exceed unity.
6.11.2 Model for an Artery
We now consider a model for growth and adaptation of an artery in response
to changes in loading conditions during development and maturity. The artery is
treated as a tube consisting of two incompressible, pseudoelastic layers representing
the contractile media (inner layer) and passive adventitia (outer layer). Developing
arteries stretch and grow longer with increasing blood pressure. Hence, rather than
specifying the axial strain, the ends of the tube are assumed to be closed but free
to move in response to the axial force caused by pressure on the end caps. Time
t = 0 corresponds to the time at which significant flow begins in the embryo, and
any initial stresses are neglected. The initial geometry is depicted in Fig. 6.24.