378
7 Remodeling
The increased moment at 4 days is consistent with the initial increase in opening
angle (Fig. 7.11b, solid red curve), but why does the angle then drop even as the
moment continues to rise? The answer lies partly in the increased wall thickness
required to return the total wall stress to its homeostatic value. While the moment
increases roughly in proportion to the thickness, the flexural rigidity for a (thin) shell
increases like the thickness cubed. Thus, the bending stiffness increases faster than
the moment, making the wall harder to bend as it opens.
As discussed in Sect. 6.7, positive opening angles are associated with residual
stress that is compressive near the lumen and tensile near the outer wall. In the
present model, smooth muscle and collagen dominate the total stress (Fig. 7.12b–
d). Growth of the muscle, which is located mostly in the media, compresses the
media and stretches the adventitia circumferentially, consistent with the computed
distributions of σ m
θ . This growth also stretches the collagen in and near the
adventitia, increasing σ c
θ in this region. These shifting stress distributions combine
to give the changes in total residual stress σ θ that affect the opening angles.
To test the model further, the opening angle is computed when either collagen or
elastin is effectively eliminated by setting c c = 0 or c e = 0 after the artery reaches
its homeostatic state. When collagen is removed, the angle increases by 54%, from
102 ◦ to 157 ◦ , in reasonable agreement with the experimental finding that degrading
collagen with collagenase leads to a 70% increase in opening angle in rat saphenous
arteries (Zeller and Skalak 1998). Removing elastin in the model, however, yields
only a small 4% increase, compared to a 53% increase when arteries are exposed to
elastase. Although the aorta has a different structure, these results suggest that the
model, with the selected parameters, underestimates the mechanical contribution of
elastin.
Finally, note that the model predicts that all collagen fibers are in a state
of compression when the homeostatic artery is unloaded (Fig. 7.12b). During
hypertension, collagen fibers in the outer part of the wall experience tension, while
those closer to the lumen remain compressed (Fig. 7.12c, d). These results are
consistent with observations that collagen fibers are wavy in unloaded arteries and
straighten with increasing stretch (Chow et al. 2014).
Modelers often assume that wavy collagen fibers are buckled and, therefore,
can sustain only relatively small compressive loads. Thus, collagen stress is taken
as zero when λ c∗ < 1. However, when this assumption is included in the
present model, the opening angle decreases with the initial rise in pressure (not
shown), contrary to the experimental results. This apparently unrealistic behavior
warrants further investigation. It may be that buckled collagen bundles embedded
in surrounding tissue can sustain higher compressive stress than commonly thought
(see Example 8.7 in next chapter).
These are just some of the complex interactions that occur as an artery grows
and remodels. It is important to emphasize that these trends depend on the model
assumptions, as well as the chosen parameter values. Clearly, more experiments
are needed to test the model. However, the reasonably good qualitative agreement
between experimental and theoretical opening angles suggests that the model
captures at least some aspects of the actual behavior.
7 Remodeling
The increased moment at 4 days is consistent with the initial increase in opening
angle (Fig. 7.11b, solid red curve), but why does the angle then drop even as the
moment continues to rise? The answer lies partly in the increased wall thickness
required to return the total wall stress to its homeostatic value. While the moment
increases roughly in proportion to the thickness, the flexural rigidity for a (thin) shell
increases like the thickness cubed. Thus, the bending stiffness increases faster than
the moment, making the wall harder to bend as it opens.
As discussed in Sect. 6.7, positive opening angles are associated with residual
stress that is compressive near the lumen and tensile near the outer wall. In the
present model, smooth muscle and collagen dominate the total stress (Fig. 7.12b–
d). Growth of the muscle, which is located mostly in the media, compresses the
media and stretches the adventitia circumferentially, consistent with the computed
distributions of σ m
θ . This growth also stretches the collagen in and near the
adventitia, increasing σ c
θ in this region. These shifting stress distributions combine
to give the changes in total residual stress σ θ that affect the opening angles.
To test the model further, the opening angle is computed when either collagen or
elastin is effectively eliminated by setting c c = 0 or c e = 0 after the artery reaches
its homeostatic state. When collagen is removed, the angle increases by 54%, from
102 ◦ to 157 ◦ , in reasonable agreement with the experimental finding that degrading
collagen with collagenase leads to a 70% increase in opening angle in rat saphenous
arteries (Zeller and Skalak 1998). Removing elastin in the model, however, yields
only a small 4% increase, compared to a 53% increase when arteries are exposed to
elastase. Although the aorta has a different structure, these results suggest that the
model, with the selected parameters, underestimates the mechanical contribution of
elastin.
Finally, note that the model predicts that all collagen fibers are in a state
of compression when the homeostatic artery is unloaded (Fig. 7.12b). During
hypertension, collagen fibers in the outer part of the wall experience tension, while
those closer to the lumen remain compressed (Fig. 7.12c, d). These results are
consistent with observations that collagen fibers are wavy in unloaded arteries and
straighten with increasing stretch (Chow et al. 2014).
Modelers often assume that wavy collagen fibers are buckled and, therefore,
can sustain only relatively small compressive loads. Thus, collagen stress is taken
as zero when λ c∗ < 1. However, when this assumption is included in the
present model, the opening angle decreases with the initial rise in pressure (not
shown), contrary to the experimental results. This apparently unrealistic behavior
warrants further investigation. It may be that buckled collagen bundles embedded
in surrounding tissue can sustain higher compressive stress than commonly thought
(see Example 8.7 in next chapter).
These are just some of the complex interactions that occur as an artery grows
and remodels. It is important to emphasize that these trends depend on the model
assumptions, as well as the chosen parameter values. Clearly, more experiments
are needed to test the model. However, the reasonably good qualitative agreement
between experimental and theoretical opening angles suggests that the model
captures at least some aspects of the actual behavior.