Question 5.8: Aneurysm, a ballooning or dilation of a blood vessel, often ......
Aneurysm, a ballooning or dilation of a blood vessel, often afflicts the abdominal aorta, a large vessel supplying blood to the abdomen, pelvis, and legs. While aneurysms can develop and grow gradually, the rupture (rapid expansion and tearing) of an aneurysm is usually catastrophic. Although the healthy abdominal aorta has a diameter of 1.2–2 cm, an aneurismal abdominal aorta may have a diameter up to 6–10 cm. Anatomy textbooks give a range of values for the thickness of artery walls, from which we choose a median value of 0.1 cm. The figure below shows a rough sketch of this anatomy. Using outside references, determine the stresses in the walls of a healthy abdominal aorta and one affected by aneurysm.
Given: Dimensions of healthy aorta and aneurysm.
Find: Stresses in the aorta walls.
Assume: The healthy aorta is a cylindrical pressure vessel and the aneurysm may be a cylindrical or spherical pressure vessel; thin wall theory applies.
We would like to model the artery as a pressure vessel, despite the many differences between a physiologically realistic blood vessel and the idealization we have just studied. If we choose a radius of 1.2 cm (or larger) for our model healthy abdominal aorta, we can call our vessel is thin-walled.
The pressure inside the artery varies from a low (diastolic) to high (systolic) value over each heartbeat. Using a typical healthy systolic pressure of 120 mm Hg (1.6 N/cm²), we can calculate the peak circumferential or hoop stress in a healthy abdominal aorta:
\sigma_{\theta \theta}=\frac{p r}{t}=\frac{1.6 \mathrm{~N} / \mathrm{cm}^2 \cdot 1.2 \mathrm{~cm}}{0.1 \mathrm{~cm}}=19 \mathrm{~N} / \mathrm{cm}^2 .
If the vessel grows to a diameter of 6 cm, the hoop stress in a cylindrical vessel becomes
\sigma_{\theta \theta}=\frac{p r}{t}=\frac{1.6 \mathrm{~N} / \mathrm{cm}^2 \cdot 3 \mathrm{~cm}}{0.1 \mathrm{~cm}}=48 \mathrm{~N} / \mathrm{cm}^2 .
If, however, the abdominal aorta remodels itself into a more spherical shape, the hoop stress will be reduced:
\sigma_{\theta \theta}=\frac{p r}{2 t}=\frac{1.6 \mathrm{~N} / \mathrm{cm}^2 \cdot 3 \mathrm{~cm}}{2(0.1 \mathrm{~cm})}=24 \mathrm{~N} / \mathrm{cm}^2 .
This crude calculation suggests that the aorta may change its shape in part to reduce the stress induced by internal (blood) pressure. It is worth noting again that this pressure pulses, too, resulting in a cyclic loading and unloading of the vessel. Other factors contributing to aneurysm development include elastin degradation, atherosclerosis, and genetics, but continuum mechanics is certainly part of the package.