Question 4.2.1: 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 to 2 cm, an aneurismal abdominal aorta may have a diameter up to 6 to 10 cm. Figure 4.20 shows a rough sketch of this anatomy.
We’d 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. Anatomy textbooks give a range of values for the thickness of artery walls, from which we choose a median value of 0.1 cm. If we choose a radius of 1 cm for our model healthy abdominal aorta, we can call our vessel 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 circumferential or hoop stress in a healthy abdominal aorta:
\sigma _1=\frac{pr}{t}=\frac{1.6 ^{\textrm{M}}/_{\textrm{m}^2}\cdot 1 \textrm{ cm}}{0.1 \textrm{ cm}}= 16 ^{\textrm{N}}/_{\textrm{cm}^2} .
If the vessel grows to a diameter of 5 cm, the hoop stress in a cylindrical vessel becomes
\sigma _1=\frac{pr}{t}=\frac{1.6 ^{\textrm{M}}/_{\textrm{m}^2}\cdot 2.5 \textrm{ cm}}{0.1 \textrm{ cm}}= 40 ^{\textrm{N}}/_{\textrm{cm}^2} .
If, however, the abdominal aorta remodels itself into a more spherical shape, the hoop stress will be reduced:
\sigma _1=\frac{pr}{2t}=\frac{1.6 ^{\textrm{M}}/_{\textrm{m}^2}\cdot 2.5 \textrm{ cm}}{2(0.1 \textrm{ cm})}= 20 ^{\textrm{N}}/_{\textrm{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’s 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.