Wednesday, November 7, 2012

Under Tension -- Laplace's Theorem

The tangential tension on a blood vessel is thought of as obeying Laplace's theorem -- tension is a product of the pressure on the wall and the radius of the vessel.

Tangential tension = Pressure x radius
T = Pr

This formula is most appropriate for measuring force per unit tube length. For a more generalized case of tangential stress against the wall at a point, the wall thickness also plays an important role (as would seem intuituve)

Tangential stress = Pressure x (radius/wall thickness)
t = P (r / w)

These formulas are fundamental to thinking about vascular aneurysms, in that, the larger the aneurysm radius, and the higher the systemic blood pressure, the more the tangential tension against the wall...

... also... if there is a global or focal weakness to the wall (e.g. cystic medial necrosis), then the effective width and tensile strength of the wall decreases, and creates an area susceptible to increased tangential stress.

... and when the tangential tension exceeds the tensile strength of the wall, the aneurysm ruptures.

According to "Rutherford's Vascular Surgery," however, although Laplace's formula is useful for a rough understanding of the tension in the wall, it's too simple to explain the complexities of actual aneurysms.  Laplace's theorem is most suitable for a sphere with a wall size that decreases with increasing radius... and it may often overestimate the amount of actual stress in a real aneurysm wall.  Mural thrombus developing along the wall of an aneurysm appears to effectively increase its wall thickness, which may reduce tangential stress and risk of rupture... but the level of protection is unclear since thrombus and arterial wall are different materials with different tensile strengths, and the actual tensile strength of the wall would seem to be a combination of wall thickness and relative arterial wall strength.  Perhaps the thrombus is developing in areas with inherently weaker arterial wall strength.

1. "Rutherford's Vascular Surgery" Cronenwett and Johnston. 7th ed. (2010)