For a circular cylinder with length l and an inner radius ri., as model of a blood vessel, P acts to push the two halves apart with a force equal to P times the area of the horizontal projection of the cylinder, being 2Iri. (Fig. 1). It acts on both halves, so the force FP is:
FP = 2PIri. (1a)
The two halves are kept together by wall stress, σw, acting in the wall only. The related force Fw, visualized in Fig. 2, is thus:
FW = 2σwld, (1b)
d the wall thickness. 2dl is two
times the longitudinal cross area of the wall itself, at which this force is
acting (Fig. 2). The force is in equilibrium and thus: 2PIri = 2σwld. This gives the form of the law of
σw = Prid─1. (2)
Stress has the dimension N/m2, the same as pressure.
Fig. 2 (a) Visualization of the forces acting on and in the wall. (b) Cancellation of the sideward directed forces due to P. The horizontal projection of the inside of the cylinder and P directly gives FP (eq. 1a).
In other words it says that pressure and wall stress are related by the ratio of radius over wall thickness. With constant d and P, but increasing ri, σw increases proportionally with ri, At a certain ri, ultimate strength is reached (see Tensile strength) and rupture occurs (along the length axis of the cylinder). With a blood vessel d decreases with increasing ri, and consequently rupture occurs with a much smaller increase of ri. This may happen with an aneurism with its small d.
Another measure is the wall tension T, the force per unit length of the cylinder:
TW = FW(2l)─1 = Pri. (3a)
For a sphere, a similar derivation holds and the result is σw = ½Pri/d and TW is = ½Pri. .
TW is = ½Pri. (3b)
In (experimental) vascular medicine and cardiology and experimental pulmology.
a simple shape such as a sphere, or circular cylinder, the law may be applied
to the ventricular wall in diastole and systole, as well as to the vessel wall.
The law of
more general form of the
1/rc = 1/r1 + 1/r2, (4)
where r1 is the radius of the largest tangent circle and r2, that of the largest one perpendicular on the plane of the former. For a sphere r1 = r2, and for a cylinder r1 = ∞ and r2, = ri. After calculating rc for the cylinder and sphere respectively (3a) and (3b) directly follow.
σw,r = Pri2 (1+ ro2/r2)/ (ro2- ri2), (5)
where ro and ri are the external and internal radius, respectively, and r is the position within the wall for which local stress is calculated.
The relevant force related to (local) wall stress of the heart muscle of the left ventricle is often calculated for the ventricular “equatorial” plane. It is F = P∙Ae, where Ae is equatorial cavity cross-sectional area and P luminal pressure. The wall stress σw is given by F/Aw, with Aw the equatorial cross-sectional area of the muscle ring. Thus σw = P∙Ae/Aw.
Many other relations between wall force or stress and ventricular pressure have been reported, but since measurement of wall force is still not possible, it is difficult to decide which relation is best.
Relation to the Young modulus
a relatively thin arterial wall (d« ri and incompressible) one can
E = (ri2/d)ΔP/Δri. (6)
For thick walls, as is often the case in arteries, the Young modulus (see Elasticity 1: elastic or Young’s modulus) is best derived from the measurement of pressure and radius:
E = 3 ri2 ro(ΔP/2Δro)/(ro2- ri2) (7)
Van Oosterom, A and Oostendorp, T.F. Medische Fysica, 2nd edition, Elsevier gezondheidszorg, Maarssen, 2001.
N. Westerhof, Noble M.I.M and Stergiopulos N. Snapshots of hemodynamics: an aid for clinical research and graduate education, 2004, Springer Verlag.