**Laplace’s
law
**

**Principle**

The
law of __surface
tension__ and consequently has the dimension N/m.

For
a circular cylinder with length *l* and
an inner radius r_{i}., 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 2*I*r_{i}. (Fig. 1). It acts on both halves, so the force
F_{P} is:

F_{P} = 2P*I*r_{i}. (1a)

Fig.
1

The
two halves are kept together by wall stress, σ_{w}, acting in the
wall only. The related force F_{w}, visualized in Fig. 2, is thus:

F_{W} =
2σ_{w}*l*d, (1b)

where
d the wall thickness. 2d*l* 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: 2PIr_{i} = 2σ_{w}*l*d. This gives the form of the law of

σ_{w} = Pr_{i}d^{─1}.
(2)

Stress
has the dimension N/m^{2}, 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 F_{P} (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 r_{i},
σ_{w} increases proportionally with r_{i}, At a certain r_{i}, 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 r_{i}, and
consequently rupture occurs with a much smaller increase of r_{i}. 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:

T_{W} = F_{W}*(*2*l*)^{─1}
= Pr_{i}. (3a)

For
a sphere, a similar derivation holds and the result is σ_{w} = ½Pr_{i}/d
and T_{W} is = ½Pr_{i}. .

T_{W} is = ½Pr_{i}.
(3b)

**Application**

In
(experimental) vascular medicine and cardiology and experimental pulmology.

Assuming
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 Info**

A
more general form of the _{W} = Pr_{c}, where r_{c},
is the (local) curvature radius. The
“mean” curvature radius of any 3D shape is defined as:

1/r_{c} = 1/r_{1} + 1/r_{2},
(4)

where
r_{1} is the radius of the largest tangent circle and r_{2},
that of the largest one perpendicular on the plane of the former. For a sphere r_{1}
= r_{2}, and for a cylinder r_{1} = ∞ and r_{2},
= r_{i}. After calculating r_{c} for the cylinder and sphere respectively (3a)
and (3b) directly follow.

The
law of __Elasticity
and Hooke's law__) material the distribution of
circumferential stress or hoop stress across the wall thickness can be
approximated by:

σ_{w,r} = Pr_{i}^{2}
(1+ r_{o}^{2}/r^{2})/ (r_{o}^{2}- r_{i}^{2}), (5)

where
r_{o} and r_{i} 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∙A_{e}, where A_{e} is equatorial cavity cross-sectional
area and P luminal pressure. The wall stress σ_{w} is given by F/A_{w},
with A_{w} the equatorial cross-sectional area of the muscle ring. Thus
σ_{w} = P∙A_{e}/A_{w}.

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*

For
a relatively thin arterial wall (d« r_{i} and incompressible) one can
use

E = (r_{i}^{2}/d)ΔP/Δr_{i}. (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 r_{i}^{2}
r_{o}(ΔP/2Δr_{o})/(r_{o}^{2}-
r_{i}^{2}) (7)

**Literature**

Van Oosterom, A and Oostendorp, T.F. Medische Fysica, 2^{nd}
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.