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Chapter 9
LAW OF LAPLACE
Description
The original law of LaPlace pertains to soap bubbles, with radius r, and gives
the relation between transmural pressure, P, and wall tension,
in a thinwalled sphere as
It can be used to calculate tension in alveoli. This
tension is directly related to surface tension and has the dimension N/m. The
form of the law of LaPlace as most often used in hemodynamics gives the
relation between pressure within the lumen of a hollow organ and the stress
in the wall. Stress has the dimension
For a circular cylinder, as model
of a blood vessel, the pressure acts to push the two halves apart with a force
equal to pressure times the area (left figure in the box). Thus force is
The two halves are kept together by wall stress,
acting in the wall only.
This force is thus
These forces are in equilibrium and thus:
which gives
This form of the law of LaPlace is more
correctly called Lame's equation. For a sphere, a similar derivation holds and
the result is
We see that pressure and wall stress are related by the ratio of radius over
wall thickness.
Applicability of the Law of LaPlace
The law of LaPlace applies to cylindrical or spherical geometries,
irrespective of whether the material is linear or nonlinear or if the wall is thin
or thick. The only limitation of LaPlace’s law is that it yields the average wall
stress and thus it cannot give any information on the stress distribution across
the wall. For cylindrical geometries, and assuming linearly elastic (Hookean)
material the distribution of circumferential stress or hoop stress across the
wall thickness can be approximated by:
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Basics of Hemodynamics
where and is the internal and external radius, respectively, and r is the
position within the wall for which local stress is calculated.
There is a large body of literature, especially for the thick walled heart,
where (local) wall stress or muscle fiber stress is related with pressure for
different complex geometries (for information see [3]).
Hefner [2] extended the Law of LaPlace for the left ventricle by showing
that the equatorial wall force (F) is
where
cavity crosssectional area and P luminal pressure. The wall stress,
is given by
with
the equatorial cross-sectional area of the muscle ring. Thus
Mirsky and Rankin [5] suggested an often used estimation of mid-wall
stress. For an ellipsoidal heart shape the mid-wall stress is:
with D and l the mid-wall diameter and mid-wall length of the ventricle (see
figure in box).
Arts et al. [1] derived a simple and practical relation between fiber stress,
and ventricular pressure,
that reads:
where
and
are ventricular lumen and ventricular wall volume,
respectively. This equation can be used in all moments of the cardiac cycle
and thus allows the calculation of fiber stress in both diastole and systole,
including the ejection phase.
Many other relations between wall force or stress and ventricular pressure
have been reported, but since measurement of wall force is still not possible
[3], it is difficult to decide which relation is best.
The law of LaPlace pertains to geometrically simple bodies but may be
applied to non-linear material. 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 LaPlace can also be
used in the contracting heart, where the force is generated in the wall and the
rise in pressure is the result of the contracting muscle.
Relation to the Young modulus
Assuming that the arterial wall is relatively thin
and incompressible,
one can use LaPlace’s Law to derive the following expression for the
incremental elastic modulus (Chapters 10 and 11):
where the
and the
are the change in internal radius and transmural
pressure.
If the wall cannot be considered thin, as is often the case in arteries, the
Young modulus is best derived from the measurement of pressure and radius
using the following expression [4].
Law of LaPlace
Physiological and clinical relevance
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The Law of LaPlace, although basic and pertaining to simple geometries,
helps in understanding cardiac and vascular function. The law is therefore of
great conceptual importance. For instance, the ratio r/h is a main determinant
of the wall stress. The radius of curvature of the left ventricle is smaller at the
apex than at the base, and so is the wall thickness h. The ratio r/h at the apex
and base is, however, the same, resulting in similar wall stress at these
locations in the wall. In hypertension the cardiac muscle cells increase in
thickness by building more contractile proteins in parallel, leading to
concentric hypertrophy. The thicker wall, but
similar lumen, i.e., decreased r/h, causes the
systolic wall stress to return to presumably
normal levels despite the higher pressure in
systole. In hypertension the large arteries
hypertrophy and the thicker wall decreases
wall stress. How stresses in the cells are
IN CONCENTRIC HYPERTROPHY, sensed is still largely unknown. In dilated
when compensated, ventricular pres- cardiomyopathy where lumen size is greatly
sure and wall thickness both are increased while wall thickness is not, wall
increased in approximately the same stresses are large. This leads to high oxygen
proportion and wall stress is similar.
demand, often too large for oxygen supply.
References
Arts T, Bovendeerd HHM, Prinzen FW, Reneman RS. Relation between left
ventricular cavity pressure and volume and systolic fiber stress and strain in the
wall. Biophys J 1991;59:93-102.
2. Hefner LL, Sheffield LT, Cobbs GC, Klip W. Relation between mural force and
pressure in the left ventricle of the dog. Circ Res 1962;11:654-663.
3. Huisman RM, Sipkema P, Westerh of N, Elzinga G, Comparison of models used
to calculate left ventricular wall force. Med & Biol Eng & Comput 1980; 18:133144.
4. Love AEH. A treatise on mathematical elasticity. 1952, London & New York,
Cambridge Univ Press, third Edn.
5. Mirsky I, Rankin JS. The effects of geometry, elasticity, and external pressures
on the diastolic pressure-volume and stiffness-stress relations. How important is
the pericardium? Circ Res. 1979; 44:601-11. Review.
1.