The use of the Laplace equation in the calculation of sub-bandage pressure

Keywords: Laplace's equation; sub-bandage pressure; compression therapy; bandage application.

Abstract

Compression has been used for many centuries in the treatment of oedema and other venous and lymphatic disorders of the lower limb and is the standard treatment of uncomplicated venous leg ulcers. Laplace's law can be used to calculate or predict sub-bandage pressure and hence the level of compression applied to the limb. The aim of this article is to explain how this equation was derived and illustrate how it may be used to predict the sub-bandage pressure in the clinical setting.

Introduction

The degree of compression produced by any bandage system is determined by complex interactions between four principle factors - the physical structure and elastomeric properties of the bandage, the size and shape of the limb to which it is applied, the skill and technique of the bandager, and the nature of any physical activity undertaken by the patient.

The pressure generated by a bandage immediately following application is a function of the tension in the fabric, the number of layers applied, and the radius of curvature of the limb. The relationship between these factors is governed by Laplace's law. This states that sub-bandage pressure is directly proportional to bandage tension, but inversely proportional to the radius of curvature of the limb to which it is applied.

Despite the fact that Laplace's Law has long been quoted in this context [1], [2], [3] it is not generally well understood. In 1990, a version of the equation was published which sought to address these issues [4]. This has since been well referenced in the literature [5], [6], [7], although many who cite the equation fail to make reference to the units of measurement or attempt to explain how the formula may be used practically to predict sub-bandage pressure.

The Laplace equation

The Laplace equation used to predict sub-bandage pressure is derived from a formula described independently by Thomas Young (1773-1829) and by Pierre Simon de Laplace (1749-1827) in 1805. This defines the relationship between the pressure gradient across a closed elastic membrane or liquid film sphere and the tension in the membrane or film [8].

In this formula Pα and Pβ α are respectively the internal and external pressures at the surface, r the radius of curvature and γ is the tension in the film. The equation indicates that the pressure inside a spherical surface is always greater than the pressure outside, but that the difference decreases to zero as the radius becomes infinite (when the surface is flat). In contrast, the pressure difference increases if the radius becomes smaller and tends to infinity as r tends to zero. However, the equation breaks down before r reaches zero, and so in practice this situation does not arise.

When calculating pressures in the wall of a cylinder, a modified formula (P=T/r) is required. This is because for a given vessel radius and internal pressure, a spherical vessel will have half the wall tension of a cylindrical vessel (http://hyperphysics.phy-astr.gsu.edu/hbase/ptens.html).

The law finds application in many branches of science including physical chemistry, chemical engineering and life and health sciences. It may variously be used to explain the properties of small particles, for calculating the surface energy of metals in the solid phase and, in medicine, for calculating the forces on blood vessels and the fluid-filled alveoli in the lungs [8].

To use the simplest form of the equation (i.e. P=T/r), it is necessary to use coherent units of measurement, i.e. units that relate directly to each other. This most commonly involves the use of the Pascal to measure pressure, the metre for linear measurements, and the Newton as the unit of force.

As none of these measurements is commonly used in medical practice, the practical value of the equation will be limited unless they are converted to more familiar units such as mmHg, centimetres and kilogram force (Kgf). In addition, the size of the cylinder (or limb) is expressed in the original equation as its radius, the direct measurement of which is virtually impossible in a clinical setting. For this reason, the more familiar measure of circumference is preferred. Conversion factors for these various units are shown in Table 1.

Table 1: Conversion units

Parameter	Coherent unit	Alternative unit	Conversion factor
Pressure	Pascal	mmHg	0.0075
Force	Newton	Kgf	0.102
Length	Metre	Centimetre	100
	Radius	Circumference	2π r = (2 x 3.142r)

The use of non-coherent units also means that a constant will have to be introduced into the formula. This is, in effect, the product of all these conversion factors, resulting in an equation with the form P=TK/r where K is a constant value yet to be derived.

For this specific application of Laplace's law it is also necessary to consider two further factors: the width of the bandage and the number of layers applied. Although these variables may not appear initially to form part of the original Laplace formula, they are essential to obtain an accurate value of tension.

Importance of bandage width

In a balloon or blood vessel, the tension in the walls acts throughout the entire area of the structure. In contrast, in the case of a single layer of bandage applied to a cylinder or limb, the pressure is only exerted upon that area covered by the bandage fabric. This pressure will be determined by the total force applied to the fabric and the bandage width in accordance with the definition of pressure, which states that Pressure = Force/unit area. This means that a 10cm wide bandage applied with a total force of 'F' Newtons, will produce only half the pressure developed beneath a 5cm wide bandage applied with the same force as the force is distributed over twice the area. Bandage tension must therefore be expressed in the Laplace equation as force per unit width, which is why a value for bandage width appears in the formula.

Number of bandage layers applied

The total tension developed in a bandage is the sum of the tension in its individual yarns. It follows, therefore, that the application of two layers of a bandage, applied with constant tension, will double the number of yarns over any particular point on the surface of the leg and thus, for all practical purposes, double the pressure applied. For this reason the number of layers of bandage applied (n) must be considered when calculating sub-bandage pressure.

Derivation of formula using alternative units of measurement

The way in which these factors inter-relate is shown in the following example. This also illustrates the derivation of a form of the equation that uses clinically relevant units of measurement. In this example all relevant measurements are first provided in coherent units with the equivalent values in the alternative units quoted in parenthesis.

Consider one layer of bandage 0.1 metres wide (10cm) applied to a limb radius 0.05 metres (31.416cm circumference) with a tension of 20 Newtons (2.04KgF). Using coherent units of measurement, the sub-bandage pressure 'P' may be calculated as follows:

The equation is expressed using the alternative units of measurement to derive a value for the constant K:

For a single layer of bandage (where n=1) the equation produces a value for K of 4620. (The difference between this and the value of 4630 quoted previously is thought to be due to an earlier rounding error [4]).

The formula to calculate sub-bandage pressure can therefore be summarised as:

Discussion

It must be recognised that the value for sub-bandage pressure obtained when using this formula only applies at the time of application. Most bandages lose a significant percentage of their initial tension over time, which will result in a reduction in applied pressure. The width of the bandage quoted in the formula also relates to the width of the fabric at the time and point of application. Some bandages reduce significantly in width as they are stretched, a phenomenon known as 'necking'. It is this measured width that should be used in any calculation, not the unstretched or nominal width of the fabric.

The other factor that plays an important role in determining initial sub-bandage pressure is the method of application. The calculations described above relate to a single turn of bandage applied at right angles to the limb. In practice most bandages are applied in the form of an overlapping spiral with the degree of overlap determining the number of layers of fabric that overlay a particular point on the surface of the limb. An overlap of 50% effectively provides two layers of bandage, but a 66% overlap will produce three layers of bandage, for example. For this reason, particular care should be taken when applying highly elastic bandages with a significant amount of tension to ensure that the edges of the bandage do not overlap excessively as this can result in localised areas of very high pressure, possibly resulting in areas of tissue necrosis.

Finally, the calculated value for sub-bandage pressure is the average pressure that will be exerted by a bandage on a limb of known circumference. If the bandage is applied to a cylinder of uniform cross section, the pressure underneath the bandage will be consistent over the entire surface area. However, a limb will exhibit marked differences in radius at various points around its circumference and pressure values determined using a direct measuring device at these locations will vary dramatically from the calculated average. The positioning of pressure sensors is therefore critical as these will produce different results depending upon where they are placed around the leg. This is why pressures determined experimentally do not always correlate well with predicted calculated values. For this reason it is usually recommended that padding is applied beneath compression bandages to fill concavities and protect more prominent areas to reduce local variations in sub-bandage pressure to acceptable values.

The clinical importance of compression and issues relating to the measurement of sub-bandage pressure have been discussed in a recent publication [9], [10].

Conclusion

Due to practical problems of measuring sub-bandage pressure it is frequently desirable to use the Laplace equation to predict the average pressure that a bandage will produce with a given level of applied tension. The use of Laplace's law has hitherto been poorly understood, including the importance of bandage width and the number of layers applied. This paper has sought to address these issues and to explain Laplace's equation as used in clinical practice.

References

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3. Blair SD, Wright D, Backhouse CM, Riddle E, McCollum CN. Sustained compression and healing of chronic venous ulcers. BMJ 1988; 297: 1159-61 (erratum appears Dec 10; 297: 1500).

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6. Nelson EA. Compression bandaging in the treatment of venous leg ulcers. J Wound Care 1996; 5(9): 415-18.

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9. Partsch H. Understanding the pathophysiological effects of compression.. In: Understanding compression therapy. EWMA Position document. London: MEP, 2003; 2-4.

10. Clark M. Compression bandages: principles and definitions.. In: Understanding compression therapy. EWMA Position document. London: MEP, 2003; 5-7.