Laplace’s Law – Its applications in critical care and peri-operative medicine

Dr Salman Yahya and authors look at Pierre Simon Laplace’s (1806) law of physics which states that “the tension on the wall of a sphere is the product of the pressure times radius of the chamber, and the tension is inversely related to the thickness of the wall [1]”

Wall tension T= (pressure × radius) ÷ (2×wall thickness)  

                       T = (P×R) ÷ 2t         

For a sphere with one liquid surface, pressure P=2T÷R  

For a sphere with two liquid surfaces, pressure P=4T÷R  

The equation for a pipe/blood vessel, wall tension T=P×R÷t [2]     

 

Definition of terms used:  

P=Pressure gradient

T=Wall tension 

t=Wall thickness  

R=Radius  

 

The law explains that as the radius of a tube or a sphere increases, the pressure gradient across the wall decreases. It also states that as the surface tension increases, the pressure gradient across the wall increases.

Laplace’s law of physics is applicable in a variety of clinical specialities including anaesthetics, critical care, cardiology, surgery, pregnancy, labour and delivery. 

 

Anaesthetic breathing system and Laplace’s Law 

The reservoir bag on the anaesthetic machine has a safety function which protects the patient’s lungs from barotrauma. It is the most flexible part of the breathing system and during gas filling, it is the reservoir bag that first shows evidence of filling. When the APL valve is closed, the reservoir bag is filled beyond the normal volume, but nevertheless it must maintain a safety pressure within limits of 60 cm H2O [3]. Laplace’s law shows the correlation between the tension T on the surface of reservoir bag, the inside pressure P and the reservoir bag radius R as follows 

           P= 2T÷ R

Pulmonary barotrauma is prevented in case of malfunction or unintentional closing of APL valve. In fact, in the presence of an overflow or a flow obstruction in the breathing system, the radius of the reservoir bag increases and, according to the above equation, the pressure inside it decreases thus preventing a dangerous rise in pressure in the entire breathing system and, consequently in the lungs. 

ETT Cuff pressure and the Laplace’s Law 

Laplace’s law describes the relationship between pressure and the wall tension of the ETT cuff and its pilot balloon. 

P= 2T÷R
or T = (P×R) ÷ 2 

It is important to mention Pascal’s Principle here: 

Given by the Blaise Pascal in 1652 [4] that “pressure is applied to a confined fluid at any point is transmitted undiminished throughout the fluid in all directions and acts upon every part of the confining vessel at right angles to its interior surfaces and equally upon equal areas”. It finds its application in the pressures across the ETT cuff and the pilot balloon. 

According to Pascal’s principle, the pressure inside the pilot balloon is the same as that in the cuff. However, what we really feel with our fingers is the wall tension of the pilot balloon (not the pressure), which is, according to the above-mentioned equation, directly proportional to the radius. Because the radius of the pilot balloon is smaller than that of the cuff, its wall tension is lower compared with the cuff. If we could touch the cuff of ETT directly (or theoretically a larger pilot balloon is incorporated), we would probably be more” accurate” in estimating its pressure. Otherwise, the so-called “finger pressure” technique fails to correctly assess the inflation pressure of the ETT cuff.  

Alveoli and the Laplace’s law 

Alveoli are prismatic or polygonal in shape i.e., their walls are flat, and Laplace’s law considerations regarding their inflation apply only to the very small, curved region in the fluid where these walls intersect. 

When the alveolar radius is small, the surfactant film is compressed, and the surface tension is small. When the alveolar radius is large, the surfactant film is expanded, and the surface tension is high. 

 Laplace’s law states that pressure P inside a sphere is directly proportional to the tension T in the walls and inversely proportional to the sphere’s radius R. 

In the diagram without surfactant, the wall or surface   tension in both large and small alveoli is about the same. As a result, a greater pressure develops in the smaller alveolus, which then proceeds to empty into the adjacent larger alveolus. 

In the diagram with surfactant, the surface tension reducing properties of the surfactant increase the individual surfactant molecules get closer together. This property counteracts the Laplace’s law and reduces the tendency for smaller alveoli to empty into the nearby larger alveoli. 

Aneurysm of aorta and the Laplace’s law 

Laplace’s law is useful in thinking about dilated tubular structures such as blood vessels e.g., aneurysm of aorta. The relationship between wall tension and the radius shows why more dilated regions of a tube develop more wall stress and therefore are at risk of perforation. 

The wall stress related to blood pressure in the non-aneurysmal aorta is relatively low and uniformly distributed, whereas within the aortic aneurysm, the region of high and low stresses is present. Increased tension stress results in progressive vessel dilatation and weakening of aortic media.                                                                                                                                                    

According to Laplace’s law, the wall tension is proportional to the vessel radius for a given blood pressure. When an artery wall develops a weak spot and expands as well, it might seem that the expansion would provide some relief, but in fact the opposite is true. The expansion subjects the weakened part of the aorta to even more tension. When the weak part of aorta dilates (radius is increased), wall thickness is reduced and the wall stress (T= P×R

÷

2× thickness) is increased which makes it more susceptible to rupture [5] 

This can be seen in Marfan syndrome, the abnormality in the fibrillin [2] gene leads to abnormally weak and thin aortic wall leading to increased radius at normal blood pressure and thus increased tension on the wall which further thins the wall and increases the tension on the wall (a vicious cycle).This accounts for the well –documented increasing risk of dissection and aneurysms as the aortic root calibre increases [2] 

Dilated cardiomyopathy and the Laplace’s law 

Dilated cardiomyopathy is predominantly a systolic dysfunction in which there is progressive enlargement of one or both ventricles of the heart. It leads to the stage that the interaction between actin and myosin myofilaments become insufficient. That causes a decrease in stroke volume and systolic impairment. The ratio between wall thickness to diameter ratio is low, that leads to a remarkable increase in ventricular wall stress i.e.  

LV wall stress T= (LV pressure ×R) ÷(2× LV thickness) 

The law determines that the LV wall stress is increased as the LV wall thickness is decreased. LV wall stress is the force acting against the myocardial cells and causes the coronary vessels to collapse which decreases the coronary perfusion, further worsening the condition i.e., demand is increased, and supply is decreased. 

In dilated cardiomyopathy, the heart becomes greatly distended and the radius(R) of the ventricle increases. Therefore to create the same pressure (P) during ejection of the blood much larger wall tension (T) has to be developed by the cardiac muscle. The dilated heart requires more energy to pump the same amount of blood as compared to the heart of normal size. As a logical extension of the same principle, partial left ventriculectomy can be performed to treat end-stage heart disease that aims to reduce the ventricular radius [6] 

 

Pregnancy, Labour and Delivery 

Laplace’s law  and Pascal’s principle dictate birth timing and pregnancy through exponential uterine wall tension(EUWT) where its malfunction alter birth timing and/or mode of delivery: A Hypothesis. 

EUWT creation, autonomic maintenance, and autonomic EUWT termination make pregnancy an autonomic cycle with constant intervals and circadian timers. EUWT malfunctions alter birth timing and/or mode of delivery. Laplace’s law and Pascal’s principle measure EUWT which might be the law of physics that determines pregnancy duration [7] 

 

Bladder Reconstruction 

In selection of intestinal segments for bladder substitution, de-tubularised bowel (i.e. more spherical and as P=2T/R for a sphere whereas it is P=T/R for a cylinder) segments provide greater capacity at lower pressure and require a shorter length of intestine than do the intact segments. They accommodate to filling more readily because, the container with greater radius will develop less pressure and thus hold larger volumes at lower pressure [8] 

 

Sigmoid Colon and diverticuli  

Sigmoid colon diameter is less compared to the rest of large bowel, so P=2T÷R. That means pressure is increased which is why diverticuli are common in sigmoid colon.

 

Conclusion 

Before start of every surgical list, it is mandatory to check the anaesthetic machine including reservoir bag, APL valve, leaks, and obstruction in the circuit (refer to AAGBI check list) [9]. Reservoir bag, the most flexible part of the breathing system, is used for monitoring respiration as well as ventilating the patient. It also acts as reservoir of gases/anaesthetic vapours and protects the patient from excessive pressure (barotrauma) within the breathing system. 

As regards the endotracheal tube, the cuff pressure should be checked straight away after intubation of trachea. The longevity of the procedure, the use of N2O, patient’s body temperature and the change in posture of the patient, all can alter the cuff pressure and the checks should be repeated. Otherwise, overinflated cuff could result in damage to vocal cords, cricoid cartilage, and later stenosis of trachea secondary to ischaemic damage. 

While managing a patient undergoing AAA repair, care should be exercised that the blood pressure does not shoot up i.e. at the time of laryngoscopy/ET intubation and surgical incision. Increase in blood pressure could result in rupture of aorta before cross clamps are applied.  

While dealing with a patient suffering with dilated cardiomyopathy, contemplate cardio-stable anaesthesia, avoid tachycardia and fluid overload. Increase in BP and/or SVR could result in ventricular dilatation and worsening of the scenario. 

More studies are needed to know the application of Laplace’s law in obstetrics. 

 

Conflict of interest: none declared. 

References: 

1. Essential Equation for Anaesthesia, Key Clinical Concepts for FRCA and EDA, pp24-25 DOI:https:doi.org/10.1017/CBO9781139565387.014
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3. International Standards Organisation. ISO11.040.11: Anaesthetic respiratory and  reanimation equipment; 2006
4. Halliday D, Resnick R, Walker J. Fundamentals of physics .Singapore; Wiley; 2001.pp.328-329(Google Scholar)
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Authors: 

1. Dr.Salman Yahya, clinical fellow anaesthetics STH NHS FT Sheffield
2. Dr.Majid Zaki, Speciality Anaesthetic Doctor, Peterborough City Hospital, NHS FT
3. Mohammad Awais Akram, Consultant Anaesthetist, THQ Hospital Mian Channu, Pakistan
4. Dr.Emad Abo Ebeid, Consultant Anaesthetist (locum), Airedale General Hospital NHS Trust
5. Sher Mohammad, Consultant anaesthetist STH NHS FT Sheffield

 Correspondence email:smyousafzai@doctors.org.uk