PHYSICAL EquILIBRIA OF HEART AND VESSELS 



87 



would be the opposite. It becomes clear that with 

 distensible tubes there is no one relation between 

 flow and pressure difference, but a whole family of 

 such relations, depending upon the transmural 

 pressure of the resistance vessels. This has been 

 demonstrated in vascular beds (7). Equation i 

 therefore becomes insufficient to define the flow- 

 pressure relations and does not contain all the neces- 

 sary data to determine the flow. We must add an 

 equation which recognizes the dependence of the 

 geometry of vessels on their transmural pressure, i.e., 





(4) 



where / represents some function, which we might 

 call the "'distensibility function."' It becomes of 

 great importance to determine this function /, i.e., 

 to study how the size of distensible vessels depends 

 upon their transmural pressures. 



3. FORCES CONCERNED WITH THE EQUILIBRIUM 

 OF THE BLOOD VESSEL WALL (5) 



-•1. The Distending Force 



Consider a unit length of a cylindrical blood 

 vessel, where first we consider the wall as very thin 

 compared to the radius r of the cylinder (fig. 2A) : 

 the intravascular pressure P,,. dynes per cm- is every- 

 where at right angles to the wall, and gives a force, 

 per unit length of the cylindrical vessel, equal to 

 2irrPiy dynes, tending to increase the radius. On the 

 outside of the vessel, the tissue pressure opposes 

 this, with a total force of 2irrPT. The total "expand- 

 ing" force is then, in dynes per centimeter; 



F = ^irriP,. - Pt) 

 B. The Constricting Force 



•2T\rPT 



(5) 



The force holding this expanding force in equi- 

 librium is that due to a "'circumferential tension" Tc, 

 in dynes per unit length of the cylindrical vessel 

 (fig. 2B and 2C). 



4. EQUILIBRIUM BETWEEN THESE FORCES: 

 THE LAW OF LAPLACE 



There are two ways of equating these forces, to 

 be found in any college physics text. Forces may be 

 resolved from figure 2B, or, in a better method, use 



..©- 



ICM. 



■ V I / ^c ^c / 



C \j 4' 'y ,' 



Fio. 2. The forces that are in equilibrium at the blood vessel 

 wall. Pi„ — intravascular pressure, Pt — tissue pressure, Ptm — 

 transmural pressure, Tc — circumferential tension in dynes/cm 

 length of the cylindrical vessel. 



is made of the "principle of virtual work." The 

 tension in the wall, T,-, may be expressed either as 

 dynes per centimeter length of the cylinder, or as 

 ergs per .square centimeter of surface of the wall 

 (just as surface tension is expressed as dyne/'cm or 

 as ergs/cm-). The "principle of virtual work" states 

 that if a small displacement is made from the equi- 

 librium, as increasing the radius r Xo r + dr, the 

 work done must equal the change in energy that 

 would result. This is simply a special case of the 

 general principle of conservation of energy. In this 

 particular case, for an increase of radius to (r + dr), 

 the pressure, which is everywhere at right angles to 

 the wall, will do work equal to 2-n-rPr.v/ dt . The total 

 energy of the wall will have increased, because the 

 surface of a unit length of the cylinder has increased 

 from a value 27rr to 27r(r -|- dr). The increase in 

 surface area is then 2-K-dr square centimeters, and 

 in the surface energy is 2irTc-dr. 

 Equating, we have 



•2TrrPrit dr = ^ttTc ■ dr 

 ■ P -^ 



(6) 



For equilibrium, then, the transmural pressure must 

 always equal the circumferential tension divided 

 by the radius. This means that the circumferential 

 tension in the wall has a sort of inechanical advantage 

 in opposing the transmural pressure. The smaller 

 the radius of the cylinder, the greater the pressure 

 that can be held in equilibrium by a given tension 

 in the wall. This principle is well known to engineers 

 who have to design piping for transport of fluids 

 under high pressures, e.g., O,, against bursting under 

 high pressure. A pipe of small radius will be safe, 

 where a larger pipe, with the same thickness of 

 metal in the wall, would easily burst. 



The same application of the principle of virtual 



