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HANDBOOK OF PHYSIOLOGY 



' CIRCULATION I 



Consider an annular cylindrical ring of the wall, 

 of unit length of vessel, between radii r and r + dr. 

 Let the tension at this radius be T' r per unit (radial) 

 thickness, i.e., in this annular ring it will be T\ dr. 

 (The T' must be used instead of the former T, since 

 this was the total tension, whereas T' is, of course, 

 proportional to the thickness of the ring of wall being 

 considered.) Then the law of Laplace applies, at 

 everv point within the wall, and there must be a 

 drop of pressure in the wall between inside this 

 ring and outside it, given by 



dP = 



Integrating 



T,-dr 



"■» r; dr 



f Ll 



(lo) 



(II) 



where r, and r„ are the inside and outside radii, 

 respectively. 



If we do not wish to introduce a knowledge of the 

 particular way in which T'^ varies through the wall 

 (since this will be determined by the elastic constants 

 and the degree of stretch of each layer, and in con- 

 tractile tissue, by the "active tension" of the muscle 

 of each layer), we may conveniently work with an 

 "average tension of the wall," Tc, defined by: 



---[' 



Trdr 



(12) 



where r is an average radius, i.e., (r, + r„)/-- This 

 brings us back to the simple equation for the law 

 of Laplace, i.e. 



n 



(.3) 



None of the conclusions reached later, e.g., as to 

 instability of blood vessel equilibrium, is altered by 

 this consideration. 



Once the value of the circumferential tension at 

 different places in the wall is known, the value of 

 the integral of equation i 1 can be found, and the 

 gradient of pressure through the wall evaluated. 

 However, up to this point, we have given only the 

 equations that determine what the total tension 

 must be, in a vessel of a certain radius, to be in 

 equilibrium with the pressure. When the wall is 

 distensible, however, this "certain radius" is not 

 fixed but determined by another relation, i.e., that 

 between the tension in the wall and the stretch of 

 the wall, which is determined b\' what is called the 



"elastic diagram" for the vessel. If the transmural 

 pressure is altered, the equilibrium will be destroyed, 

 and the radius will change until the tension, under 

 the new degree of stretch, once more obeys the 

 equation of Laplace. The solution of the problem 

 then depends upon an analysis of the elastic behavior 

 of the vessel wall. When this has been discussed, it 

 is possible to return to the solution of equation 10 

 for some simple idealized cases, and to investigate 

 the distribution of tension through the thickness of 

 the wall, e.g., which layers will play the major role 

 of holding the pressure in check; and we can also 

 determine the drop of pressure through the thickness 

 of the wall. These solutions are important for an 

 imderstanding of the particular architecture of the 

 walls of blood vessels. The pressure gradient in the 

 wall obviously also has a bearing on the possibility 

 of "vasa vasorum," the blood vessels within the 

 arterial wall, remaining open against the pressure 

 in the tissues where they lie. The solution of equation 

 1 I will be given for special cases in the Appendix. 



7. N.\TURE OF THE TENSION IN THE \V.\LL, 

 EL.'\ST1C AND ACTIVE TENSIONS 



The law of Laplace tells us the magnitude of the 

 circumferential tension, and the second calculation 

 the longitudinal tension, that must exist in the wall 

 to hold the transmural pressure in ccjuilibrium. 

 These calculations can tell us nothing of the origin 

 of these tensions. The reason for the tension in the 

 wall of living blood vessels is twofold. First, it may 

 be due to the stretch of the wall, with the property 

 of "elasticity" of the tissue. This elasticity would be 

 present also in the "dead" vessels, or indeed in a 

 tube of any "elastic" material. In the second place, 

 it may be due to contraction of li\ing smooth muscle 

 in tiie wall. The latter may be called the "active 

 tension," T^. The total tension in the wall: 



T = Ta + Te (14) 



It is important to define these two types of tension in 

 terms of their dependence on stretch, not with respect 

 to any particular tissue. Elastic tension is defined as 

 tension which depends on the degree of stretch of 

 the wall, i.e. 



Te = /(') (15) 



while acti\e tension is, by definition, independent 

 of stretch, and dependent on the physiological activity 

 of the tissue (vasomotor activity .-4), i.e., 



T.^ = f{A) (16) 



