PHYSICAL EQUILIBRIA OF HEART AND VESSELS 



95 



FIG. 13. Equilibrium diagram 

 for a blood vessel wall under 

 elastic tension alone. 



FIG. 14. Equilibrium under 

 active tension alone (soap film). 

 [From Burton (5).] 



Radius r 



(tangent of the angle) is equal to the transmural 

 pressure, Ptm- The point A, where the '"Laplacian 

 line" and the curve intersect, represents the point 

 of equilibrium. Its coordinates give the elastic tension 

 in the wall and the radius of the vessel, that will 

 pertain to the particular transmural pressure. If the 

 pressure were reduced, a Laplacian line of reduced 

 slope would result, intersecting the curve at D, 

 indicating the reduced radius that would result at 

 the reduced transmural pressure. 



The equilibrium under transmural pressure and 

 elastic tension alone is completely stable, if this is 

 the shape of the tension-length diagram. Suppose 

 that the radius were somehow to be increased from 

 r, corresponding to point A, io (r -\- dr), correspond- 

 ing to point B. The tension in the wall, i.e., the 

 ordinate of point B, would now be greater than the 

 value required for equilibrium (point B'), so the 

 net force would tend to reduce the radius back to the 

 original value. Similarly, if the radius were supposed 

 to diminish to point C, a radius (r — dr), the tension 

 would now be less than that required for equilibrium, 

 and the net force would tend to increase the radius 

 back to the original value. The intersection at A 

 represents therefore "stable equilibrium." 



12. THE PHENOMENON OF BLOWOUT 



However, it must be recognized that if the trans- 

 mural pressure is great enough, equilibrium may 

 not be possible. The curve of tension vs. stretch for 

 arteries does not continue to increase in slope, but 

 becomes a straight line when the stretch is enough 

 to have reached the unstretched length of all the 

 collagenous fibers in the wall. If the Laplacian line 



has a slope great enough to be parallel to this final 

 slope of the elastic line, no intersection is possible. 

 The vessel radius will increase until the vessel bursts. 

 This is the phenomenon of "blowout," familiar with 

 the rubber inner tube of tires. The transmural pressure 

 required for blowout can be calculated by equating 

 the slope of the Laplace line to the final slope, i.e.. 



p = 



* max 



dTc 

 dr 



(18) 



The increase in the circumferential tension in the 

 wall, per unit length of the vessel, is given, in terms 

 of Young's Modulus: 



dT, 



Y. t. 



dr 



(19) 



where / is the tliickness of the wall, dr is the increase 

 in radius, and ro is the unstretched length of the 

 fibers. When all the fibers are stretched, Y will reach 

 its maximum value, Fmax- The product of Y and 

 thickness may be called the "elastance" of the wall 

 E, in dynes per unit elongation, i.e.. 



dTc _ 

 ~d^ ~ 



Then the blowout pressure, 

 P 



E 



ro 



dynes/cm2 



(■20) 



(2l) 



i.e., to the ratio of the maximum elastance of the 

 wall to the radius. This is, of course, provided the 

 yield point or elastic limit were not reached, as the 

 stretch increased, before the lines of figure i 3 became 

 parallel. 



Roach & Burton (24) on autopsy specimens of 



