PHYSICAL EQUILIBRIA OF HEART AND VESSELS 



105 



TABLE 3. Ratio., as a Percentage, of the Elongation of the 

 Fibers in the Vessel Wall as Radius r to that of the Innermost 

 Fibers, at Radius r, , for Various Values of the 

 Percentage Elongation {a) of the Innermost Fibers 



thickness of the wall compared to the radius of lumen is less, 

 the gradient is even more linear. 



Case 2: Purely Elastic Artery 



Here we assume that the tension is purely due to stretch, 

 and the variation of tension with radius will depend, in addi- 

 tion, on how the degree of stretch varies through the thickness 

 of the wall. It has perhaps not been generally realized that this 

 degree of stretch must be very different for different layers in a 

 thick-walled elastic tube, where, as with the tissues of the wall 

 of blood vessels, the material is practically incompressible. 

 Let us assume, for simplicity, that when the transmural pressure 

 increases the artery increases in diameter, but not in length. 

 Fenn (12) has shown that this is \ery closely the case for arteries, 

 though it is very far from true for some other \essels (such as 

 the abdominal vena cavaj. In this case the cross-sectional area 

 of the wall from inside to a given radius, r, must remain con- 

 stant. Suppose that when the inner radius ri increases to ari 

 (the innermost circumferential fibers will be stretched in the 

 ratio of a: i), r becomes 0r. Then, because of the incompressi- 

 bility, we must have : 



7r(r= - r.2) = ■ni^-'r- - a-n-) 



(5) 



The degree of stretch (3 of the fibers in the wall will be pro- 

 gressively less as we proceed thi-ough the thickness of the wall. 

 From equation 5 we can construct a table showing the per- 

 centage of stretch of fibers at radius r over the percentage of 

 stretch of the innermost fibers, of radius r, , for \arioui ratios 

 of r/n , and degrees of stretch a of the innermost fibers. 



The table shows that the ratio of elongation is only slightly 

 altered by the degree of stretch, and is given quite well by the 

 approximation to equation 5 obtained for small degrees of 

 stretch, i.e. 



a = I + 5 where i -^ i 



a^ ^ I + 2«, ;32 - I -I- 26(r,/r)2, (6) 



,3 = I -f «(r,/r)'(/3 - I) ^ (a - i)(r,/r)2. 



Equation 6 gives a very simple rule. In a thick-walled vessel 

 the degree of elongation ((3 — i ) of the outer layers is very 



much less than that of the inner layers. Correspondingly, the 

 elastic tension in the inner layers will be much higher than 

 that of the outer layers, and the tissue pressure will fall off 

 much more rapidly in the inner layers than in the rest of the 

 wall. 



The next step is to translate the degree of stretch in the 

 different layers of the wall into the elastic tension it will pro- 

 duce. The simplest case is to assume that the elastic diagram 

 follows Hooke's law, though this is very far from applying to 

 the arterial wall. 



T\ = k{0 - i) 



= k 



\~^) "'J 



-r-'] 



(7) 



where .1 = [a'- — i)r,- 



Inserting this value into equation i gives 



= -'/(^i^-;) 



dr 



(8) 



Fortunately, this integral has a standard from which can easily 

 be evaluated 



where 



P = h\ln{i -1- a)—a\ + Constant. 

 (.42 -f r2)i(2 



(9) 



By inserting boundary conditions for any given case, the curve 

 of fall of pressure through the wall can be determined (fig. 21). 

 It is much more marked in the innermost layers, so that the 

 pressure has fallen to half in the first 28 per cent of the wall 

 thickness. 



A simpler solution is to tise the approximate solution, equa- 

 tion 6, which gives 





kr? 



X — + Constant. 



and a parabolic type of curve of fall of pressure. 



These solutions, however, do not approach reality for arteries 

 where the tension-length diagram is very far from linear. Resort 

 may be had to graphical integration, using an actual tension- 

 length diagram of an artery (cf fig. 1 1). The result is an even 

 steeper fall of pressure in the inner layers of the wall (fig, 2 1 j for 

 the young vessels, and yet steeper for the old vessels. It is clear 

 that the tissue pressure in the wall of arteries falls to half the 

 intravascular pressure in a small inner proportion of the thick- 

 ness of wall, possibly in the first 10 per cent. The exact curve 

 depends on the degree of stretch. The more the vessel 

 is stretched, the more the steep gradient is shifted to the inner 

 layers. 



The above calculations, even the final more sophisticated 

 one, assume uniform distribution of elasticity through the wall, 

 which can hardly be the case. The usefulness of the theory lies 

 in the possible insight it may give as to why the various ele- 

 ments of different elasticity, elastin and collagen, are arranged 

 as they are in different arteries. 



