PH\'SICAL EqUILIBRIA OF HEART AND VESSELS 



89 



the maximum and minimum \aiues of all the radii 

 so found (8, 28). 



For a sphere, equation 8 becomes equation 7, 

 where ri = u = r, and becomes equation 6 for a 

 cylinder, where r2 = oc . For surfaces where the 

 two principal curvatures are in the same "sense," 

 i.e., the centers of curvature are on the same side 

 of the surface, the terms i/Vi + i/Vo add together. 

 These are called ".synclastic surfaces" (fig. 3). The 

 walls of ventricles of the heart are "synclastic." 

 Other surfaces may have the principal radii of 

 curvature in opposite sense, and are called "anti- 

 clastic" surfaces. The wall of the inside of the arch 

 of the aorta is an example. Here the two terms sub- 

 tract from each other. In contrast, the wall on the 

 outer curvature of the arch is synclastic (fig. 5). 

 As a consequence of equation 8 and the above, the 

 tension required in the wall on the inside of the 

 arch, to hold the aortic pressure in equilibrium, is 

 much greater than that required on the outside of 

 the arch, and it is no surprise to find that the wall is 

 correspondingly thicker at the bottom of the arch 

 than at the top.' 



CIRCUMFERENTIAL 

 Kz) AREA » tsq. cms per unit l«ngth 



4 





y ^ ^' Tr . p. R. 



LONGITUDINAL 



Thickn«st t cmi. 



Radius R. cms. 



Fores /Arsa> PR/t 

 dynss / sq. cm . 



'• ^\ Fores ■ n R ' P. 



P. R / 2 t 

 dynss / sq. cm 



FIG. 6. Comparison between the calculations for the circum- 

 ferential tension and the longitudinal tensions in the wall. 

 [From Roach & Burton (23).] 



forgotten that the longitudinal tension is reckoned 

 per unit length of the circumference of circular cross 

 section of the wall, while the circumferential tension 

 is reckoned per unit length of wall, parallel to the 

 cylindrical axis). 



5. EQUILIBRIUM FOR THE LONGITUDINAL TENSION 



In addition to the circumferential tension, there 

 must be a longitudinal tension in the wall of a cylin- 

 drical vessel, if it is to be in equilibrium with the 

 transmural pressure. No matter what the shape of 

 the rest of the vascular bed, to which the particular 

 cylindrical segment considered is connected, it can 

 be shown that the result of all the pressure forces 

 acting on the walls of the rest of the bed is equivalent 

 to the force that would be exerted on a plane parti- 

 tion at right angles to the axis, closing off the end 

 of our segment. This force would be wi- X Ptm- 

 If Tl is the longitudinal tension in the wall, reckoned 

 per unit length of the circumference, the total force 

 over the cross section of the wall is 27rr X T ,^ (fig. 

 6). Equating: irr- X Ptm = ^rr X T,, i.e.. 



Ptm 

 2r 



(9) 



Thus the longitudinal tension Tl is half the cir- 

 cumferential tension, T^ (though it must not be 



' For those who love mathematical formulas, the ratio of 

 thickness at the bottom to thickness at the top of the arch is 

 given by (n — i) (n -f 2)/(n -f- i) (n — 2) where n is the ratio 

 of the radius of the arch (to the axis of the aorta) to the radius 

 of the aorta. For n = 3, the ratio of thickness is 2.5, for n = 4; 

 1.8, and for n = 5j 1.2. 



6. MODIFICATION OF THE LAW OF LAPLACE FOR 

 THICK-WALLED VESSELS 



Some doubts have been expressed in the physi- 

 ological literature that the law of Laplace can be 

 applied to any vessel where the thickness of the wall 

 is not small compared to the radius of the lumen. 

 This ratio in blood vessels runs from less than 3 per 

 cent for veins, 20 per cent only for the aorta, and 

 up to unity or even more for the thick-walled 

 arterioles. In the heart also, particularly at the 

 apex, the thickness of the ventricular wall may be 

 considerable, compared to the radii of curvature. 

 There is no basis whatever for these doubts, though 

 the law must be applied, where the ratio of thickness 

 to radius is considerable, with the aid of the cal- 

 culus, to successive concentric layers of the wall. 

 This results in an interesting insight into the way in 

 which the tissue pressure within the wall itself must 

 vary from the inner to the outer wall. Even without 

 this type of analysis we may use the device of using 

 an "average tension" in the wall, to allow the simpler 

 form of the law to be used. Take the case of a thick- 

 walled cylindrical vessel of inside radius r; and out- 

 side radius r„. Instead of using P,,,, the intravascular 

 pressure, and Pt, the tissue pressure outside, we may 

 substitute Ptm for the inside pressure and zero for 

 the outside tissue pressure. 



