8 m 



HANDBOOK OF PHYSIOLOGY 



CIRCULATION II 



but the value for Sa would not be the same as that 

 for S . It the changes are very small, 



and 



„ - Ilia 



</ dr 



(4) 



Next, let us express the stress in terms of pressure. 

 This is usually clone by taking the pressure (P) as 

 equal to T/r (21). As pointed out by Frank (29), and 

 stressed recently by Peterson et al. (91), when the wall 

 is relatively thick in relation to the internal radius, 

 the more proper expression would be P = Ta/r, 

 where a is wall thickness. With an imposed stretch: 

 P + AP = a (T + AT), '(r + Ar). When P = o 

 and T = o, this becomes AP{r„ + Ar)/a = AT. 

 Substituting for AT in equation 1 : 



__ AP(r„+ Ar)r 

 o a Ar 



(5) 



If Ar and AP are very small, their product will be 

 infinitesimal, so that 



5 - dPr o 



a dr 



And from equations 3 and 4 we obtain: 

 AP(r d + Ar)r d 



a Ar 



and 



dPr/ 



" a dr 



(6) 



(7) 



- (8) 



Now to express the radius change in terms of 

 volume: 



V= Ivr 2 , and V + AV * lir(r + Ar) Z 



' lir(r 2 + 2rAr + Ar 2 ), or 



AV = lrr(2rAr + Ar 2 ), or 

 AV 



Ar 



Iw(2r + Ar) 



Substituting for Ar in the denominator and then for 

 lief- in the numerator of equation 5: 



_ AP(r„ + Ar)r l*-(2r + Ar) 

 o 



a AV 



(9) 



.. 2dPV a r n 

 adV 



(10) 



Substituting for Ar in the denominator of equation 7, 

 and following through as in equations 9 and 10: 



and 



a AV 



2dPV d r d 

 a dV 



(II) 



(12) 



A commonly used, but incomplete, modulus (see 

 equation 7) is: 



. APr, 



5, -- 



d Ar 

 And another (see equation 11) is: 



b < AV 



(13) 



(14) 



The relationship between these moduli could be 

 illustrated by taking a hypothetical tube with an 

 unloaded radius of 10 mm, in which the radius 

 increased 1 mm for each 1 g per cm 2 increase in 

 tension. The stress-strain relation will then be as 

 shown in figure iA. The value for the modulus 

 (equation 1 ) would be 10. The calculated pressure- 

 radius and pressure-volume curves would not be 

 linear (fig. 2P and C). 



A modulus calculated on the basis of a loaded 

 initial radius (equation 3) will increase as rrf increases 

 (table 1). A modulus based on pressure change would 

 give the constant value of 10 if equation 5 is used, but 

 if equation 6 is employed, the S value decreases as 

 the strain becomes larger, so that even a 1 per cent 

 change in strain produces a decreased value. Curi- 

 ously enough, the value of Sd calculated from equa- 

 tion 7 shows a constant value, while that from equa- 

 tion 8 progressively decreases. 



Converting the radius changes to volume increases 

 makes the formulas very cumbersome. Once again, 

 equation 10 gives a changing modulus for S ot as 

 does equation 1 2 for Sj. There is no simple way of 

 converting the moduli values obtained from the 

 different formulas to each other. 



Based on studies with rubber, King (62, 63) 

 introduced another measure of extensibility, /3, which 

 is the ratio of the unstretched length L to the maxi- 



