574 
HEMODYNAMICS 
radius (R) to thickness (h) is around lO.^ Typi- 
cal blood vessel values of this ratio, at physio- 
logic strains, are : 8 to 10 for MDTA and CA, 
and 6 to 8 for LCCA. The degree of error as- 
sociated with this assumption cannot be easily 
evaluated. 
Assuming the vessel to be a uniform cylinder 
implies that the diameter and thickness remain 
constant with length. The CA and LCCA seg- 
ments approximated a cylinder satisfactorily. 
The descending aorta, however, tapers and the 
difference in end diameters of the 8 cm long 
segment was 1 to 2 mm. Thus, for a cylindrical 
blood vessel it is convenient to use the cylindri- 
cal coordinate system (r, 9, z) as a frame of 
reference. That is, the z coordinate corresponds 
to the center line of the lumen, the r coordinate 
to a vessel radius, and the 6 coordinate to the 
vessel circumference. 
The linearity assumption is based on the con- 
sideration that incremental strain perturbations 
around the physiologic state of strain produce 
linear responses provided these incremental 
strains remain within 4% of the initial strain. 
That is, within this 4% limit, the radius-pres- 
sure-longitudinal force relations and the length- 
pressure-longitudinal force relations are linear. 
Patel and Fry ^ have shown that under phys- 
iologic loading, the shearing strains can be neg- 
lected. That is, the blood vessel may be treated 
as a curvilinearly orthotropic tube. This as- 
sumption reduces the number of independent 
constitutive parameters from 21 to 6, thus sim- 
plifying the analysis. 
The homogeneity assumption implies that the 
mechanical properties over the vessel segment 
are uniform. Histologically, homogeneity is 
more justified for the circumferential and longi- 
tudinal directions than for the radial directions. 
Nevertheless, this assumption imposes a restric- 
tion on the results in that they represent aver- 
age properties. 
The final assumption of incompressibility has 
been justified by Carew, Vaishnav and Patel.* 
Incompressibility implies that when a vessel 
undergoes a change in strain, the volume of the 
vessel wall remains constant. 
The external forces acting on the blood vessel 
segment are the intraluminal pressure (p) and 
an additional longitudinal force (F). With this 
kind of loading, and because the segment is 
homogeneous and a thin-walled cylinder, the 
only non-zero stresses are the normal stresses 
Se, and Sr along the coordinate lines 0, z and 
r, respectively, and are given by 
2\ h 
— P 
1 + 
27rRh 
(la) 
(lb) 
(Ic) 
Equations (la) and (lb) result from equilib- 
rium considerations, and equation (Ic) equates 
the radial stress to the average of the pressure 
which is — p on the inner surface and zero on 
the outer surface. 
The state of strain corresponding to the 
above state of stress is described by the exten- 
sion ratios Kg, K and in the 9, z and r direc- 
tions respectively. Because of orthotropy (3), 
under the given state of stress, there are no 
shearing strains. These ratios are given by 
R 
Ro 
Lo 
: ho 
(2a) 
(2b) 
(2c) 
where Ro, Lo and ho are the dimensions of mid- 
wall radius, segment length and vessel thick- 
ness, respectively, when p = F = 0 (unstressed 
state). L is the resulting length of the segment 
when the forces p and F are applied. Thus, the 
extension ratio is the ratio of a stressed dimen- 
sion value to its unstressed value. Since the 
vessel wall is incompressible, the stretches 
must satisfy the condition 
X(l\Xr — 1 
(3) 
Now suppose the segment is subjected to a 
small sinusoidally fluctuating pressure incre- 
ment (Ap) and longitudinal force increment 
