576 
HEMODYNAMICS 
where dj (i,j = r, d, z) are complex incre- 
mental proportionality constants that relate 
the component of strain along the i direction to 
the corresponding stresses in the three j direc- 
tions. The negative signs for Cij (i j) have 
been chosen for convenience. Inherent to these 
relations is the assumption of linearity within 
the range of the incremental strains. Also, 
note that this formulation is such that it al- 
lows the material to exhibit an incremental 
orthotropic response. 
Assuming that Cij in equations (15-17) obey 
the symmetry conditions 
Czr — Crz ; CzS — C^z > C^B — C 
(18) 
and using equation (7), we have from equa- 
tions (15-17) 
Ceo — Cz(9 — C,.|9 =Czz — Cziy — Cz,- = 
C„ - Cre - Crz = 0 (19) 
which is used to express Cy (i j) in terms of 
(i = j). Thus, 
Czr — V2 (Czz -f Crr — Cffe) 
CrS = V^(C,.,- + Ced — Czz) 
Cfe = V2 (Cw + C^z — C,.,-) 
(20) 
(21) 
(22) 
Finally, using equations (18) and (20-22), 
we can write equations (15-17) in terms of 
Cffff, Czz and Cn.. That is. 
state of incremental strain, around the same 
initial strain, is required to provide two more 
independent complex equations. These two 
states of incremental strains should yield sig- 
nificantly different values of the strain ratios 
— and — . This requirement of two strain 
Cz ei- 
states formed the basis of our experimental 
procedure. It should be noted that we now 
have four complex equations in three complex 
unknowns of which any three equations could 
be used. However, it is also possible using a 
method of least squares to use all four equa- 
tions. We decided to use the latter method. 
When presenting results, complex incremental 
elastic moduli, similar to the more familiar 
Young's modulus, are given instead of dj. 
These moduli are calculated from the complex 
incremental proportionality constants using 
the following general relations: 
e: =- 
E." 
C'- 
(Cii)2 + (Cii)2 
c;; 
(c;i)2+ (c;;)2 
(26) 
where the prime and double prime quantities 
denote the real and imaginary parts, respec- 
tively, of the complex quantities. E' is referred 
to as the storage modulus and E", the loss 
modulus. 
EXPERIMENTAL PROCEDURE 
ee= (P, - i/aPz - i/2Pr)C«« + 
(Pr - P.) Czz + (Pz - Pr) Cr.. (23) 
ez = i/2(Pr-P*) C,, + 
(Pz - I/2P, - l/2Pr)Czz + 
1/2 (P^ - Pr)C„ (24) 
er = 1/2 (P. - P^) Coe + 1/2 (Pe - Pz) Czz + 
(P,- 1/2P, - i/2Pz)C„ (25) 
Only two of these three complex equations are 
independent. Thus, we have three complex un- 
knowns and two complex equations. In order 
to solve for all unknowns another independent 
At attempt will be made to keep the pro- 
cedural description general. If specific detail 
is required, the reader is referred to references 
5 and 6. The vessel to be studied was suffi- 
ciently exposed so that a reasonably uniform 
segment of length, L, could be measured and 
marked. During this time, the aortic pressure 
was continuously monitored. Two cylindrical 
plugs, chosen to snugly fit the vessel lumen 
at each end of the segment, were inserted and 
secured. Figure 1 depicts the experimental 
setup used to study the LCCA and CA. Both of 
these segments were removed from the dog and 
stripped of their surrounding tissues up to the 
adventitia. A hollow metal rod (N) was 
