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Bioimpedance methods 155

The electrical conductivity of the blood suggested by Gaw et al. (2008) is
1 2 H
σ b 5 σ pl ; ð5:15Þ
1 1 C 2 1ÞH
ð
where σ pl and σ b are plasma and blood conductivities, respectively, and H is the
hematocrit. Equation (5.15) introduces a nondimensional aspect factor, C. In the
round duct model, C is a function of the tube radius, r 0
Cðr 0 Þ 5 f ðr 0 ÞUC b 1 ½1 2 f ðr 0 ފUC r ; ð5:16Þ

C r 5 C a 1 2C b Þ=3; C a 5 1=M; C b 5 Cr 0 ðÞ 5 2= 2 2 Mð Þ: ð5:17Þ
ð
In the above, f(r 0 ) is the orientation rate of the RBCs, r 0 is the tube radius, and M
is a function the RBCs shape, presumed ellipsoids with a and b (a , b) axes

3
M 5 cosϕU ϕ 2 sin2ϕÞ=sin ϕ; cosϕ 5 a=b: ð5:18Þ
ð
Setting the average value a/b 5 0.38, used for RBCs, yields the simplified expres-
sion M a/b. An expression that approximates the orientation rate is

21
n θ ðÞ
r
frðÞ 5 0 ; ð5:19Þ
21 21
r
r
n 0 θ ðÞ 1 θ ðÞ
d 0
with r is the radius of the cylindrical duct, n is the volume density of the RBCs frac-
tion that are stably aligned with the flow, n 0 is the volume density of RBCs, θ 0 is the
cell orientation from random to lined up with the flow time constant, and θ d is the
cells randomization (or cell disorientation) time constant. In our model r 5 r 0 ,
Morega et al. (2012, 2016), θ 0 scales inverse proportionally with the shear rate, and θ d
scales inverse proportionally with the inverse of the square root of the shear rate.

Hemodynamic of larger vessels
The RBCs deformation is due to shear stress. In round tubes fully developed flows,
2
the average friction factor (Bejan, 1993) is a measure of the shear rate, τ w [N/m ]

U du
4η 5 τ w 5 μ 2 ; ð5:20Þ
r 0 dr r5r 0
where u is the velocity, U the average velocity, and η the dynamic viscosity.
These important results may not be applied directly because, when more realistically
rendered through reconstruction, the aorta departs from the tube geometry. To circumvent
this difficulty, an aorta-equivalent round tube is introduced. The cylinder radius r 0 is com-
puted out of the average cross-section area of the aorta segment considered in the model.