Page 74 - Computational Modeling in Biomedical Engineering and Medical Physics

Shape and structure morphing of systems with internal flows 61

circulation—from the arteries that convey the deoxygenated blood to the lungs, and
ends with the pulmonary veins that transport back the oxygenated blood to the left side
of the heart—and the systemic circulation—that carries the oxygenated blood to the
muscles, where O 2 is delivered, and the veins that return the deoxygenated blood to the
heart.
The working chambers (left, L; and right, R) that pump the blood from the
core to the capillaries of the pulmonary circulatory system contract simultaneously
in the interval t 1 . The heart muscle contracts both chambers, increasing the pres-
sure of the blood, which returns to the heart, from P 0 (reference) to P 0 1 ΔP,
where

ΔP 5 r s U s 5 r p U p ; ð2:22Þ

U s and U p are the average velocities of the blood ejected from the heart and r s , r p
are the resistances posed by the systemic and pulmonary flow paths, dominated by
the contribution of the smallest blood vessels and capillaries. Consequently, the
vascularized tissues may be homogenized as porous media saturated with fluid,
with laminar Darcy flow (Chapter 1: Physical, Mathematical, and Numerical
Modeling).
Mass conservation law yields

_ m L 5 _m R 5 _m;

where
ρV R;L
_ m R;L 5 5 ρA p;s U p;s ; ð2:23Þ
t 1
and ρ is the blood density, A s and A p are the effective areas of the cross-sectional sur-
faces related to U s and U p . The contractions of both chambers are, hence, the same,
V L 5 V R 5 V.
The per cycle mechanical average power required by the myocardium is

2VΔP 2r s V 2
_ W 5 5 ; ð2:24Þ
t 1 1 t 2 A s t 1 t 1 1 t 2 Þ
ð
where t 2 is the rest time between two consecutive heartbeats. In this time interval O 2
diffuses from the alveoli to the lung capillaries and from the systemic capillaries to the
muscle.
These mass diffusion transfer is governed by the scale relations order jBDΔC=δ,
1=2 , mB DΔC
δB Dt 1 Þ 1=2 At 1 , with t 2 instead of t 1 where A is the area of the surface of
ð
ð Dt 1 Þ
mass transfer for all vessels capillaries, pulmonary, and systemic. The amount of O 2
transferred in the interval t 2 is proportional to t 1=2 , and the average mass transfer rate is