Translational Research: Multi-Scale Models of the Pulmonary Circulation in Health and Disease - Multiscale Computer Modeling in Biomechanics and Biomedical Engineering - page 265

of blood flow, which for analyses of perfusion distribution is a reasonable
approach [ 39 ]. However, there are conditions under which the pulsatility of blood
flow may be critical, and the assumption of laminar flow in the pulmonary blood
vessels may not be appropriate in the largest vessels under certain pathologies.
Another class of pulmonary blood flow models has emerged in recent years, which
is more inclusive of the fluid-dynamic and geometric details of the largest blood
vessels, and simplified at smaller scales (beyond 5 or 6 generations). This approach
has been led by the group of Taylor et al. (e.g. [ 57 ]) who solved a one-dimensional
time dependent model for blood flow in the largest pulmonary vessels, assuming
that the flow profile was parabolic. They used the same elasticity law given in
Eq. 3 and solved the following partial differential equations representing conser-
vation of mass and momentum

þ s
q

2 Q
oz 2

Q 2
s

o Q

ot þ 4

o
oz

o p

dz ¼ 8plQ

þ l
q

o

ð 7a Þ

;

3 qs

o s

ot þ o Q

oz ¼ 0 ;

ð 7b Þ

where t is time, z is axial distance, s is vessel cross-sectional area (s = pD 2 /4),
and q is the density of blood. The flow equations were solved using a finite
element scheme (a Galerkin/least squares stabilization in space and a Galerkin
method in time), with a periodic flow waveform in the main pulmonary artery and
a vascular impedance boundary condition at all terminal vessels (in this case the
segmental vessels). Similar methods were followed by Clipp and Steele in a study
of blood flow in the lamb vasculature [ 58 ], but this study included a gravitational
influence depending in which 'zone' a lumped parameter group of arteries resided.
This type of model provides important steps toward modeling the temporal
interaction between pulmonary blood vessels and airways. However, like the
steady-state models described above, it lacks a description of mechanical and
fluid-dynamic features of the pulmonary microcirculation. As it is at this micro-
scale that gas exchange and drug delivery will occur, and this is a scale that is
hard to investigate in the in vivo environment, we now turn to discussion of how
computational modeling has been used to explore function in the pulmonary
capillary beds.

3 The Microcirculation

In a typical human lung, the branching airway network terminates in an astounding
*480 million alveoli, with a measured range (across six adult subjects) of 274-
790 million [ 59 ]. Wrapped over the surface of each of these alveoli is a dense
network of pulmonary capillaries; a total number of 280 billion has been esti-
mated. This vast number of vessels provides a gas exchange surface area of about