Biomedical Engineering Reference
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volume meshes of the lung using the fitting methods described by Fernandez et al.
[ 53 ]. The lung-air matrix within this volume was assumed to be a compressible
non-linearly elastic continuum with homogeneous and isotropic material proper-
ties [ 54 ]. Stress and strain in the tissue were related by the strain energy density
function
W ¼ c
2 e ð aJ 1 þ bJ 2 Þ ;
ð 4 Þ
where W is the strain energy, J 1 and J 2 are the first and second invariants of the
Green strain tensor, and a, b and c are constants with values set such that a uniform
expansion of tissue from a theoretical stress and strain reference state (half of
functional residual capacity—FRC) resulted in physiological expansion pressures
of *5 cmH 2 O at FRC and *25 cmH 2 O at total lung capacity (TLC) [ 54 ]. Pre-
dictions of lung tissue density distribution (local expansion) in the supine posture
were validated against MDCT data for the subjects that were studied. Once stress
and strain have been calculated, regional elastic recoil pressure (P e ) can be esti-
mated using
P e ¼ ce c
2k ð 3a þ b Þð k 2 1 Þ ;
ð 5 Þ
where k is the isotropic stretch from reference volume to the lung volume of
interest, and
c ¼ 3
4 ð 3a þ b Þð k 2 1 Þ 2 :
ð 6 Þ
The tethering pressure acting on a vessel or airway is then estimated locally as
being equal and opposite to local P e [ 29 , 55 ]. Burrowes and Tawhai [ 29 ] used
predictions of local tethering pressure and tissue deformation coupled to the model
of the pulmonary macrocirculation described previously to distinguish the separate
contributions of arterial vascular geometry and gravitational tissue deformation to
perfusion distribution. This is an example of the strength of computational mod-
eling in comparison to experiment: the ability to 'switch-on-and-off' features of a
biological system and analyze their relative or compound effect on a phenomenon.
In this case, the model of Burrowes and Tawhai showed that the deformation of
lung tissue contributes significantly to observed gravitational gradients of blood
flow (removal of this effect halved the predicted gradient), consistent with the
concept of the lung behaving as a 'Slinky TM '[ 56 ]. However it was acknowledged
that the model was still lacking a description of the capillary vessels (the micro-
circulation), whose external pressure is air, rather than tissue tethering pressure
[ 29 ]. A multi-scale approach would be needed to address this limitation of the
model (see Sect. 4 ).
While the models described above incorporate a large arterial domain, they
must take a simplified approach to the fluid dynamics of pulmonary blood flow.
The use of the Poiseuille equation effectively provides a time-averaged description
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