Biomedical Engineering Reference
In-Depth Information
3. Each set of seed points is split in two (by the splitting plane) to define two new
seed point sets.
4. The center of mass of the two new seed point sets is calculated and two vessels
are generated along the vectors that run between the parent vessel and the new
centers of mass. The length of the new vessels is defined to be 40 % of the
distance between the end of the parent vessel and the center of mass.
5. If the new vessel is less than a pre-defined length (typical the length of a
terminal bronchiole or the arteries and veins accompanying it), or there is only
one seed point in its set, the vessel is defined as a terminal vessel and the seed
point closest to that vessel is removed from the set of seed points.
Classifying each vessel in a generated tree by order allows a convenient defi-
nition of the caliber of each vessel. Morphometric studies of pulmonary vessels
have shown a linear relationship between the logarithm of vessel diameter and the
Strahler order of the vessel [
37
,
44
,
45
]. The Strahler ordering system defines
terminal branches (in this case the terminal bronchioles of the airway tree and the
vessels accompanying them) as order 1. Then, the order of the parent branch
proximal to two daughter vessels is defined as the highest of the orders of the
daughter branches if these orders are different, or the order of the daughter
branches plus one if the order of the daughter branches is the same. Then,
logD
ð
x
Þ¼ð
x
N
Þ
logR
d
þ
logD
N
;
ð
1
Þ
where D(x) is the diameter of a vessel/airway of Strahler order x, N is the total
number of Strahler orders in the tree, R
d
is the Strahler diameter ratio (effectively
the rate of decrease of vessel/airway diameter), and D
N
is the diameter of the
vessel/airway of the highest order in the tree. The value of D
N
can be estimated
from morphometric parameters or from imaging data. The geometric model
developed by Burrowes et al. [
22
] matched well with anatomical data from cast
based studies [
37
,
44
].
By assuming that the blood vessels are rigid and adopting a steady (time-
averaged) laminar flow, Burrowes et al. [
22
] predicted the blood flow and pressure
distribution through the isolated arterial tree. Equations describing Poiseuille
resistance with a gravitational dependence (dominating in the axial direction) were
solved along with flow conservation equations at bifurcations. That is,
;
Q
¼
pD
4
128l
DP
L
þ
q gcos h
ð
2a
Þ
Q
P
¼
Q
D1
þ
Q
D2
;
ð
2b
Þ
where Q is the rate of blood flow through a vessel, D is the diameter of the vessel,
l is the viscosity of blood, DP is the pressure drop through the vessel, L is the
length of the vessel, q is the density of blood, g is gravitational acceleration, h is
the angle the vessel makes with the direction of gravity, Q
P
is the flow through a
parent vessel, and Q
D1
and Q
D2
are the flows through each of the daughter vessels
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