Biomedical Engineering Reference
In-Depth Information
FIGURE 5.14
Flow rate at a bifurcation would be divided
into two daughter branches. The geometry (angles and radii)
of the bifurcation determines the division of the flow within
the two daughter branches. By assuming that the work of
supplying tissue with oxygen for each vessel should be mini-
mized, one can derive a relationship between the vessels.
B
Q
2
Q
1
O
θ
A
φ
C
Q
3
changes in the blood vessel radius (Hagen-Poiseuille equation). Let us take the simplest
case, where one vessel branches into two daughter vessels (
Figure 5.14
). The flow in the
main branch has to be divided into the two daughter branches. The question remains,
how is the bifurcation designed to optimize the amount of work that the blood vessel does
in order to supply the tissue with sufficient blood (distance of tissue from each capillary),
compared to the rate of energy used by the tissue?
The first and easiest way to optimize the delivery of blood through this bifurcation is
for points
A
,
B
, and
C
to all lie within one plane. Therefore, the blood vessels between the
points could potentially be straight, thus minimizing the length of the vessel (however,
realistically points
A
,
B
,
C
, and
O
would not be in the same plane). What is also known is
that the lengths of each vessel are dependent on the location of the center of the bifurca-
tion point. For instance, by moving the branch point (
O
) to the right (but keeping it in the
same plane, which is represented by the textbook's sheet of paper), branch 1
elo
ngates and
bra
nches 2 and 3 reduce in length. If the branch point is moved along the
OC
line or the
OB
line, various combinations of changes in the vessel lengths/angles can be obtained
and optimized. In a study conducted in 1926, C.D. Murray proposed a mathematical
representation to optimize the work at a bifurcation. These equations became known as
Murray's law for the minimum work, and subsequently, they have been related to the vas-
cular system. They are represented as
r
1
5
r
2
1
r
3
r
1
1
r
2
2
r
3
cos
θ
5
2
r
1
r
2
r
1
2
r
2
1
r
3
ð
5
:
26
Þ
cos
φ
5
2
r
1
r
3
r
1
2
r
2
2
r
3
cos
ðθ
1
φÞ
5
2
r
2
r
3
These formulas have been validated with experimental studies, and the fit is exceptionally
well. Now the important point to remember is, although we can optimize the work needed
throughout a bifurcation, there are still many fluid dynamics principles that are not optimized.
Example
Calculate the radius of one daughter branch knowing that the radius of the parent vessel is
175
μ
m and the radius of the other daughter branch is 125
μ
m. Draw the bifurcation, to scale,
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