Biomedical Engineering Reference
In-Depth Information
where we have divided the sum into regions according to the different values of
C; C
<
is for the area-preserving branching that minimizes cardiac output for pul-
satile flow in large vessels, and C
>
is for the area-increasing branching that
minimizes the energy loss for Poiseuille flow in small vessels. Further,
V
N
is the
volume of a single capillary, and
k
is the number of levels from the transition
level between the two regions to the capillaries, which is the same for all mam-
mals, i.e., it does not scale with organism size (12). Substituting the scaling rela-
tionships for H, C
<
, and C
>
into Eq. [5], we find that the first term in the square
brackets scales as
n
N
/3
and that the second term is independent of
N
. Thus, for
N
>> 1 the first term dominates, and the leading-order behavior for the blood
scales as
n
4
N
/3
V
N
. By assumption (iii), capillaries are invariant units, so
V
N
is in-
dependent of
M
, which implies
V
b
n
4
N
/3
. Also by assumption (iii), the metabolic
rate per capillary,
B
N
, is invariant, which implies
B
=
n
N
B
N
n
N
V
b
3/4
. Since en-
ergy minimization requires
V
b
M
, this implies
B
M
3/4
. This matches the well-
known empirical result.
Many other relations follow from this theory, including the determination of
the scaling of the radii and lengths of all vessels and the blood flow and pulse
rate in each of them. Furthermore, it shows that the 3/4 exponent is only an ap-
proximation, and that deviations can be expected for small mammals where the
number of vessels that can support a pulse is small compared to that of a large
mammal (13). Such deviations are indeed suggested by the data (3,42).
Metabolic energy is conserved as it flows through cells and mitochondria,
which may possess hierarchical networks of cellular transport and chemical re-
actions respectively. Surprisingly little is known about intracellular transport
networks, and an important challenge, both theoretical and experimental, is the
construction of realistic models. The continuity of flow of metabolic energy
through this series of sequential networks, a "hierarchy of hierarchies," imposes
boundary conditions between each level that lead to constraints on densities of
mitochondria and respiratory complexes (both considered invariant terminal
units at their appropriate level) (13). This explains why there are typically a few
hundred mitochondria per cell in vivo, but several thousand in vitro, and why
there are several thousand respiratory complexes per mitochondrion (and not ten
or ten million).
2.2. Body Temperature Dependence
Rates of chemical reactions,
R
, depend crucially upon temperature (29,30),
as first demonstrated by Arrhenius (43) with his famous equation:
l
dR
E
=
.
[6]
dT
kT
2
