Biomedical Engineering Reference
In-Depth Information
3.
BIOMEDICAL APPLICATIONS
Many medical studies are performed on animals other than humans, espe-
cially on mice and rats. Understanding how to interpret data from animal studies
and how to compare them with similar conditions in humans requires the use of
scaling relationships. In this section we briefly describe several areas in which
progress has been made in applying the theory of biological scaling to biomedi-
cal problems.
3.1. Ontogenetic Growth and Developmental Times
The theory of biological scaling naturally leads to a general growth equation
applicable to all animals (50,51),
dN
BNB E
dt
=+
c
.
[10]
cc
c
Here,
N
c
is the total number of cells in the organism,
B
c
is the metabolic rate
given to maintenance per cell, and
E
c
is the energy needed to create a fully
grown, new cell. Metabolic energy is transported through the network to cells
where it is used either for maintenance (the first term on the right side of Eq.
[10]), including replacement of cells, or for growth of new cells (the second
term on the right side of Eq. [10]). Substituting
N
c
=
m
/
m
c
into Eq. [10], where
m
is the ontogenetic mass at time
t
after birth and
m
c
is the average cell mass, gives
an equation for the rate of growth of an organism:
¬
¬
dm
Bm
B
-
-
=
0
-
m
3/4
®
,
-
m
[11]
c
c
-
-
-
-
-
-
dt
E
E
®
c
c
where
B
0
is a taxon-dependent normalization constant for the scaling of meta-
bolic rate. The parameters of the resulting growth equation are determined solely
by fundamental cellular properties, such as their metabolic rate and the energy
required to create them. The model gives a natural explanation for why animals
stop growing: the number of cells supplied (
N
c
m
) scales faster than the num-
ber of supply units (since
B
N
N
m
3/4
) and leads to formulae for the asymptotic
mass of the organism:
M
= (
B
0
m
c
/
B
c
)
4
. Eq. [11] can be solved analytically to de-
termine
m
(
t
), leading to a classic sigmoidal curve. From the ensuing equations, a
universal scaling curve for growth is derived that is well fit by data from many
different organisms (Figure 4). Ontogenetic development is therefore a universal
phenomenon determined by basic cellular properties. Furthermore, this theory
