Biomedical Engineering Reference
In-Depth Information
(a)
(b)
(c)
(d)
Figure 1.6
Ossification of a long bone:
(a) hyaline cartilage model with surface
ossification only; (b) development of tra-
becular bone network; (c) development of
medullary cavity; (d) after achievement of
bone maturity, only articular cartilage and
epiphyseal plate are left from the cartilage
tissue, while the medullary cavity is en-
larged. Cartilage part is denoted by dots
and the bone part is denoted by black color.
After [66].
Remarkable symmetry characterizes crystals and organic forms, because both are
subdued primarily to the topological laws of close packing by the Descartes-Euler
theorem.
1.1.3
Planarity of Biological Structures
The development of living world is accomplished in two-dimensional structures.
Such structures are easily accessible to environment and external influence. But it
seems that the fundamental reason is geometrical topology. The Descartes-Euler
theorem on polyhedra in three-dimensional space assures that, for a polyhedron,
the following relation among the number of vertices
N
V
,numberofedges
N
E
,and
number of faces
N
F
is satisfied
N
V
2.
In two dimensions, this relation becomes more sharp, as only two independent
numbers are left. Therefore, all processes with phase transformation are accom-
plished more easily in two dimensions. The Descartes-Euler Law forces the
compensation of positive and negative curvatures in planar tissue: 3
−
N
E
+
N
F
=
×
p
3
+
2
×
+
=
+
×
+
×
+
−
×
>
p
4
9where
pN
are
percentages of
N
-sided cells and each cell gives 6-
N
units of curvature.
p
5
p
7
2
p
8
3
p
9
sum of (
N
6)
pN
for
N
