Biomedical Engineering Reference
In-Depth Information
A OBA is the rate of elimination of active OBA
A OCA is the rate of elimination of active OCA
A OST is the apoptosis rate of active OST
All the constants and their descriptions used here can be found in Table 7.1.
Equations (7.6)-(7.9) represent cell balance equations where changes of each
cell population (i.e., OBP, OBA, OST, and OCA) are caused by the addi-
tion and removal of cells of the respective lineage. Addition of cells occurs
naturally by proliferation and differentiation of precursor cells, and elimi-
nation of cells is caused by apoptosis (or by external extraction of cells).
Differentiation and apoptosis are regulated by several activator and repres-
sor functions—for example, TGF-β binding to its receptors on uncommitted
osteoblast progenitors promoting their differentiation Π β
T
, or TGF-β
binding to its receptors on osteoblast precursor cells inhibiting their differ-
entiation Π β
(
),
actOBU
T
, .
Note that in this hierarchical model there are two different time scales.
A short time scale, which is less than 12 hours, is required to describe the
production of NO and PGE 2 , and a longer time scale of several months (up
to 12 months) is needed to capture the effects of these two factors and others
on bones.
Bone matrix is the largest source of TGF-β in the body [52]. Indeed, the
content of TGF-β in dried bone powder is approximately 1000-fold greater
than the levels required for osteoblastic stimulation [53]. TGF-β, as well as
other growth factors and specific components embedded in the bone matrix,
is released by osteoclasts during bone resorption [54]. The effect of TGF-β on
osteoblasts is bidirectional, depending upon the state of maturation of the
osteoblasts [23].
On the one hand, TGF-β has the potential to stimulate osteoblast recruit-
ment, migration, and proliferation of osteoblast precursors (meaning OBPs
in our model) [54]. On the other hand, TGF-β inhibits terminal osteoblastic
differentiation into OBAs [55]. Based on the preceding discussion, the release
rate of TGF-β can be evaluated as follows [3]. Considering the fact that the
osteoblast lineage originates from a large population of OBU, the progenitors
express a specific TGF-β receptor that, once activated, leads to differentiation
of those progenitors into OBP [3]. In the model presented in Pivonka et al. [3],
they assumed that the release rate of TGF-β is proportional to OCA:
(
)
rep
OBP
dT
dt
()
β =α⋅
KOCA
DT S
⋅β+
T
( 7.10 )
res
T
β
β
where S T β is a source/sink term for TGF-β, α is a proportionality constant
expressing the TGF-β content stored in the bone volume, K res is the relative
bone resorption rate, and
D T is a constant degradation rate.
In the model described here, it is assumed that the release rate of TGF-β
from the bone matrix is constant, and the binding of TGF-β to its receptors is
β
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