Biomedical Engineering Reference
In-Depth Information
6.3 Bone Remodeling Formulation
In this section, formulations presented in references 2 and 21 are briefly
reviewed. In the model shown in Figure 6.1, cellular interactions are carried
out via activation of cell receptors. The receptors either bind molecules
secreted by other cell types called paracrine, or with molecules secreted by
the same cell called autocrine, or with other transmembrane molecules via
direct cell-to-cell contacts. The different cell types represented in the model
respond to the activation of their receptors by producing new molecules,
differentiating, or dying [21]. The mathematical formulation of the model is
primarily influenced by physiological events involving receptor binding and
intracellular signaling modeling [41,42].
Without considering the osteoblastic interactions, the reaction scheme of
the binding of PTH with its receptor is represented as follows:
p
k
p
5
PP
+
PP
(6.1)
r
r
d
p
k
6
where p p and d p are abbreviations for PTH production and destruction fluxes.
Applying the law of mass action [21] to reaction equation (6.1), the follow-
ing ordinary differential equations used to describe the reactions of recep-
tors and corresponding ligands, including PTH ( P ) with its receptor ( P r ), are
obtained:
dP
dt
(
)
)
(
(
)
P
=++
SI
kP PkRPPP BR kP
·
• −
− •⋅
⋅ +−⋅
(6.2)
p
P
6
r
5
T
r
P
(
)
dP P
dt
(
)
r
P
=
kR
− •⋅ −⋅
PPPkPP
(6.3)
5
T
r
6
r
where the small dot stands for the multiplication and the large dot above
represents a receptor-ligand complex. For example, P r P is the complex
formed by PTH and its receptor. The meaning and values of the parameters
S p , I p , k 5 , k 6 , and k p can be found in Table 6.1.
It should be mentioned that, depending on the profile of administration,
I p may be constant or may be time dependent. R P is the number of PTH
receptors per cell. This number is supposed to be constant, implying that no
significant synthesis, degradation, internalization, or recycling of receptors
occurs over the time span for which the model applies. B and R are the
concentrations of AOBs and ROBs, respectively. The PTH binding reaction
(Equation 6.1) equilibrates much more rapidly than the time it takes for the
cell populations of the model to change noticeably. That is also the case for
all other binding reactions of the model. Consequently, only the steady states
of these molecular events really alter the cell dynamics in the model.
 
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