Biomedical Engineering Reference
In-Depth Information
osteoclastogenesis, bone resorption, OPG, RANKL, and M-CSF concentra-
tions in marrow culture. Specifically, in this experiment, the authors used
three different electric field intensities of PEMF (4.8, 8.7, and 12.2 μV/cm)
and observed that the recruitment of osteoclast-like cells was inhibited
by approximately 33% and increased by approximately 10% when electric
field intensities of PEMF were 4.8 and 12.2 μV/cm, respectively. No signifi-
cant differences across all time points were observed compared with the
control group.
8.2.2
Mathematical Model
In the cell population dynamics model presented in Wang and Qin [1], three
cell populations (see osteoblastic and osteoclastic lineages in Figure 7.1) are
considered in model equations including OBP, OBA, and OCA. OBUs and
osteoclastic precursors (OCPs) work as reservoirs whereby the cells differ-
entiate into functional cells such as osteoblasts and osteoclasts, respectively.
Their numbers are much larger than the functional cells OBP, OBA, or OCA.
As a result, OBU and OCP are assigned a very large constant compared with
other cell numbers in the model (i.e., OBU = OCP = 1 × 10
-2
pM).
In a manner similar to the treatment in the previous chapter, the Hill equa-
tions (7.1), (7.3), and (7.4) are again used to describe the activation and repres-
sion of the receptor-ligand interactions.
The equations governing the evolution of the number of osteoblastic and
osteoclastic cells in each maturation stage are simply balance equations
[38], which means that each cell stage is fed by an entering flow and is emp-
tied by the outgoing flow of differentiated or apoptotic cells (see Figure 8.1).
As a result, utilizing Figure 8.1 and based on the formulation in Pivonka
et al. [25], the bone cell population dynamics can be formulated as follows:
dOBP
dt
T
β
T
β
=
D
⋅
BU
⋅Π
−
D
⋅
BP
⋅Π
(8.1)
OBU
OBP
actOBU
,
rep OBP
,
dOBA
dt
T
β
=
D
⋅
BP
⋅Π
−
A
⋅
BA
(8.2)
OBP
OBA
rep OBP
,
dOCA
dt
RL
⋅Π
β
T
=
D
⋅
CP
⋅Π
−
A
⋅
CA
(8.3)
OCP
act OCP
,
OCA
actOCA
,
dBV
dt
()
()
()
()
=⋅
k
BA tOBA
−
0
−⋅
k
CA tOCA
−
0
(8.4)
for
res
where
D
OBU
,
D
OBP
,
D
OCP
,
A
OBA
, and
A
OCA
, are explained after Equation (7.9).
BV
represents bone volume in percentage (%), and
k
for
and
k
res
are the relative
bone formation and bone resorption rates, respectively. The simulation starts
from a so-called “steady state,” whereby
BV
is 100% and
dBV
/
dt
= 0, corre-
spondingly;
OBA
(
t
) is
OBA
(0) and
OCA
(
t
) is
OCA
(0) . Equations (8.1)-(8.3) are
