Page 300 - Advances in Biomechanics and Tissue Regeneration

298 15. COMPUTATIONAL SIMULATION OF CELL BEHAVIOR FOR TISSUE REGENERATION

has a normal unit vector n in the direction of e pol ,and sðX Þ is a position vector of an arbitrary node located on ∂Ω (see
0
0
Fig. 15.7). Hence, the projection of s in the direction of n can be defined by
δ ¼ n s (15.30)
Thereby, nodes with negative δ are located on the rear cell surface ∂Ω while nodes with positive δ belong to the front
0

+
+
cell ∂Ω , satisfying ∂Ω ¼ ∂Ω [∂Ω .
0
0
0
0

15.2.6 Cell Migration, Considering Constant Cell Shape
When we are interested in some cell behavior such as cell migration, differentiation, proliferation, and/or apoptosis,
it is more efficient to consider a constant cell shape during cell activities. Therefore, at each time step, the cell displace-
ment is calculated by Eq. (15.27) and the new cell center coordinates are calculated. When the cell displacement is high
enough (i.e., if it is greater than the length of one element of the working domain), the cell centroid is moved to the
center of the next neighbor element (1 of the 24 existing neighboring elements). Otherwise, the cell will keep its position
until the accumulation of the calculated displacement is equal to or more than one element size.

15.2.7 Cell Migration, Considering Cell Shape Change and Remodeling
During cell migration, the cell extends a protrusion from the membrane vertex whose position vector is approxi-
mately in the direction of cell polarization. On the contrary, it retracts the trailing end from the surface vertex whose
position vector is totally in the opposite direction of cell polarization. Thus, the maximum value of δ (Eq. 15.30) delivers
the surface node located on ∂Ω 0 + from which the cell must be extended while the minimum value of δ represents the
ex
surface node located on ∂Ω from which the cell must be retracted. Assume e 2 Ω is the finite element that the surface
0

re
node with the maximum δ belongs to it, and e 2 Ω is the finite element that the surface node with the minimum value
0
re
of δ belongs to it. To update the cell shape changes during migration, simply, e is moved from the Ω domain to the Ω
0
domain; in contrast, e ex is eliminated from the Ω domain and is included in the Ω domain [126, 127].
0
Note that, based on [128], the cell is prevented from obtaining an infinitely thin shape during migration, considering
that the cell can extend approximately 10% of its whole volume as pseudopodia.
It is interesting to evaluate the cell elongation, E elong , and the cell morphological index (CMI) during its migration.
Therefore, they can be estimated by
p ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
l min l med
(15.31)
E elong ¼ 1
l max
(15.32)
SðtÞ
CMIðtÞ¼
S in
where S in denotes the initial area of the cell membrane (spherical cell shape).
15.2.8 MSC Differentiation and Apoptosis
Many experimental studies reported that sensing of the mechanical properties of the ECM is essential for different
cell types to undergo differentiation, proliferation, and apoptosis [85, 129–131]. For instance, a determined deforma-
tion experienced by an MSC stimulates its differentiation into a specific cell type, which is due to the differences in their
tissue of origin [91, 132, 133]. Besides, the magnitude and duration of the mechanical signal sensed by the cell and the
degree of preconditioning are also considered important factors. Experimental works demonstrate that a typical cell
may undergo apoptosis when it senses a high deformation [129, 131]. For instance, Kearney et al. [131] demonstrated
that specific tensile strain applied on MSCs can trigger cell apoptosis, mediated by the cell internal strain. In vitro
experiments showed that mechanical injury beyond 20% of the tensile cyclic strain causes significant neuronal cell
death [129] and a 10% tensile cyclic strain imposed continuously in a 2D MSC culture leads to a considerable apoptosis.
Guided by well-known experimental works [85, 129–131], it is here assumed that an MSC, m, is able to differentiate
into a certain cell type i, i 2{s, c, l}, where s, c, and l represent osteoblast, chondrocyte, and neuroblast lineage speci-
fications [134, 135]. Mechanoregulation of cell differentiation is presented by cell internal deformation (the mechanical
signal) in the cell polarization direction. Therefore, the deformation of each node located on the cell surface in its polar-
ization direction can be obtained by

II. MECHANOBIOLOGY AND TISSUE REGENERATION