Partial Differential Equations - Modelling and ... - ResearchGate

212 J. Hao et al.
The backward reaction rate k r in [KH01] is given as follows:
[ ]
k r = kr 0 r0 F
exp ,
k b T
where kr 0 is the reverse reaction rate when the spring length is at its equilibrium
length, r 0 is the reactive compliance, F is the force on the bond and is
equal to σ(|x b |−λ), k b is the Boltzmann constant and T is the temperature.
The ratio of the forward reaction rate and the reverse reaction rate at any
separation distance is given:
k f
= k0 f
k r kr
0 exp
[− σ(|x b|−λ) 2 ]
2k b T
where kf 0 is the forward reaction rate when the spring length is at its equilibrium
length. Then the forward reaction rate in [KH01] takes the form
k f = k 0 f exp [σ(|x b |−λ)(2r 0 − (|x b |−λ))/(2k b T )] .
The strength of the adhesion of each cell (or number of bonds formed via
the above dynamical process) depends on the densities of ligans and receptors
in the contact region between the cell and surface, the area of the contact region,
and two reaction rates. For the hyaluronan-mediated adhesion, the above
dynamical bonding approach is a good model. But for the integrin-mediated
adhesions of chondrocytes reported in [SBBR + 02], we can apply the above
model to form bonds in a probabilistic way with two different considerations:
(1) having larger string constants since focal adhesions and fibrillar adhesions
are much stronger than the hyaluronan-mediated adhesions, (2) after the number
of bonds reaches its plateau, we switch to the deterministic approach to
decide when the bond should be break off by checking whether its length is
longer than a chosen one.
3 A Fictitious Domain Formulation
for the Fluid/Particle Interaction and Its Discretization
3.1 Fictitious Domain Formulation
In this section we briefly discuss a fictitious formulation for the fluid-particle
interaction in shear flow and discretization in space and time developed
[PG02]. Let Ω ⊂ R 2 be a rectangular region (three-dimensional cases have
been discussed in [PG05]). We suppose that Ω is filled with a Newtonian
viscous incompressible fluid (of density ρ f and viscosity µ f ) and contains a
moving neutrally buoyant rigid particle B centered at G = {G 1 ,G 2 } t of density
ρ f (see Fig. 2); the flow is modeled by the Navier–Stokes equations and
the motion of B is described by the Euler–Newton equations. We define