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Cell Adhesion and Detachment in Shear Flow 219 Table 1. Simulation parameters. Parameters Definition simulation value R cell radius 4.0–5.0 µm N r receptor number 780 N L ligand density 10 6 –10 8 /cm λ equilibrium bond length 0.2 µm σ spring constant 0.016 dyne/cm µ viscosity 0.01–0.014 g/cm·s ρ fluid density 1.0 g/cm 2 U max shear rate 20–80/s H c cut-off length 0.4 µm T temperature 310 K kf 0 forward reaction rate 100.0/s kr 0 reverse reaction rate 10.0/s r 0 reactive compliance 0.02 µm with the long semi-axis r a equal to 0.5 and the short semi-axis r b equal to 0.4. The velocity boundary conditions are as follows: a given constant on the top boundary, zero on the bottom boundary, and periodicity in the horizontal direction. The fluid and cells are at rest and the cells are in the contact region initially (see Fig. 6(a)). We assume that the densities of fluid and cells are 1 g/cm 3 . The mesh size h for the flow field is 1/48 and the time step △t is 0.001 (unit: 0.1 second). The parameters used in the simulations are given in Table 1. We observed the simulations up to t = 100 (10 s), long enough for the flow to be fully developed. The simulations were conducted at different shear rates and dynamical viscosities, and the results are summarized in Table 2. From the table, we can see, no cells were detached from the wall by the observed time when the shear rate is 20/s for the dynamical viscosity of 0.01 g/cm-s; while the detachment percentage increases from 10% to 40% when the shear rate increases from 30/s to 40/s. All the 20 cells were detached from the wall when the shear rate is greater than 80/s. Figure 5 shows the effect of shear rate on cell detachment. This observation qualitatively agrees with the in vitro experiment [SKE + 99]. We also observed that the detachment percentage increases from 10% to 35% when the dynamical viscosity is increased from 0.01 to 0.014 (g/cm-s). Figure 6 shows the snapshots of positions of 20 cells at t =0,5,5.35, 6.06, 9.49, and 10 (s), for the simulation with the dynamical viscosity equal to 0.01 (g/cm-s) and the shear rate of 30 (/s). The snapshots quite clearly depict the process of cell detachment from the wall. All the cells adhered to the wall at t = 5 s; one cell was about to be detached at t =5.35 s; one cell was completely detached from the layer at t =6.06 s. We found that during the early stage of detachment the percentage of the detached cells is highly
220 J. Hao et al. Table 2. The calculated detachment percentages at t =10s. Dynamical viscosity (g/cm-s) Shear rate (/s) Detachment (%) 0.01 20 0 0.01 30 10 0.014 30 35 0.01 40 40 0.01 80 100 100 90 percentage of detached cell (%) 80 70 60 50 40 30 20 10 0 20 30 40 50 60 70 80 shear rate ( /s) Fig. 5. The effect of shear rate on cell detachment (viscosity= 0.01 g/cm-s). linearly correlated with the observed time. This observation was also found in in vitro experiments [SBBR + 02]. We have used our models and algorithms to simulate adhesion and detachment of chondrocytes. The simulations successfully depicted the process of cell detachment from the wall. The numerical results qualitatively agree with the experiments in the literature. Since there are few publications on modeling chondrocytes for this problem, our modeling and simulation are quite preliminary. More work is needed in modeling and in investigating parameters for the cell adhesion at different stages as discussed in [ZBCAG04]. Acknowledgement. We acknowledge the helpful comments and suggestions of R. Bai, S. Canic, E. J. Dean, R. Glowinski, J. He, H. H. Hu, P. Y. Huang, G. P. Galdi, D. D. Joseph, and Y. Kuznetsov. We acknowledge also the support of NSF (grants ECS-9527123, CTS-9873236, DMS-9973318, CCR-9902035, DMS-0209066, DMS-0443826) and DOE/LASCI (grant R71700K-292-000-99).
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Partial Differential Equations
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Partial Differential Equations Mode
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Dedicated to Olivier Pironneau
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VIII Preface computers has been at
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Contents List of Contributors .....
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List of Contributors Yves Achdou UF
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List of Contributors XV Claude Le B
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Discontinuous Galerkin Methods Vive
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Discontinuous Galerkin Methods 5 2
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Discontinuous Galerkin Methods 7 g
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Discontinuous Galerkin Methods 9 (
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Discontinuous Galerkin Methods 15 W
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Let a h and b h denote the bilinear
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Discontinuous Galerkin Methods 21 l
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Table 1. Primal DG for transport Di
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Discontinuous Galerkin Methods 25 [
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Mixed Finite Element Methods on Pol
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Mixed FE Methods on Polyhedral Mesh
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4 Hybridization and Condensation Mi
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is symmetric and positive definite,
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Domain Decomposition and Electronic
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