| HOME | HELP | FEEDBACK | SUBSCRIPTIONS | ARCHIVE | SEARCH | TABLE OF CONTENTS |
Department of Radiation Oncology, Massachusetts General Hospital and Harvard Medical School, Boston, Massachusetts
Correspondence: Address reprint requests to Lance L. Munn, Dept. of Radiation Oncology, Mass. General Hospital, 3409 Bldg. 149, Charlestown, MA 02129. Tel.: 617-726-4085; Fax: 617-726-1962; E-mail: lance{at}steele.mgh.harvard.edu.
 |
ABSTRACT |
|---|
 TOP ABSTRACT INTRODUCTIONMETHODSRESULTSDISCUSSIONCONCLUSIONSACKNOWLEDGEMENTSREFERENCES |
|---|
|
INTRODUCTION |
|---|
|
TOPABSTRACTINTRODUCTIONMETHODSRESULTSDISCUSSIONCONCLUSIONSACKNOWLEDGEMENTSREFERENCES |
|---|
), graft rejection, and autoimmune dysfunction rely on leukocyte rolling. Although intense research has characterized the adhesive mechanisms of leukocyte-endothelium interactions, relatively little is known about the mechanical events that cause a cell to begin rolling in the vasculature. Human blood is a concentrated suspension of near-colloidal-size cells in a complex aqueous solution. The red blood cells (RBCs) dominate the cell population on the basis of number and volume concentration. The leukocytes (white blood cells, WBCs) are less easily deformed than red cells and are far fewer in number, with a ratio of about one leukocyte to every 1000 RBCs.
In previous studies, we have shown in vitro (Melder et al., 1995; Munn et al., 1996; Yuan et al., 2001) and in vivo (Melder et al., 2000) that the presence of RBCs greatly enhances leukocyte adhesion to activated endothelium. Mathematical simulations suggest that the mechanism for this involves an increase in leukocyte dispersion and wall contact frequency (Migliorini et al., 2002; Munn et al., 1996). Once rolling is achieved, the RBCs near the plasma rich zone also encourage the continued rolling of the leukocyte, adding a normal force component and torque (Migliorini et al., 2002). But these previous studies were performed in channels, tubes, and vessels of constant diameter, whereas the diameter of a real vessel changes along its length.
The expansion of a capillary into a postcapillary venule is an important example. It has been shown that leukocytes preferentially roll and adhere in postcapillary venules, and that specific adhesive mechanisms are involved (Asako et al., 1992; Bienvenu and Granger, 1993; Johnston et al., 1996; Ley et al., 1993; oude Egbrink et al., 2002; Tsai et al., 1992). But the fluid dynamics of an RBC suspension at a capillary-postcapillary junction are complex, and may also contribute to the abundance of leukocyte-endothelium interactions in these regions.
In capillaries, white blood cells tend to flow with a lower velocity than red blood cells. This is due to the larger volume of the white blood cells, their spherical shape, and the smaller deformation during flow in narrow blood vessels. As a result, red blood cells often accumulate upstream of a white cell in a capillary in single file, whereas downstream of the white cell, a red-cell depleted region is formed. In an interesting large-scale model system, Schmid-Schönbein and co-workers studied this phenomenon and the resulting particle-fluid dynamics at expansion zones, which represented capillary-postcapillary junctions (Schmid-Schönbein et al., 1980). Their results suggest that when a white cell enters an expansion, the trailing red cells pass the slower white cell and thereby displace it away from the vessel axis toward the wall. This RBC-WBC interaction could potentially be responsible for the initiation of cell rolling in postcapillary venules and the propensity for leukocytes to interact in these regions.
It is likely that the hematocrit and RBC aggregation state are important determinants of cell rolling initiation. Pearson and Lipowsky analyzed the effect of RBC aggregation on leukocyte-endothelium (L-E) interactions in mesenteric postcapillary venules, and found enhanced adhesion with larger aggregates and lower flow rates (Pearson and Lipowsky, 2000). But because of the complex fluid dynamics, rigorous analysis of the particle-particle interactions and the forces involved have not been addressed to date.
In this study, we characterize the interactions and estimate the forces involved as RBCs and WBCs interact in capillaries and in postcapillary expansions. Our results suggest that expansion geometry and RBC organization are critical factors that determine the initiation of rolling in postcapillary venules.
|
METHODS |
|---|
|
TOPABSTRACTINTRODUCTIONMETHODSRESULTSDISCUSSIONCONCLUSIONSACKNOWLEDGEMENTSREFERENCES |
|---|
; Qian et al., 1992). Unlike conventional computational methods based on macroscopic continuum equations, the LBM uses the mesoscopic Boltzmann equation to determine macroscopic fluid dynamics. The LBM is flexible, has broad applicability, and may be straightforwardly adapted for parallel computing. Due to the ease of incorporating complex boundary conditions, it has been successfully applied to multiphase and multicomponent fluids, flows through porous media, and solid particle suspensions (for a review, see Chen and Doolen, 1998).
|
 |
) form of the Boltzmann equation for the particle density distribution:
 | (1) |
 |
j is the BGK collision operator, 
is a relaxation time, x is the location of the lattice node, cj is the particle velocity, and fjeq is the equilibrium distribution that is determined by the fluid density, 
, and the momentum, v. For a two-dimensional, 9-speed LB model, fjeq has the following form (see Aidun et al., 1998):
 |
 |
 |
, and momentum, v, using:
 | (2) |
; Qian et al., 1992; Sun, 2000):
 |
 |
is the kinematic viscosity. The Navier-Stokes equations are nonlinear partial differential equations which are much more difficult to solve than the discrete, linear equations of the LBM (Eq. 1).
|
, who modified the original algorithm proposed by Ladd (1994). The method solves the lattice Boltzmann equations on a square lattice with nine directions for the fluid phase (Qian et al., 1992), which is coupled with the Newtonian rotation and translation of solid particles suspended in the fluid through solid-fluid interactions. The cell motion follows Newton's law:
 | (3) |
is the angular velocity, T is the torque, I is inertia, and F is the net force acting on the cell. For RBCs, F consists only of the hydrodynamic force. For leukocytes, F includes not only the hydrodynamic force but also the ligand force FL, and an empirical repulsive van der Waals-like potential interaction Fc which is defined in Eqs. 5 and 6.
At each iteration, the position of the cell is determined by:
 | (4) |
p is the rotational angle of the cell. Equations 3 and 4 are solved using a Runge-Kutta method. The cell's current position and the fluid boundary are then updated.
The repulsive wall interaction is implemented using a Derjaguin approximation (Bongrand and Bell, 1984; Israelachvili, 1985), and prevents the leukocyte from moving into the wall, which would lead to a mathematical singularity. The following force (per unit length) normal to the wall is applied to the center of the leukocyte (Bongrand and Bell, 1984; Israelachvili, 1985):
 | (5) |
is the separation between the surfaces and A is the Hamaker constant. This repulsive van der Waals-like potential exhibits the same behavior as more rigorous expressions that include steric and electrostatic forces (Bongrand and Bell, 1984).
Receptor-ligand model for leukocyte rolling
If the leukocyte approaches within 40 nm of the vessel wall, it can be influenced by stochastic interactions of receptors distributed on the leukocyte with ligands on the wall (Migliorini et al., 2002; see also Fig. 1, this article). The endothelium is assumed to contain a high density of receptor molecules and is uniformly reactive (Bell, 1978; Chang and Hammer, 1996). We use an adhesive dynamics model to describe cell attachment, rolling, and firm adhesion in flow (Hammer and Apte, 1992), which has also been used in similar form by other groups to describe cell-surface interactions (Chang and Hammer, 1996; Dong et al., 1999; Zao et al., 2001). The receptor-ligand bonds are modeled as springs, which provide a force along the bond direction given by:
 | (6) |
is the equilibrium bond length, and 
is the spring constant. The forces contributed by individual ligand-receptor bonds are summed to determine the total bond force and the torque acting on the cell during the period 
t.
If the receptor-ligand distance Yl is less than a critical value Hc (40 nm), a new bond can form with finite probability Pf (Chang et al., 2000; Dong et al., 1999):
 | (7) |
A preexisting bond is broken with probability Pr, which increases with its current load:
 | (8) |
* is the transition state spring constant. The procedure for updating the status of receptor-ligand bonds is as follows: the bond status at time t is checked first for each receptor on the WBC. If the receptor is bound to a ligand on the wall at time t, a random number is generated between 0 and 1. If the random number is inferior to Pr determined by Eq. 8, the receptor will disengage at the next time step; otherwise, it will remain bound. If the receptor is free at time t and the receptor distance from the wall is inferior to Hc, a random number is also generated between 0 and 1. If the random number is inferior to Pf determined by Eq. 7, the receptor will bind to a ligand on the wall at the next time step. Otherwise, it will remain free.
The receptor density correctly influences initial attachment and rolling behavior. For example, increasing the receptor density in Case B2 (described in the Results section) by a factor 4 decreases the rolling velocity by 5.4%. When the receptor density is increased by a factor 10, the rolling velocity decreases dramatically and the WBC actually stops, forming firm adhesion. On the other hand, when the receptor density is decreased by a factor 4 or 10, the rolling velocity increases 7.3% and 9.8%, respectively. But because the current study does not directly address the rolling behavior of the cells after initial attachment, all simulations were performed using a highly reactive surface typical in inflammation to focus on the initial step of rolling.
|
RESULTS |
|---|
|
TOPABSTRACTINTRODUCTIONMETHODSRESULTSDISCUSSIONCONCLUSIONSACKNOWLEDGEMENTSREFERENCES |
|---|
). In vessels with diameter <15 µm, red cells may travel either in single- or multifile, influenced by vessel diameter and tube hematocrit (Skalak and Chien, 1987). In capillaries smaller than the diameter of a RBC, red cells tend to deform in parachute-like shapes and travel in single file groups (Skalak and Brånemark, 1969; Tsukada et al., 2001). As capillary diameter increases, they prefer to travel stacked in multifile (at sufficiently high hematocrit). We first simulated this well-defined case as a test of the model and to provide the initial condition for cells exiting the capillary and entering the postcapillary venule. These simulations were performed for capillaries of 10.2 µm in diameter with three RBCs after a WBC. Because the leukocyte does not contact the wall, no receptor-ligand binding occurs in the capillaries of our simulations.
Case A: leukocyte with three RBCs in a capillary of diameter 10.2 µm
Calculations were carried out using a 600 x 34 lattice grid corresponding to 180 µm x 10.2 µm. The computational domain was folded in the x-direction using a uniform velocity profile µL = 1 mm/s as the inlet boundary condition and a stress-free condition at the tube outlet (Aidun et al., 1998). In essence, the inlet and outlet are at the same location, which remains 250 grids upstream of the cells. With this folded domain technique we can simulate flow in a very long tube using a relatively small number of lattice nodes in the x-direction. The lattice Boltzmann relaxation time was 1, corresponding to a time step of 1.25 x 10-8 s. Initially, a leukocyte and three red cells move in the capillary with velocity µL. The red cells enter the center of the capillary with edge-on orientation. The white cell is one grid-point off-center. The simulation corresponds to an actual time interval of 0.45 s, which is normalized to 1 in the figures.
Fig. 2 shows the initial position of the cells before flow commences and snapshots at normalized time t = 0.111, 0.222, 0.333, 0.667, and 1.0 (note that the x-axis range increases with time). The velocity distribution (represented by small arrows) is parabolic except when disturbed by the cells. As expected from experimental observations, red blood cells flow with a higher velocity than the leukocyte and accumulate upstream of the leukocyte. From t = 0.333 until the end of the simulation, the red cells maintain a stacked, single file orientation behind the leukocyte. Because it is slightly below the tube axis, the white cell rotates with a small angular velocity.
|
Simulations at expansions
We next performed three sets of simulations (Cases B, C, and D) to demonstrate the effect of RBCs on WBC margination at the expansion of a tube with a sudden change in diameter from 10.2 µm to 22.8 µm (Case B), to 24.6 µm (Case C), and to 26.4 µm (Case D), respectively.
The inlet geometry and boundary conditions were the same as in Case A. The Reynolds number was 8.5 x 10-3, and the lattice Boltzmann relaxation time was 1, corresponding to a time step of 1.25 x 10-8 s. The simulations correspond to 0.45 s in real time, which was normalized to 1 in the figures. The forces and the torques acting on the WBC were calculated for a cell of 9 µm diameter.
As in the previous case, the WBC starts the simulation one grid-point below center. If the WBC and RBCs are all initially centered about the tube axis, the RBCs catch up with, but never pass, the WBC. They travel down the tube in-line in a persistent, yet unstable configuration. This is because of the ideal geometry of the numerical simulation, which does not allow for small perturbations in trajectory that would upset the unstable configuration and allow the faster RBCs to pass. In a real vessel, the deformability of the cells and the nonsmooth nature of the wall would undoubtedly lead to small fluctuations in the trajectories, thus justifying our initial placement of the WBC slightly off-center.
Case B: expansion from 10.2 µm to 22.8 µm
Cases B1–B2 were carried out using an 800 x 76 lattice grid corresponding to 240 µm x 22.8 µm. The vessel has a diameter of 34 grid-points (10.2 µm) at the inlet and expands to a diameter of 76 grid-points (22.8 µm) at x = 150 grid. In Case B1, we examine a WBC with no RBCs. In Case B2, three RBCs trail the WBC.
Case B1, leukocyte in the absence of RBCs
Fig. 3 a shows snapshots of the WBC at normalized time t = 0, 0.056, 0.333, 0.5, and 0.667. Due to continuity, the plasma disperses as the cross section increases, resulting in a decrease in axial velocity. Driven by hydrodynamic forces, the WBC decelerates and moves slightly downward as it enters the expansion and the velocity profile expands to fill the larger tube. After the initial deviation resulting from the relaxation of the flow field, the net force is essentially zero (Fig. 3, b–d), and the leukocyte moves along a straight path, in agreement with experiment (Schmid-Schönbein et al., 1980). Figs. 3, e–g, show the resulting x-, y-, and angular velocities.
|
|
160 µm/s which slows to 142 µm/s after all the RBCs have passed (Fig. 4 e). Upon attachment, the angular velocity of the leukocyte increases (Fig. 4 g); the negative value corresponds to clockwise rotation. The oscillations in force and velocity after attachment are due to stochastic formation and dissociation of individual receptor-ligand bonds. The next set of simulations shows that if the expansion diameter is increased, three RBCs are not sufficient to initiate WBC rolling.
Case C: expansion from 10.2 µm to 24.6 µm
Now we increase the expansion diameter by six grids. Cases C1–C3 were performed using an 800 x 82 lattice grid corresponding to 240 µm x 24.6 µm. The vessel has a diameter of 34 grid-points (10.2 µm) at the inlet and expands to a diameter of 82 grid-points (24.6 µm) at x = 150 grid. We follow one WBC with three, four, and five RBCs in Cases C1, C2, and C3, respectively.
Case C1, leukocyte with three RBCs
The initial cell configuration is the same as Case B2, i.e., the leukocyte is initially one grid-point off-center toward the bottom wall and the RBCs are at the center. However, the expansion diameter is larger than that of Case B2.
Fig. 5 a shows snapshots at normalized time t = 0, 0.056, 0.333, 0.5, 0.667, and 0.944. Similar to the two preceding cases, the leukocyte slows and moves downward as it enters the expansion. The hydrodynamic force induced by the RBCs pushes the leukocyte closer to the bottom wall as it enters the expansion of the vessel. Its radial movement is further increased as the three RBCs pass over the leukocyte. However, in this case, the WBC does not come close enough to the wall (i.e., <40 nm) to engage receptor-ligand binding because of the larger expansion. Fig. 5, b–g presents the net forces and torque acting on the leukocyte and its velocity components.
|
Fig. 6 a shows snapshots at normalized time t = 0, 0.056, 0.333, 0.5, 0.667, and 0.944. Compared to Case C1, the additional RBC passing over the WBC provides sufficient force to push the WBC close enough to the wall for ligand bond to engage at t = 0.5. The WBC then begins rolling and slows down. The force and velocity profiles of the WBC plotted in Fig. 6, b–g, are similar to those of Case B2, which has one fewer RBC and a smaller expansion diameter. Since the flow rate is the same in all cases, the larger postcapillary diameter has a lower shear rate (see Table 2). Because of this decrease in shear rate, the rolling velocity drops to 116 µm/s from 142 µm/s (see Fig. 6 e) when the postcapillary diameter increases to 24.6 µm from 22.8 µm.
|
|
|
|
|
In the next example, we demonstrate that if the expansion diameter is increased further, the same 5-cell rouleau of RBCs can induce WBC rolling.
Case D: expansion from 10.2 µm to 26.4 µm
Here we increase the expansion diameter again by six grids. Cases D1–D2 were performed using an 800 x 88 lattice grid corresponding to 240 µm x 26.4 µm. The vessel has a diameter of 34 grid-points (10.2 µm) at the inlet and expands to a diameter of 88 grid-points (26.4 µm) at x = 150 grid. We introduce one WBC followed by five RBCs in both Case D1 and Case D2. But in Case D2, the RBCs are ellipsoidal rather than rectangular to investigate the effect of RBC shape.
Case D1, leukocyte with five RBCs
The initial configuration of the WBC and five RBCs and the inlet fluid conditions in the capillary are the same as that of Case C3. The expansion diameter is six grids larger than that of Case C3.
Fig. 8 a shows snapshots at normalized time t = 0, 0.056, 0.333, 0.5, 0.667, and 0.944. Compared to Case C3, all five RBCs pass over the WBC in this larger expansion and provide sufficient force to push the WBC close enough to the wall for ligand bonds to engage at t = 0.6, at which point the WBC decelerates and starts rolling. The force and velocity profiles of the WBC are plotted in Fig. 8, b–g. They are similar to those of Cases B2 and C2 in which WBC adhesion took place. Again, because the tube diameter is larger, the rolling velocity is lower, 98 µm/s (Fig. 8 e). This value of rolling velocity is in agreement with experimental studies (Munn et al., 1996).
Case D2, leukocyte with five ellipsoidal RBCs
Since it appears that maintaining the stacked arrangement of RBCs is important in inducing adhesion, we next simulated ellipsoidal RBCs, which are less stable in a stacked configuration. The number of RBCs and the geometry of the vessel are the same as Case D1.
Fig. 9 a shows snapshots at normalized time t = 0, 0.056, 0.222, 0.333, 0.5, and 0.667. Fig. 9, b–g, presents the net forces and torque acting on the leukocyte and its velocity components. Because the train of ellipsoids is less stable than the previous stacks of flat RBCs (such as in Case D1), it tends to bend (see the snapshots at t = 0.056 and 0.222) and pushes the WBC to the wall with more force at the beginning. But the train disrupts quickly and the disorganized group of ellipses passes the WBC more easily (compare Figs. 9 a and 8 a at t = 0.333 and 0.5). This results in a lower time of interaction and less influence on the WBC. Therefore, the WBC does not contact the wall although it comes in close proximity (0.18 µm from the wall).
Table 2 summarizes the results from the various simulations. Listed are the expansion diameter, the number of RBCs, the adhesion status, the closest approach distance between WBC and the wall, and the corresponding WBC velocities (Vx) at this position. We also calculated an equivalent rolling velocity (Vroll = Rc*), where Rc is the radius of the WBC and is the average angular velocity at the minimum gap. If Vroll is equal to Vx, the WBC is rolling (cases B2, C2, and D1). In the other cases, the WBC velocity is much higher than the equivalent rolling velocity.
|
DISCUSSION |
|---|
|
TOPABSTRACTINTRODUCTIONMETHODSRESULTSDISCUSSIONCONCLUSIONSACKNOWLEDGEMENTSREFERENCES |
|---|
, and represents an "ideal" expansion with diameter ratio representative of that found in capillary-postcapillary junctions. It is interesting to speculate that the expansion geometry in vivo is not arbitrary, but has evolved to an optimal configuration that encourages leukocyte attachment.
Table 2 summarizes the influence of expansion diameter and the number of RBCs on WBC adhesion. For a given expansion diameter there appears to be an optimal configuration of the trailing RBCs required to drive the WBC to the wall. In our simulations, WBC adhesion occurred in Case B2 (expansion diameter 22.8 µm, three RBCs), Case C2 (expansion diameter 24.6 µm, four RBCs), and Case D1 (expansion diameter 26.4 µm, five RBCs). Thus, larger expansions require more RBCs. But too many RBCs cannot maintain a stacked arrangement, and become disorganized too early before they pass the WBC. Our results show that a coherent rouleau of RBCs is more effective at pushing the WBC to the wall than a loosely-associated group of cells. Previous studies have concluded that RBC aggregation is an important determinant of L-E interactions in postcapillary venules (Mayrovitz et al., 1987; Pearson and Lipowsky, 2000). Although we did not directly address RBC aggregation in the current study, our results suggest that groups consisting of 3–5 RBCs separated by plasma will lead to greater radial motion of the leukocyte. The formation of these groups should depend on the relative numbers of RBCs and WBCs in the blood, the fluid dynamics at the capillary inlet, and the propensity of the RBCs to aggregate.
The shape of the RBCs in our simulations affected their ability to stay in a stacked formation. RBCs are actually biconcave disks, but in profile they appear as rectangles with round ends. As our simulations show, a rouleau of flat RBCs behaves quite differently from a train of ellipses of the same size. Whereas flattened shapes can easily form rouleaux and therefore provide sufficient force to initiate WBC rolling, the ellipsoidal RBCs tended to "roll" against one another, leading to disruption of the initial stacking arrangement.
Another interesting observation is that the blood vessels in solid tumors, as opposed to normal tissues, do not follow the same hierarchical progression from artery to vein. Instead, tumor angiogenesis results in a chaotic, malformed network with no apparent organization (Less et al., 1991). Thus they do not have normal capillary-postcapillary expansions. Tumors are also notoriously proficient at evading immune surveillance by leukocytes. It is possible that this is, at least in part, due to the inability of the leukocytes to initiate rolling within tumor vessels.
|
CONCLUSIONS |
|---|
|
TOPABSTRACTINTRODUCTIONMETHODSRESULTSDISCUSSIONCONCLUSIONSACKNOWLEDGEMENTSREFERENCES |
|---|
|
ACKNOWLEDGEMENTS |
|---|
|
TOPABSTRACTINTRODUCTIONMETHODSRESULTSDISCUSSIONCONCLUSIONSACKNOWLEDGEMENTSREFERENCES |
|---|
This work was supported by National Institutes of Health grant R01 HL64240 (to L.L.M.).
|
FOOTNOTES |
|---|
Submitted on August 20, 2002; accepted for publication March 21, 2003.
|
REFERENCES |
|---|
|
TOPABSTRACTINTRODUCTIONMETHODSRESULTSDISCUSSIONCONCLUSIONSACKNOWLEDGEMENTSREFERENCES |
|---|
Asako, H., P. Kubes, B. A. Baethge, R. E. Wolf, and D. N. Granger. 1992. Colchicine and methotrexate reduce leukocyte adherence and emigration in rat mesenteric venules. Inflammation. 16:45–56.[Medline]
Bell, G. I. 1978. Models for the specific adhesion of cells to cells. Science. 200:618–627.
Bernardin, D., O. E. Seroguillaume, and C. H. Sun. 1991. Multispecies 2D lattice gas with energy-level-diffusive properties. Physica D. 47:169–188.
Bhatnagar, P., E. Gross, and M. Krook. 1954. A Model for collision processes in gases. 1. Small amplitude processes in charged and neutral one-component systems. Phys. Rev. 94:511–525.
Bienvenu, K., and D. N. Granger. 1993. Molecular determinants of shear rate-dependent leukocyte adhesion in postcapillary venules. Am. J. Physiol. 264:H1504–H1508.[Medline]
Bongrand, P., and G. I. Bell. 1984. Cell-cell adhesion: parameters and possible mechanisms. In Cell Surface Dynamics: Concepts and Models. A. Perelson, C. DeLisi, and F. W. Wiegel, editors. Marcel Dekker, New York.
Chang, K., and D. A. Hammer. 1996. Influence of direction and type of applied force on the detachment of macromolecularly-bound particles from surfaces. Langmuir. 12:2271–2282.
Chang, K., D. F. J. Tees, and D. A. Hammer. 2000. The state diagram for cell adhesion under flow. Proc. Natl. Acad. Sci. USA. 97:11262–11267.
Chapman, G. B., and G. R. Cokelet. 1997. Model studies of leukocyte-endothelium-blood interaction. Biorheology. 34:37–56.[Medline]
Chen, H., S. Chen, and W. H. Matthaeus. 1992. Lattice Boltzmann model for simulating flows with multiple phases and components. Phys. Rev. A. 45:5339–5342.
Chen, S., and G. D. Doolen. 1998. Lattice Boltzmann method for fluid flows. Annu. Rev. Fluid Mech. 30:329–364.
Dong, C., J. Cao, E. J. Struble, and H. H. Lipowsky. 1999. Mechanics of leukocyte deformation and adhesion to endothelium in shear flow. Ann. Biomed. Eng. 27:298–312.[Medline]
Fung, Y. C. 1984. Biodynamics: Circulation. Springer-Verlag, New York.
Hammer, D. A., and S. M. Apte. 1992. Simulation of cell rolling and adhesion on surfaces in shear flow: general results and analysis of selectin-mediated neutrophil adhesion. Biophys. J. 63:35–37.
Israelachvili, J. N. 1985. Intramolecular and Surface Forces. Academic Press, New York.
Johnston, B., T. B. Issekutz, and P. Kubes. 1996. The alpha 4-integrin supports leukocyte rolling and adhesion in chronically inflamed postcapillary venules in vivo. J. Exp. Med. 183:1995–2006.
Ladd, A. C. J. 1994. Numerical simulations of particulate suspensions via a discretized Boltzmann equation. I. Theoretical foundation. J. Fluid Mech. 271:285–309.
Less, J. R., T. C. Skalak, E. M. Sevick, and R. K. Jain. 1991. Microvascular architecture in a mammary carcinoma: branching patterns and vessel dimensions. Cancer Res. 51:265–273.
Ley, K., T. F. Tedder, and G. S. Kansas. 1993. L-selectin can mediate leukocyte rolling in untreated mesenteric venules in vivo independent of E- or P-selectin. Blood. 82:1632–1638.
Mayrovitz, H. N., S. J. Kang, B. Herscovici, and R. N. Sampsell. 1987. Leukocyte adherence initiation in skeletal muscle capillaries and venules. Microvasc. Res. 33:22–34.[Medline]
Melder, R. J., L. L. Munn, S. Yamada, C. Ohkubo, and R. K. Jain. 1995. Selectin- and integrin-mediated T-lymphocyte rolling and arrest on TNF-a-activated endothelium: augmentation by erythrocytes. Biophys. J. 69:2131–2138.
Melder, R. J., J. Yuan, L. L. Munn, and R. K. Jain. 2000. Erythrocytes enhance lymphocyte rolling and arrest in vivo. Microvasc. Res. 59:316–322.[Medline]
Migliorini, C., Y. Qian, H. Chen, E. Brown, R. Jain, and L. Munn. 2002. Red blood cells augment leukocyte rolling in a virtual blood vessel. Biophys. J. 83:1834–1841.
Munn, L. L., R. J. Melder, and R. K. Jain. 1996. Role of erythrocytes in leukocyte-endothelial interactions: mathematical model and experimental validation. Biophys. J. 71:466–478.
oude Egbrink, M. G., G. H. Janssen, K. Ookawa, D. W. Slaaf, R. S. Reneman, X. H. Wehrens, K. J. Maaijwee, N. Ohshima, H. A. Struijker Boudier, and G. J. Tangelder. 2002. Especially polymorphonuclear leukocytes, but also monomorphonuclear leukocytes, roll spontaneously in venules of intact rat skin: involvement of E-selectin. J. Invest. Dermatol. 118:323–326.[Medline]
Pearson, M. J., and H. H. Lipowsky. 2000. Influence of erythrocyte aggregation on leukocyte margination in postcapillary venules of rat mesentery. Am. J. Physiol. Heart Circ. Physiol. 279:H1460–H1471.
Qian, Y. H., D. d'Humieres, and P. Lallemand. 1992. Lattice BGK models for Navier-Stokes equations. Europhys. Lett. 17:479–483.
Schmid-Schönbein, G. W., S. Usami, R. Skalak, and S. Chien. 1980. The interaction of leukocytes and erythrocytes in capillary and postcapillary vessels. Microvasc. Res. 19:45–70.[Medline]
Skalak, R., and P.-I. Brånemark. 1969. Deformation of red blood cells in capillaries. Science. 164:717–719.
Skalak, R., and S. Chien. 1987. Handbook of Bioengineering. McGraw-Hill, New York.
Sun, C. H. 2000. Adaptive lattice Boltzmann model for compressible flows: viscous and conductive properties. Phys. Rev. E. 61:2645–2653.
Tsai, A. G., D. Nolte, K. Messmer, and M. Intaglietta. 1992. Effect of oxygen (perflubron emulsion) on leukocyte-endothelial interaction in postcapillary venules. Biomater. Artif. Cells Immobilization Biotechnol. 20:959–961.[Medline]
Tsukada, K., E. Sekizuka, C. Oshio, and H. Minamitani. 2001. Direct measurement of erythrocyte deformability in diabetes mellitus with a transparent microchannel capillary model and high-speed video camera system. Microvasc. Res. 61:231–239.[Medline]
Warnock, R. A., S. Askari, E. C. Butcher, and U. H. von Andrian. 1998. Molecular mechanisms of lymphocyte homing to peripheral lymph nodes. J. Exp. Med. 187:205–216.
Yuan, J., R. J. Melder, R. K. Jain, and L. L. Munn. 2001. A lateral view flow system for studies of cell adhesion and deformation under flow conditions. Biotechniques. 30:388–394.[Medline]
Zao, Y., S. Chien, and S. Weinbaum. 2001. Dynamic contact forces on leukocyte microvilli and their penetration of the endothelial glycocalyx. Biophys. J. 80:1124–1140.
This article has been cited by other articles:
 |
|
 |
|
  C. Sun and L. L. Munn Particulate Nature of Blood Determines Macroscopic Rheology: A 2-D Lattice Boltzmann Analysis Biophys. J., March 1, 2005; 88(3): 1635 - 1645. [Abstract] [Full Text] [PDF]
|
|
| ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| HOME | HELP | FEEDBACK | SUBSCRIPTIONS | ARCHIVE | SEARCH | TABLE OF CONTENTS |
