^{1}

### Abstract

The physiological inflammation response depends upon the multibody interactions of blood cells in the microcirculation that bring leukocytes (white blood cells) to the vessel walls. We investigate the fluid mechanics of this using numerical simulations of 29 red blood cells and one leukocyte flowing in a two-dimensional microvessel, with the cells modeled as linearly elastic shell membranes. Despite its obvious simplifications, this model successfully reproduces the increasingly blunted velocity profiles and increased leukocyte margination observed at lower shear rates in actual microvessels. Red cell aggregation is shown to be unnecessary for margination. The relative stiffness of the red cells in our simulations is varied by over a factor of 10, but the margination is found to be much less correlated with this than it is to changes associated with the blunting of the mean velocity profile at lower shear rates. While velocity around the leukocyte when it is near the wall depends upon the red cell properties, it changes little for strongly versus weakly marginating cases. In the more strongly marginating cases, however, a red cell is frequently observed to be leaning on the upstream side of the leukocyte and appears to stabilize it, preventing other red cells from coming between it and the wall. A well-known feature of the microcirculation is a near-wall cell-free layer. In our simulations, it is observed that the leukocyte’s most probable position is at the edge of this layer. This wall stand-off distance increases with velocity following a scaling that would be expected for a lubrication mechanism, assuming that there were a nearly constant force pushing the cells toward the wall. The leukocyte’s near-wall position is observed to be less stable with increasing mean stand-off distance, but this distance would have potentially greater effect on adhesion since the range of the molecular binding is so short.

The author is grateful for fruitful discussions with R. D. Moser early on in this effort and with H. Zhao who helped formalize the area preserving constraint.

I. BACKGROUND

II. THE MODEL SYSTEM

A. Physical details

B. Numerical flow solution

1. Boundary integral discretization

2. Wall boundaries

3. Cell area constraint

III. RESULTS

A. Visualizations

B. Mean velocity profiles

C. Leukocyte trajectories

IV. DISCUSSION

A. Marginated leukocyte environment

B. Fåhræus-Lindqvist effect and the mean leukocyte standoff distance

V. SUMMARY

### Key Topics

- Cell membranes
- 25.0
- Cell adhesion
- 23.0
- Aggregation
- 14.0
- Cell communication
- 10.0
- Lubrication
- 10.0

## Figures

System schematic.

System schematic.

Sample visualizations of cases from Table I: (a) , (b) , (c) , and (d) . Plotted are the discrete quadrature points, interpolated from the control points, which define the cells as discussed in Sec. II B. Flow is from left to right.

Sample visualizations of cases from Table I: (a) , (b) , (c) , and (d) . Plotted are the discrete quadrature points, interpolated from the control points, which define the cells as discussed in Sec. II B. Flow is from left to right.

(a) Mean velocity profile for case , —; the show the velocity profile calculated in the absence of cells, just using the forces from (14) with parabolic fit . (b) Case , — is contrasted with the slower flowing case - - -.

(a) Mean velocity profile for case , —; the show the velocity profile calculated in the absence of cells, just using the forces from (14) with parabolic fit . (b) Case , — is contrasted with the slower flowing case - - -.

(a-d) Leukocyte centroid trajectories and probability distributions across the model microvessel for the cases as labeled; (e) shows the same for case as in (c) but for a red cell. In most cases only a part of the simulation time is shown.

(a-d) Leukocyte centroid trajectories and probability distributions across the model microvessel for the cases as labeled; (e) shows the same for case as in (c) but for a red cell. In most cases only a part of the simulation time is shown.

Probability of leukocyte being within of either wall for the constant cell properties -marked cases in Table I and for corresponding cases with mean hematocrit 0.33 and . The - - - curve is for cases with (see text).

Probability of leukocyte being within of either wall for the constant cell properties -marked cases in Table I and for corresponding cases with mean hematocrit 0.33 and . The - - - curve is for cases with (see text).

Marginated probability of for all cases. For each point the bar length is as measured on the vertical axis. The solid correspond to the cases shown in Fig. 5.

Marginated probability of for all cases. For each point the bar length is as measured on the vertical axis. The solid correspond to the cases shown in Fig. 5.

Contours of mean -direction velocity averaged on the condition that for cases (a) , (b) , (c) , (d) , (e) , and (f) . All plots show contours at intervals of 0.001. Negative contours are dashed and the contour is not shown.

Contours of mean -direction velocity averaged on the condition that for cases (a) , (b) , (c) , (d) , (e) , and (f) . All plots show contours at intervals of 0.001. Negative contours are dashed and the contour is not shown.

Local red cell volume density averaged when for cases (a) , (b) , (c) , (d) , (e) , and (f) . The circle shows the approximate location and shape of the leukocyte, though it does move and deform slightly in the course of the simulation.

Local red cell volume density averaged when for cases (a) , (b) , (c) , (d) , (e) , and (f) . The circle shows the approximate location and shape of the leukocyte, though it does move and deform slightly in the course of the simulation.

Visualizations of leukocyte’s neighborhood for strongly and weakly marginating cases for relatively flexible and stiff red cells.

Visualizations of leukocyte’s neighborhood for strongly and weakly marginating cases for relatively flexible and stiff red cells.

(a) Red cell density for case ; (b) randomly selected visualizations of the leukocytes neighborhood. The statistical sample is smaller in this case than those in Fig. 8, which leads to choppier contours.

(a) Red cell density for case ; (b) randomly selected visualizations of the leukocytes neighborhood. The statistical sample is smaller in this case than those in Fig. 8, which leads to choppier contours.

(Top) Probability distribution of leukocyte wall distance; (bottom) red-cell density close to the wall. The lines show the -cases in Table I: –– , –– , , - - -, and — .

(Top) Probability distribution of leukocyte wall distance; (bottom) red-cell density close to the wall. The lines show the -cases in Table I: –– , –– , , - - -, and — .

Leukocyte wall distance versus its speed for the -marked cases in Table I with fit — . The show the same for the single red cell test cases discussed in the text with fit - - - .

Leukocyte wall distance versus its speed for the -marked cases in Table I with fit — . The show the same for the single red cell test cases discussed in the text with fit - - - .

Probability distribution of leukocyte distance from the wall for case —– and case with and tripled .

Probability distribution of leukocyte distance from the wall for case —– and case with and tripled .

## Tables

Cases simulated. The -marked cases indicate a series with constant cell properties.

Cases simulated. The -marked cases indicate a series with constant cell properties.

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