The deformations of neutrophils as they pass through the
pulmonary microcirculation affect their transit time, their tendency to
contact and interact with the endothelial surface, and potentially their degree of activation. Here we model the cell as a viscoelastic Maxwell material bounded by constant surface tension and simulate indentation experiments to quantify the effects of
(N-formyl-L-methionyl-L-leucyl-L-phenylalanine (fMLP)-stimulation on its mechanical properties (elastic shear modulus
and viscosity). We then simulate neutrophil transit through individual
pulmonary capillary segments to determine the relative effects of
capillary geometry and fMLP-stimulation on transit time. Indentation
results indicate that neutrophil viscosity and shear modulus increase
by factors of 3.4, for 10
9 M fMLP, and 7.3, for
106 M fMLP, over nonstimulated cell values, determined to
be 30.8 Pa·s and 185 Pa, respectively. Capillary flow results
indicate that capillary entrance radius of curvature has a significant effect on cell transit time, in addition to minimum capillary radius
and neutrophil stimulation level. The relative effects of capillary
geometry and fMLP on neutrophil transit time are presented as a simple
dimensionless expression and their physiological significance is discussed.
 |
INTRODUCTION |
Neutrophils are larger in average diameter than
about 40% of capillary segments in the human lungs (Doerschuk
et al., 1993
), and transit between 50 and 100 segments in a
single pass through the extensively interconnected pulmonary
microcirculation (Hogg et al., 1994). These factors,
together with their decreased deformability relative to erythrocytes,
cause neutrophils to be delayed for seconds or even longer before they
deform sufficiently to transit individual pulmonary capillary segments,
and result in an increased concentration of the cells in the pulmonary
capillary blood with respect to erythrocytes (Lien et al.,
1987, 1990,
1991; Doerschuk, et al.,
1993, 2001;
Hogg et al., 1994). Chemoattractants such as
N-formyl-L-methionyl-L-leucyl-L-phenylalanine
(fMLP), which have been shown to decrease neutrophil deformability,
also increase neutrophil concentration in the lungs (Worthen et
al., 1989; Lipowsky et al., 1991).
We recently used a computational network model of the pulmonary
microcirculation to estimate pressure drops in the lung. Normal segmental pressure drops were found to be ~10 Pa (0.1 cmH20), increasing locally by 100-300% when a local
capillary segment was blocked by a deforming neutrophil (Huang
et al., 2001). In that study, the transit time of a neutrophil
through an individual capillary segment was based in part on results of
micropipette aspiration experiments (Fenton et al.,
1985) and in part on theoretical predictions of cell entrance
into a micropipette (Yeung and Evans, 1989) (neutrophil
"entrance time" and "transit time" are considered synonymous in
this study because we are only concerned with transit through
capillaries that are relatively short and narrow, so that the bulk of
the cell's transit time consists of its entering into the narrowest
part of the capillary segment). Though the expression incorporated the
separate effects of driving pressure and minimum capillary radius on
neutrophil transit time, it was based on studies using blunt-ended
micropipettes. We noted, however, that capillary geometry could
significantly influence neutrophil transit times and that entrance
curvature and lack of axisymmetry were important effects to consider.
This provided the primary motivation for the present study, namely to
obtain a more general expression for neutrophil transit time that
addresses the effects of entrance geometry. We also sought to better
understand the relative importance of cell activation level on transit
time through the pulmonary capillaries.
To address these issues, an existing neutrophil model was selected from
the literature and appropriate model parameter values were determined
for its nonstimulated behavior and for two levels of stimulation
produced by treatment with fMLP (109 and
106 M). We modeled the cell as a homogeneous viscoelastic
Maxwell material with a constant bounding surface tension of 31 pN/µm (Dong et al., 1988). The Maxwell model parameter values
(elastic shear modulus Gcell and viscosity
µcell) were determined by simulating previously performed
indentation experiments (Worthen et al., 1989) on
nonstimulated and fMLP-stimulated neutrophils, and varying the
parameters to best fit the experimental data. In the subsequent phase
of the study, we modeled and simulated neutrophil transit through
individual, axisymmetric pulmonary capillary segments. Minimum
capillary radius and capillary entrance geometry (more specifically,
the radius of curvature of the constriction) were varied to quantify
their respective effects on transit time. For this purpose, we
performed a fully coupled fluid-solid interaction finite element
analysis using a nonlinear kinematic description for the cell and an
arbitrary Lagrangian-Eulerian (ALE) formulation for the surrounding
plasma. Effects of cell-capillary wall adhesion and time-varying
cellular behavior during transit, such as actin polymerization, were neglected.
In what follows, we begin by presenting the continuum mathematical
models used for nonstimulated and fMLP-stimulated neutrophils (they are
geometrically and constitutively identical, differing only in the
specific values of their constitutive constants determined from the
indentation experiments), the indentation experiment, and pulmonary
capillary segments. We then present the essential features of the
finite element methods used to obtain their solutions. Indentation
results are presented next, followed by our predictions of
dimensionless capillary transit times. A simple, closed-form expression
that fits the data in the Newtonian, or viscous deformation-dominated, limit is presented, as well as a correction that accounts for the
effects of cellular elasticity. Dimensionless and dimensional transit
times are considered in a physiological context in the Discussion that
follows, in which the relative effects of fMLP and capillary geometry
on cell transit time are addressed.
| |
MODELING |
The neutrophil
Of the various mechanical models that have been proposed for the
neutrophil, three are most widely accepted. Although each treats the
cell as a homogeneous, incompressible, deformable sphere, only two
consider explicitly the cortical region, treating it as a bounding
membrane that exerts a small constant surface tension of order 30 pN/µm. In one model, the cell is treated as a linear standard
viscoelastic solid (Schmid-Schönbein et al., 1981;
Sung et al., 1988; Lipowsky et al.,
1991), in another, as a Newtonian fluid bounded by constant
surface tension (Evans and Yeung, 1989; Frank and
Tsai, 1990; Needham and Hochmuth, 1990;
Tran-Son-Tay et al., 1991; Hochmuth et al.,
1993; Drury and Dembo, 1999), and in the last,
as a linear Maxwell material, also bounded by constant surface tension
(Dong et al., 1988; Skalak et al., 1990).
Of these three models, the Maxwell model with constant surface tension
was selected for use in the present study. The model incorporates the
elastic restoring effects of the actin-rich cortical layer lining the
periphery of the cell, lying just below its external lipid bilayer,
with the ability to capture both the elastic, solid-like short
time-scale behavior of the cell and its viscous, fluid-like long
time-scale behavior (Schmid-Schonbein et al., 1981;
Evans and Kukan, 1984; Dong et al.,
1988). Additionally, it is capable of predicting the
experimentally observed linear compressive stroke force-displacement
relation exhibited by neutrophils during indentation, provided the
material property values are appropriately chosen. Despite their
predictive capabilities, the Newtonian model neglects cellular
elasticity and predicts a concave downward force-displacement relation
during indentation (Zahalak et al., 1990), and the
standard solid model neglects the cell's cortical tension and predicts a static deformation limit to constant applied loads, in contradiction to the fluid-like behavior exhibited by neutrophils during micropipette aspiration (Evans and Yeung, 1989). We chose not to
adopt one of the more recent inhomogeneous neutrophil models that treat the multi-lobed nucleus as a distinct entity from the cytoplasm (Tran-Son-Tay et al., 1998) due to their added
complexity and because the limited amount of indentation data available
to us precluded a unique determination of cytoplasmic and nuclear
mechanical properties. Cortical shell bending effects were also
neglected, because they have been shown in a micropipette aspiration
study to be important only for very small pipette radii, less than 1 µm (Zhelev et al., 1994).
Unlike nonstimulated neutrophils, for which numerous published models
exist, fMLP-stimulated neutrophils have not been modeled extensively.
For this reason, it was necessary to determine an appropriate model for
fMLP-stimulated cells, and to determine appropriate model parameter
values corresponding to various levels of stimulation. For consistency
with the nonstimulated cell model, and due to its ability to accurately
predict the force-displacement indentation data for fMLP-stimulated
neutrophils, the Maxwell model with constant surface tension was also
used to model the cell in its fMLP-stimulated state. Indentation
experiments on nonstimulated and two levels of fMLP-stimulated
neutrophils (109 and 106 M) were simulated
to determine their appropriate model parameter values (Worthen,
et al., 1989).
For the nonstimulated neutrophil, a constant cortical tension of 31 pN/µm was assumed (Dong et al., 1988). For the
fMLP-stimulated neutrophils, two assumptions regarding the effects of
fMLP on neutrophil mechanical properties were made. First, the cortical tension of the cell was assumed to be unaffected by fMLP. Second, the
characteristic viscoelastic decay time of the cell
(µcell/Gcell) was assumed to be
unaffected by fMLP concentration and equal to the value found for the
nonstimulated cell, 
s. The former assumption was made in the
absence of quantitative information regarding the specific effects of
fMLP on cortical tension, though it might be expected that cortical
tension would increase in the presence of fMLP due to the high actin
content of the cortical region and the fact that the primary effect of fMLP is to increase actin assembly (Worthen et al.,
1989; Motosugi et al., 1996). (This is
particularly the case when one considers that fMLP has been found to
localize F-actin in the submembraneous region of circulating
neutrophils [Saito et al., 2002] and that cytochalasin-D, which is known to disrupt actin filament organization, has been found to reduce cortical tension by 40-60%
[Ting-Beall et al., 1995].) The latter assumption was
made because we lacked an empirical basis for any other assumption, as
might have been derived from the entire indentation force-displacement
relation exhibited in studies of the type performed by Worthen
et al., (1989).
Mathematical model
The updated Lagrangian Hencky formulation was used to describe the
kinematics of the neutrophil, accounting for the large displacements,
rotations, and strains present in the model. The nonlinear kinematic
formulation uses Cauchy stresses and their conjugate measure of
deformation, Hencky (natural, logarithmic, or true) strains. For
simplicity, we present here the Maxwell material constitutive law
assuming small strain and small displacement conditions, and refer the
reader to (K. J. Bathe, 1996) for details on how the
extension is made to the fully nonlinear kinematic formulation used in
the simulations.
We define the traceless stress deviator, T', representing
the state of pure shear stress at a point as
|
(1)
|
where T is the Cauchy stress tensor, I is
the identity tensor, and the summation convention is used. The
traceless strain deviator, E', representing the state of
pure shear strain at a point is defined as
|
(2)
|
where E is the small strain tensor
|
(3)
|
and u is the material displacement vector.
Stress and strain in the homogeneous cell interior are related through
the linear Maxwell constitutive law, written in differential form for
the deviatoric response as
|
(4)
|
where µcell is the constant coefficient of
viscosity, Gcell is the constant elastic shear
modulus, and a superimposed dot denotes time rate of change. Defining
the mechanical pressure, p, to be the average compressive
stress (p 
Tkk), the bulk response of a Maxwell material can be expressed as
|
(5)
|
where 
cell is the bulk modulus. In this study, we
assume that the cell is nearly incompressible, with a Poisson ratio of 0.499, so that the bulk modulus is related to the shear modulus by,
cell = 500Gcell.
The cortical layer lining the cell was modeled as a uniform thickness
axisymmetric shell with constant in-plane equibiaxial tensile prestress
(a two-dimensional state of constant negative pressure). A linear
elastic stress-strain law with a vanishingly small elastic modulus
(E = 104 Pa) was chosen to model the
constitutive behavior of the shell, and a very small shell thickness (1 nm) was assumed to ensure that bending and deformation-induced membrane
effects on the response of the cell were negligible. These modeling
assumptions approximated a constant, uniform surface tension bounding
the cell that can be expressed as
|
(6)
|
where Scell denotes the cell's bounding
surface, T(cell) is the Cauchy stress tensor in
the cell, T(ext) is the Cauchy stress tensor in
the body that is external to the cell (indenter or plasma),
n is the local outward-directed unit normal to the cell
surface, 
denotes the constant coefficient of surface tension, and
H denotes the mean local curvature of the cell surface,
assumed positive when the center of curvature lies in the direction of
the normal, and negative otherwise. The mean local curvature of the
cell surface, H, can be expressed as the mean of the
curvatures, 1/R1 and
1/R2, of the surface in any two orthogonal
planes containing the surface normal, where the curvatures have the
same sign convention as H,
|
(7)
|
Neutrophil indentation
The neutrophil-indentation model mimics the experimental study of
Worthen et al., (1989) and is consistent with a previous finite-element simulation of the experiment (Zahalak et al.,
1990). The experiment consisted of indenting individual
neutrophils at a constant rate of indentation with a micrometer-scale
glass rod (the indenter) and measuring the time course of the resultant force. During indentation, the cell was supported by a substrate that,
like the indenter, was effectively rigid. The indentation experiments
were conducted at room temperature using 30-82 cells from each of five
donors, providing a statistical dataset for the neutrophil models to be
based upon.
Mathematical model
The undeformed cellular radius, Rcell, the
indenter radius, Rindenter, and the radius of
curvature of the indenter corner, 
, (Fig.
1) have values of 4, 1, and 0.15 µm,
respectively. The axisymmetric equations of equilibrium used to
integrate the time-dependent viscoelastic response of the cell can be
written as
|
(8)
|
where we note that mass conservation was automatically satisfied
by the use of a Lagrangian formulation.
Indentation velocity, 
, was prescribed at the top
centermost point of the indenter and corresponded to the experimental value of 5.1 µm/s. Velocity was constant and equal in magnitude during the indentation and retraction strokes, and the indenter reached
a maximum depth of 
max, corresponding to 1.5 µm.
Although the experimental value of max was 2.2-2.6
µm, 1.5 µm was deemed sufficient for the simulations due to the
linearity of the indenter force-displacement relationship during the
downward, indentation stroke, observed both experimentally and
numerically. Frictionless contact was assumed to occur between the
indenter-cell and the substrate-cell surfaces, neglecting possible
effects arising from biochemical adhesion. The kinematical contact
condition satisfied at all times in the analysis by the contacting
bodies (ensuring impenetrability) can be expressed as
|
(9)
|
where v(cell) is the material velocity of
the cell, v(ext) is the material velocity of the
external, contacting body, n is a unit vector normal to the
contact surface, and Scontact is the relevant
contact area, cell substrate or cell indenter. The normal stress jump
present at the cell surface due to surface tension effects is as given
in Eq. 6.
Capillary transit
Pulmonary capillaries vary in radius from 1 to 7.5 µm in humans,
with a mean of 3.7 µm (Doerschuk et al., 1993). They
constitute a vast and complex network of interconnected segments, each
of which is highly irregular in both cross-section and length,
providing a formidable challenge from a modeling standpoint
(Weibel, 1963). Because the primary aim of this study
was to characterize the effects of capillary entrance geometry on
neutrophil transit time, extending the relationship known for
blunt-ended micropipette geometries (Yeung and Evans,
1989; Huang et al., 2001), and because we
confined our analyses to rigid axisymmetric geometries, we modeled a
typical capillary constriction as having a constant radius of curvature
in the plane containing the vessel axis (the constriction radius of
curvature is denoted a in Figs. 2 and 3). This approximation
is one of many that could have been used, but it is well motivated by
the geometric nature of pulmonary capillaries as exhibited in the
micrograph shown in Fig. 2, where
capillary cross-sections are seen to have entrance geometries that are
reasonably well characterized by a constant radius of curvature.
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|
FIGURE 2
SEM showing an interior view of the pulmonary
capillaries in a rat lung, illustrating the motivation for modeling a
typical capillary entrance as having a constant radius of curvature
(labeled a above and in Fig. 3) (Reproduced by permission
from Guntheroth et al., 1982).
|
|
It was also critical that we consider the glycocalyx. In the present
model, the near-wall region is treated as a rigid layer, 100 nm thick
and highly permeable to plasma. This is achieved by placing a
frictionless contact surface 100 nm from the capillary wall. This
assumption was made as a compromise between having no layer, in which
case the neutrophil approaches to within several nanometers of the
wall, and a representation of the glycocalyx as proposed in several
recent models of red cell motion through capillaries (Pries et
al., 1997; Damiano, 1998; Feng and
Weinbaum, 2000). Note that, despite the recent progress made
toward accurately modeling the glycocalyx, there is still considerable
debate over the precise nature of the physical interaction between the
glycocalyx and individual blood cells. Moreover, although it is likely
that the pulmonary endothelium possesses a glycocalyx, to our knowledge it has not been directly observed and hence its thickness is unknown. Finally, although the thickness of the rigid layer used in our simulations was somewhat arbitrary, we numerically confirmed that the
cellular transit times we obtained were independent of the specific
value used within the range of 50 to 200 nm. The insensitivity of the
cell's transit time to the layer thickness within this range is due to
the facts that the pressure drop remained concentrated across the cell
and the retarding shear stress imposed on the cell by the plasma in the
layer region remained negligible (see Appendix).
Mathematical model
The axisymmetric capillary model is shown in Fig.
3, where the radii of curvature of the
capillary wall and the contact surface are (a 
)
and a, respectively, and the minimum constriction radius is
Rmin. The layer thickness, chosen to be 100 nm,
is denoted , and the unobstructed portions of the capillary were
assumed to be cylindrical with radii Rcapillary.
Choosing the right-handed cylindrical coordinate system
(r, 
, z) shown in Fig 3, the contact surface
constriction radius, R(z), can be expressed analytically as
|
(10)
|
where z denotes position along the capillary axis, and
l is the axial half-length of the contact surface. The
upstream and downstream straight sections of the capillary model were
chosen to be between 8-15 and 23-26Rcell,
respectively.
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|
FIGURE 3
Schematic of the model capillary geometry with the cell
in its initial, undeformed configuration (the cell is moving from left
to right).
|
|
The dynamical equations of motion were solved incrementally in time for
both the cell and plasma, in a fully coupled manner. For the cell, the
equations of motion are expressed in Lagrangian form as
|
(11)
|
where cell is the constant cell density and the
operator D/Dt denotes material time rate of change. For the
plasma, the equations of motion are expressed in Arbitrary Lagrangian-
Eulerian (ALE) form (K. J. Bathe et al., 1995). The ALE
formulation was needed to maintain the integrity of the finite element
mesh in the plasma domain as the cell underwent large displacements
through the capillary constriction. In the ALE formulation, the
equations of motion are expressed as
|
(12)
|
where plasma is the constant plasma density,
vm is the mesh velocity,
v/t is the time rate of change of the
plasma velocity as measured at a moving mesh point, and all velocities are measured with respect to an inertial frame of
reference. The equation of continuity for the plasma, treated as
an incompressible, Newtonian fluid with viscosity
µplasma, can be expressed as
|
(13)
|
For the plasma, a kinematical condition of no-slip was assumed on
the rigid capillary wall,
|
(14)
|
and constant normal tractions were applied on the plasma at the
capillary inlet and outlet
|
(15)
|
|
(16)
|
where tinlet and
toutlet are the magnitudes of the prescribed
inlet and outlet traction vectors, respectively. Fully developed flow
at the inlet and outlet of the model resulted in the fluid pressure
being equal to the applied normal tractions, and hence the total
pressure drop across the capillary model, denoted P, was
simply equal to the difference between the normal traction values.
Trans-capillary pressure drop was assumed to vary between the
physiological values of 20 and 80 Pa (Huang et al.,
2001) and, although the very low Reynolds number of the
flow ensured that inertial effects were negligible, the inertial terms
were retained in the momentum equations solely for numerical purposes.
Velocity and shear stress continuity were satisfied at the cell-plasma
interface throughout the analysis, whereas there was a jump in the
normal stress between the cell and plasma due to the effects of surface
tension, as expressed earlier in Eq. 6.
When the cell enters the constriction, a portion of its surface
contacts the capillary-constriction contact surface used to model
cellular interactions with the glycocalyx (Fig. 3). While in contact,
an additional normal traction is applied to the cell to ensure that the
gap between the contact surface and capillary wall remains constant and
equal to . During this time, the normal contact traction applied to
the cell by the contact surface is superimposed upon the traction
vector exerted onto the cell by the plasma (which generally consists of
normal and shear components), so that a force balance is maintained at
the cell-plasma-contact
surface interface.
Dimensional analysis
The transit time, T, required for the model cell to
transit through the capillary constriction can be expressed in its most general form as a function of all the dimensional parameters in the
model,
|
(17)
|
where T is defined as the time from which the leading
edge of the cell crosses the capillary constriction inlet to the time when the trailing edge of the cell crosses the capillary constriction outlet. To reduce the number of independent dimensional parameters in
Eq. 17, several simplifying assumptions are made. First, it is assumed
that T will be nearly independent of the upstream and downstream capillary radius, Rcapillary,
provided that T 
conv, where
conv is the convective time scale of the cell when it is freely traveling in the capillary
(~Rcell/
, where
is the average plasma velocity in the unconstricted
capillary). In this limit, the bulk of the transit time consists of the
time spent by the cell squeezing through the capillary constriction,
during which the constant pressure drop is applied across the cell, and is independent of the upstream and downstream capillary radii. Second,
it is assumed that in the same limit, the capillary transit time will
be insensitive to variations in plasma viscosity,
µplasma, and gap thickness, , between the constriction
contact surface and the wall. As the cell squeezes through the
constriction, there are only two retarding forces balancing the axial
pressure gradient: one is due to the axial component of the normal
contact traction applied to the cell by the constriction contact
surface and the other is due to the Couette component of the shear
stress in the gap. As shown in the Appendix, for the parameter ranges
explored in this study, the retarding force due to the Couette
flow-induced shear stress is negligible, validating the assumption that
the transit time is independent of µplasma and .
Finally, we have excluded the upstream and downstream lengths of the
capillary model from the transit time expression because viscous losses in those regions were negligible during transit. Thus, the applied trans-capillary pressure drop (P) remained concentrated
across the cell independently of their specific lengths, within the
range provided in the mathematical model section.
Using the above arguments, the number of independent variables in the
dimensional transit time equation above (Eq. 17) is reduced to seven,
|
(18)
|
Choosing µcell/P as the characteristic
time scale, the dimensional transit time equation can be written in
dimensionless form as
|
(19)
|
where T* (TP/µcell), G* (Gcell/P), R* (Rmin/Rcell),
a* (a/Rcell), and * [/(PRmin)] measures the effects of the
bounding surface tension on reducing the effective pressure drop
driving the cell into the constriction.
| |
NUMERICAL SOLUTION OF THE GOVERNING EQUATIONS |
Solutions of the nonlinear governing equations for the indentation
and capillary flow models presented in the previous sections were
obtained using a commercially available finite element program (ADINA,
Version 7.5, Watertown, MA).
Finite element formulation
The cell interior was discretized in space with 4/1 quadrilateral
axisymmetric mixed (u/p) solid elements. The elements use bilinear
displacement interpolation and a constant pressure degree of freedom to
analyze effectively nearly incompressible media (Sussman and
Bathe, 1987). Two-node axisymmetric shell elements were used to
discretize the cortical region bounding the cell, with two displacement
and one rotation degree of freedom per node, consistent with the
bilinear elements used for the cell interior. Surface tension effects
were modeled by imposing an initial state of uniform biaxial stress on
the shell elements (see Modeling: The neutrophil).
The updated Lagrangian Hencky kinematic formulation was used to analyze
the cellular response in both the indentation and capillary flow
models, and the constraint function method was used to satisfy the
nonlinear contact conditions (K. J. Bathe, 1996).
Temporal discretization of the finite element equations used the
Newmark method for the indentation simulations and the Euler backward
method for the fluid-solid interaction (cell-capillary) simulations.
Three-node triangular axisymmetric elements were used for spatial
discretization of the plasma. The elements interpolate pressure and
velocities linearly and use an additional bubble function for the
velocities. Both the fluid and solid elements satisfy the inf-sup
(infimum-supremum) or Babu
ka-Brezzi condition for stability and
optimality in nearly or exactly incompressible analysis (Brezzi
and Fortin, 1991).
Numerical solution of the governing solid and fluid equations of the
capillary model required the use of the fluid-solid interaction analysis capabilities of ADINA (Rugonyi and K. J. Bathe,
2001). The solution obtained was fully coupled in the sense
that the kinematical conditions of displacement, velocity, and
acceleration continuity across the no-slip fluid-solid (cell-plasma)
interface were satisfied at all times during the analysis, and the
conditions of shear stress continuity and normal stress discontinuity
due to surface tension effects presented in the previous section.
Computations were performed on Silicon Graphics Incorporated Origin and
Octane workstations on single 300-MHz CPUs. Memory requirements were
minimal because the models were two-dimensional. CPU times varied
depending upon the model type and the temporal and spatial
discretizations used, but generally were between 10-15 min and 1.5-2
h for the indentation and capillary flow models, respectively. We note
that convergence difficulties were encountered during the final, exit
phase of the capillary flow simulations due to the large acceleration
of the cell during that time.
Finite element solution validation
Validation of the finite element results consisted of two parts.
First, mesh and time-step refinements were performed to ensure convergence of the results to the solution of the underlying continuum models (see Modeling). It was verified that the solution variables of
interest (indentation force and cell transit time for the indentation and capillary flow models, respectively) changed by less than 5% when
the spatial density of nodes was doubled in the solid and fluid models
and when the time-step size was halved. Second, to further validate the
methods used in the neutrophil-capillary analysis, we simulated a
problem similar to one previously analyzed by Tran-son-tay et
al. (1994). In that study, the authors simulated the flow of a
Newtonian droplet with surface tension through a converging tapered
tube in an effort to model the flow of a neutrophil through a tapered
glass capillary tube. In what follows, we present a brief overview of
the tapered tube model along with a results comparison, and refer the
reader to Tran-son-tay et al. (1994) for details of the
original study.
Tapered tube model
The original tapered tube model of Tran-son-tay et al. consists of
a highly viscous Newtonian droplet of viscosity, µ, bounded by
constant surface tension, , flowing through an axisymmetric tube of
constant converging half-angle, 
, under a constant driving pressure
drop, P. The original droplet diameter is denoted
D0 and the deformed axial end-to-end length of
the droplet L. In their model, the external fluid is assumed
to be inviscid, resulting in a uniform pressure applied to the droplet
at its upstream and downstream ends and zero shear stress acting along
its conical section (within the gap). Lastly, a time-varying normal
pressure is applied along the conical section of the droplet to satisfy axial equilibrium, and the normal component of the droplet velocity is
constrained to be zero there to satisfy the geometric constraint imposed by the wall. The axisymmetric creeping-flow equations are
solved numerically in time employing the methods used earlier by
Yeung and Evans (1989) to study neutrophil aspiration
into glass micropipettes.
In the present study, we simulate these conditions employing the same
methods as those used for the capillary flow model (see Capillary
Transit,) changing the geometry to be that of the tapered tube.
The gap thickness and external fluid viscosity are chosen such that the
retarding shear stress on the droplet is negligible and the pressure
drop applied across the ends of the tapered tube is concentrated across
the droplet. The ratio of external to internal viscosity was
µext/µ = 106 and the dimensionless
gap thickness was /D0 = 0.015. We
simulate the droplet's Newtonian constitutive response by choosing the elastic shear modulus of the Maxwell droplet to be 1000 times the
driving pressure drop (G* 1), ensuring that elastic
deformations are negligible. Though the zero wall shear-stress
assumption is satisfied along the conical section of the droplet due to
the low viscosity of the external fluid, no assumption is made as to
the form of the wall (contact) normal stress distribution acting on the
droplet. The relevant dimensionless model parameters are time,
t* = tP/µ, surface tension 
* = /(PD0), and droplet end-to-end length,
L/D0.
Tapered tube results
Three model conditions (* = 0.015, = 3°; * = 0.015, = 6°; and * = 0.15, = 3°) were analyzed in
the original study to illustrate the effects of taper angle and surface
tension on entry rate, and each is simulated here for validation. The
entry rate of the droplet decreases sharply with increasing taper angle from 3° to 6°, while * is held constant at 0.015 (Fig. 4
A), the effects being
significant even for small deformations (L/D0 
1). Increasing surface tension by a factor of 10 from * = 0.015 to 0.15 while is held constant at 3° (Fig. 4 B) is
seen to have the opposite effect, namely to decrease entry rate (due to
the reduction in effective driving pressure caused by the different radii of curvature at the leading and trailing ends). Differences are
<5% even at the time of maximum deformation (t* = 2.7) and attributable to differences in the underlying mathematical models and
the numerical methods used to solve them.
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FIGURE 4
Comparison between the tapered tube results of
Tran-Son-Tay et al. (1994) and the present study
illustrating (A) the effects of taper angle on droplet entry
rate (* = 0.015; = 3°, 6°) and (B) the effects of
surface tension on entry rate ( = 3°; * = 0.015, 0.15).
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RESULTS |
Neutrophil indentation
The objective of the indentation simulations was to
determine appropriate Maxwell model parameter values for the neutrophil in its nonstimulated state and after stimulation with two
concentrations of fMLP that induce neutrophil activation. Instantaneous
finite-element mesh configurations of the deformed, nonstimulated cell
model are shown superimposed upon the original, undeformed mesh at two times: when the cell is maximally indented to 1.5 µm (0.294 s) (Fig.
5 A) and at the instant
when the indenter has returned to its original, zero displacement
position (0.588 s) (Fig. 5 B). Note that the cell remains
significantly deformed even after the indenter has returned to its
original position. The reasons for this are twofold. First, the
viscoelastic relaxation time scale of the cell
[relax ~ (µcell/Gcell) = 0.167 s] is
on the order of the indentation time scale [indent ~ (max/) = 0.294 s], so that
much of the work done by the indenter is dissipated in the cell during
indentation, rather than elastically stored and transmitted back to the
indenter during retraction. Second, the cell-recovery time scale
[recovery ~ (µcellRcell/) = 3.97 s] is long compared to the indentation time scale, meaning that
the surface tension force is too small to restore the cell to its spherical shape in the time it takes the indenter to retract.
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FIGURE 5
Superimposed depictions of the indentation model
(nonstimulated cell) in its original and two deformed configurations,
corresponding to (A) the time of maximum indentation
(t = 0.294 s) and (B) the instant when the
indenter returns to its original, zero-displacement position after
indentation (t = 0.588 s).
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The Maxwell model parameters, Gcell and
µcell, determined by the cell indentation simulations are
listed in Table 1 for each level of fMLP
stimulation, and the force-displacement curves computed with each
model are compared with the experimental results in Fig.
6. The indentation force-displacement
relationship is nearly linear in all cases, and significant hysteresis
is exhibited as the indenter is withdrawn. The effects of fMLP on
cellular viscosity and shear modulus are significant (Table 1),
purportedly due to increased actin assembly (Worthen et al.,
1989). Viscosity varies from a low of 30.8 Pa·s in the
nonstimulated case to 225 Pa·s in the maximally stimulated case,
while the cell's shear modulus increases from 185 to 1350 Pa.
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FIGURE 6
Maxwell model fit to mean experimental indentation data
for nonstimulated, 1 × 109 M fMLP-stimulated, and
1 × 106 M fMLP-stimulated neutrophils incubated at
room temperature. Symbols, experimental data from Worthen et al.
(1989); solid lines, finite element model results. Mean
experimental indentation force-displacement slopes (or stiffnesses)
and best-fit Maxwell model parameter values are listed in Table 1.
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Capillary flow
The indentation results can be used with the modeling data
presented (see Modeling) to define physiologically relevant dimensional and dimensionless parameter ranges for the capillary flow simulations. Considering that Rmin ranged between 2.5 and 3 µm, a varied between about 8 and 180 µm, P
ranged from 20 to 80 Pa, was equal to 31 pN/µm, G
ranged from 185 to 1350 Pa, and Rcell was
assumed to be 4 µm, the following dimensionless parameter ranges of
interest are determined
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(20)
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In carrying out the simulations, * was fixed in the narrower
range of 0.26 < * < 0.31 (corresponding to the physiological values of P = 40 Pa, = 31 pN/µm, and
Rmin = 2.5-3 µm), To reduce the total
parameter space spanned. This modeling assumption is justified by the
following observations. First, we found that the effects of * on the
cell's transit time were significant only for the smallest
a* and R* due to the relatively mild deformations incurred in this study (for example, in the extreme constriction case
of R* = 0.625, increasing * by more than ten-fold from
0.04 to 0.62 increased the transit time by 22% for a* = 2.19, whereas it increased the transit time by only 7% for
a* = 44.2). Second, we found that the effects of * on
reducing the effective driving pressure drop cannot be accounted for by
the simple expression, Peff = P Pcrit = P 2[(1/Rmin) (1/Rcell)], applicable to micropipette
aspiration (Evans and Yeung, 1989). The reason is that
during flow into the capillary constrictions analyzed here, the radii
of curvature of the leading and trailing end caps of the cell change
continuously, whereas, in micropipette aspiration, they remain
approximately constant (particularly the leading radius of curvature
interior to the pipette) for the duration of flow. For these reasons,
we chose to investigate the dependence of T* on
a*, R*, and G* for the physiologically
relevant range of * specified above, as a compromise between
practicality (limiting the number of simulations required) and the lack
of a simple expression to account for the effects of surface tension on
the characteristic time scale µcell/P. We
also note that the lower bound on R* is decreased to 0.50 only for the viscous deformation-dominated limit G* 1, to broaden the scope of the study; the minimum value of G*
had to be increased from 2 to 8 due to convergence difficulties; and
that µcell does not appear in the independent
dimensionless groups because it is only used to scale T.
Typical deformed finite-element mesh configurations of two different
cell-capillary models are depicted in Fig.
7. In each case G* = 80 and
R* = 0.625. The models differ only in that a* = 2.19 and 44.2 in Fig. 7 A and B respectively.
Although the entire capillary constriction is visible in Fig. 7
A due to the small constriction radius of curvature, only a
small region of the constriction is visible in Fig. 7 B due
to its large radius of curvature. In each model, the portions of the
cell that are internal and external to the capillary constriction are
nearly spherical, as expected due to the effects of surface tension.
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FIGURE 7
Typical deformed mesh configurations of the cell
(G* = 80) entering two model capillary constrictions with
the same minimum constriction radii (R* = 0.625) but
different constriction radii of curvature, corresponding to
(A) a* = 2.19 and (B) a* = 44.2. The cell is moving from left to right.
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For each value of R* and a* analyzed,
T* exhibits a clear limiting behavior with respect to
G* (Fig. 8 A, B, and
C). In each case,
T* reaches a plateau for large G*, the transition
occurring in the vicinity of G* 
20. The limiting
behavior represents the parameter range in which the shear modulus of
the cell is significantly greater than the driving pressure drop, so
that the Maxwell model behaves essentially as a purely Newtonian fluid
(i.e., elastic deformation during transit is negligible compared to
total viscous deformation). For G* < 20, however, elastic
deformation of the cell becomes significant and the reduction in
transit time grows sharply as G* 0, particularly in the
mildest constriction case (Fig. 8 A, R* = 0.75). This result
is intuitive because in the limit of small G* and
R* approaching one, the cell will flow through the
constriction purely elastically, without any viscous deformation at
all. Although T* depends strongly on R*, it also
exhibits a significant dependence on a* for all values of
R* simulated (Fig. 8, A, B, and C).
Additionally, the dependence of T* on a* appears to be independent of G* in the viscous deformation-dominated
limit G* 1.
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FIGURE 8
Dimensionless cell transit time, T* [Tp/µcell], versus
dimensionless cell elastic shear modulus, G* [Gcell/p], for dimensionless
minimum constriction radii, (A) R* (Rmin/Rcell) = 0.75, (B) R* = 0.6875, and (C) R* = 0.625, and three constriction radii of curvature, a*
(a/Rcell). Comparison between numerical
results (symbols) and Eq. 21 (curves).
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Within the approximate range of parameter values specified by Eq. 20, the following dimensionless relationship was found to fit the data
(R2 = 0.997) (Fig. 8 A, B, and
C)
![T*=0.58(a*)<SUP>−0.50</SUP> [(R*)<SUP>−5.0</SUP>−1]](/content/vol83/issue4/fulltext/1917/img024.gif)
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(21)
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In light of the observation that T* becomes independent
of G* for G* 1, we now focus on this
(Newtonian) regime, in which the relation for T* (Eq. 21)
simplifies to (R2 = 0.997),
![T*=0.58(a*)<SUP>−0.50</SUP> [(R*)<SUP>−5.0</SUP>−1].](/content/vol83/issue4/fulltext/1917/img026.gif)
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(22)
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Figure 9 illustrates the computed
dependence of T* on R* and a*, where
each simulation data point has been obtained from the maximum
G* data point shown in Fig. 8, and, as noted earlier, additional data have been generated to represent the case of
R* = 0.50.
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FIGURE 9
The dependence of dimensionless transit time,
T*, on dimensionless constriction radius of curvature,
a*, and minimum constriction radius, R*, for the
viscous deformation dominant regime (G* 1). Data
points are from the viscous limit simulations (G* = 80) and
lines are produced from Eq. 22.
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Though the transit time expressions (Eqs. 21 and 22) fit the simulation
data within the ranges of the dimensionless parameters given in Eq. 20,
in the limit as a* 0, corresponding to a "sudden contraction" capillary geometry, Eqs. 21 and 22 suggest that
T* increases without bound, a result that is clearly in
error. T* must exhibit a limiting behavior well before the
singular limit of a* = 0. Although the precise value of
a* at which this transition takes place is not currently
known, scaling arguments would point to a value on the order of one,
when the constriction radius of curvature is equal to the cellular
radius. This result is confirmed by the agreement between the
analytical transit time relationship derived by Huang et al. for the
flow of a Newtonian droplet into a micropipette and Eq. 22 for
a* = 1 shown in Fig. 10.
Note that, in Fig. 10, we compare T* = TP/µcell of Eq. 22, which is
strictly only valid for 0.26 < * < 0.31, with
T*pipette = TPeff/µcell of the
micropipette aspiration model, which, as mentioned earlier, analytically accounts for the effects of on the transit (or aspiration) time by defining an effective driving pressure drop, Peff P 2[(1/Rmin) (1/Rcell)] (Yeung and Evans,
1989). This leads us to further observe that, for the
micropipette limit, the effect of on T is explicitly
known, and that there exists a critical pressure drop,
Pcrit = 2[(1/Rmin) (1/Rcell)], below which T becomes
infinite. Although a similar critical pressure drop must exist for the
capillary model analyzed here, it is complicated by the different
geometry and cannot be described by such a simple algebraic expression.
We can say, however, that the critical pressure drop in our gradual
entrance geometry will be less than the corresponding value in a
micropipette, because the external radius of curvature will be
Rcell.
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FIGURE 10
Comparison between the Newtonian limit (G*
1) transit time expression of this study (Eq. 22) for a* = 1 and the micropipette aspiration time expression for a Newtonian
droplet with surface tension (Eq. 41 from Huang et al.,
2001), indicates that the capillary model transit time
expressions presented in this study lose their dependence on
a* when a*  1.
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DISCUSSION |
In this study, we analyze neutrophil transit through individual
capillary segments in the pulmonary microvascular network and present
several new results. First, we have demonstrated that a Maxwell model
with constant bounding surface tension is capable of reproducing the
mechanical behavior of neutrophils not only during micropipette
aspiration (Dong et al., 1988; Skalak et al., 1990), but also during experiments of neutrophil indentation. Second, the magnitude of the changes in cellular mechanical properties resulting from various levels of fMLP-stimulation has been estimated using the model. Finally, in an independent series of simulations, we
have used the model to examine the importance of radius of curvature at
the entrance to a capillary segment on neutrophil transit time and
found it to be a critical factor. Simple algebraic expressions were
derived that fit the numerical predictions over a wide range of
parameter values. Future studies of neutrophil transit will need to
take entrance geometry into account if accurate estimates are to be
made. Taken together, these results suggest that a Maxwell model is
capable of capturing neutrophil mechanics under a variety of
deformations, and that such models can be useful in calculating
capillary transit times provided capillary geometry is adequately described.
The neutrophil model
Comparison to previous studies
Nonstimulated neutrophil response has been studied intensively
during the past two decades (Schmid-Schönbein et al.,
1981; Dong et al., 1988; Evans and Yeung,
1989; Needham and Hochmuth, 1990; Zahalak
et al., 1990; Lipowsky et al., 1991;
Skalak et al, 1990; Tran-Son-Tay et al.,
1991; Warnke and Skalak, 1992; Tsai et
al., 1993; Zhelev et al., 1994; Drury and
Dembo, 1999) primarily using micropipette aspiration and
recovery techniques (incurring small and large deformations, imposing
high and low rates of deformation, and using small and large pipettes)
in conjunction with analytical and numerical methods to determine
appropriate neutrophil models and their parameter values. Although the
range in viscosities found in small deformation studies (Table
2) is quite significant (6.5 to 89 Pa·s), the nonstimulated cellular viscosity found in the present
study using indentation (30.8 Pa·s) is seen to be in good agreement.
Studies using the Maxwell model with cortical tension have reported
nonstimulated cellular shear moduli as low as 28.5 Pa (Dong et
al., 1988) for very small deformations and as high as 250 Pa
(Dong and Skalak, 1992) for larger deformations (Rmin/Rcell = 0.5),
bracketing our result of 185 Pa. Zahalak et al., (1990)
studied the mechanical response of nonstimulated and fMLP-stimulated
neutrophils using a finite element model of the same experimental
indentation data that this study is based on, though they neglected
viscous effects in treating the cells as purely elastic. They found
that treatment with 108 M fMLP resulted in an increase in
cellular shear modulus of almost a factor of four, from 118 to 448 Pa,
in reasonable agreement with the results of the present study (Table
1). Only one study with which we are familiar quantified the effects of
fMLP both on leukocyte viscosity and shear modulus (Lipowsky et
al., 1991), finding that topical application (107
M) of the chemoattractant resulted in a 15-fold increase in cellular viscosity and a 5.6-fold increase in shear modulus over nonstimulated cell values. In terms of the shear modulus, this result is consistent with the increases by factors of 3.4 for 109 M and 7.3 for 106 M found in the present study.
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TABLE 2
Published neutrophil viscosity values for small
deformation, nonstimulated neutrophil studies performed at room
temperature
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Limitations of the model
Any model attempting to mimic the structural properties of the
cell using a homogeneous, isotropic continuum description is open to
criticism. Homogeneous representations fail to explicitly account for
the detailed microstructure of the cell, and certainly neglect its
capability for biological remodeling in response to a variety of
factors. Despite their limitations, however, homogeneous, continuum
descriptions have proved useful as a means of simulating cellular
deformations under forcing, perhaps surprisingly so in view of the
simplifications that they imply. To explain the underlying basis for a
cell's mechanical properties, however, clearly requires a model that
takes into account explicitly the microstructure and the biology.
Even within the constraints of a continuum model, however, there are
further limitations in the development of a realistic model. Due to the
absence of quantitative experimental data on the degree of hysteresis
exhibited by neutrophils in indentation and the effects of fMLP on
cortical tension, we needed to assume that the time constant of the
cell's response in indentation
(µcell/Gcell) remained constant
and that the cortical tension was unaffected by the chemoattractant.
These assumptions are in contrast to the observations that hysteresis
actually increases with fMLP concentration (Worthen et al.,
1989) and that fMLP induces F-actin formation beneath the
plasma membrane (Saito et al., 2002), whereas
cytochalasin-D disrupts F-actin formation and also decreases cortical
tension in neutrophils by up to 60% (Ting-Beall et al.,
1995). Although the results of the neutrophil indentation
studies are somewhat limited by these considerations, the dimensionless
transit-time expressions (Eqs. 21 and 22) are independent of any
specific material properties (within the constraints of the expression
given in Eq. 20). In the case that the dimensionless cortical tension
exceeds the maximum limit explored in this study, however, it may be
noted that, in the viscous-dominated limit (G
P) the effect of solely increasing the cortical (surface)
tension of the cell will generally be to increase the cell's transit
time, because the effect of the tension is to reduce the effective
pressure driving the cell through the constriction. The degree of the
effect, however, will depend upon the radius of curvature of the
capillary examined (the effect is larger for smaller radii of curvature).
Several recent studies have focused on modeling the highly viscous,
multi-lobed nucleus as a distinct entity within the neutrophil, with
different mechanical properties from the less viscous cytoplasm (Tran-Son-Tay et al., 1998; Kan et al.,
1998). This contrasts with the more traditional approach of
treating the cell inter
and
TOP
ABSTRACT
INTRODUCTION
