| HOME | HELP | FEEDBACK | SUBSCRIPTIONS | ARCHIVE | SEARCH | TABLE OF CONTENTS |
|
|
|
|||||||
|
|||||||||
METHODS IN CELL PHYSIOLOGY
1Harvard Medical School and 2Department of Ophthalmology, Massachusetts Eye and Ear Infirmary, Boston, Massachusetts 02114; and 3Institute of Mechanics, Faculty V, Technical University, 10623 Berlin, Germany
Submitted 12 November 2003 ; accepted in final form 26 November 2003
 |
ABSTRACT |
|---|
 TOP ABSTRACT EXISTING MODELSMATERIALS AND METHODSRESULTSDISCUSSIONAPPENDIX A: THEORETICAL...APPENDIX B: SOLUTION OF...APPENDIX C: ESTIMATING ERROR...REFERENCES |
|---|
autoperfused flow chamber; intravital microscopy; inflammation; thrombus formation
Previously we introduced (29) the local catheter technique, which expands the capabilities of intravital microscopy of the mouse cremaster muscle by allowing the upstream insertion of reagents directly into the microcirculation. However, despite the advancement of this technique, it shares the limitations of most in vivo models, which derive from having all blood components and the full complexity of the vascular wall. Therefore, we discuss here the existing in vivo and in vitro models and the need for an ex vivo model that would bridge the gap between them. Subsequently, we describe the autoperfused ex vivo flow chamber that fulfills this function.
|
EXISTING MODELS |
|---|
|
TOPABSTRACTEXISTING MODELSMATERIALS AND METHODSRESULTSDISCUSSIONAPPENDIX A: THEORETICAL...APPENDIX B: SOLUTION OF...APPENDIX C: ESTIMATING ERROR...REFERENCES |
|---|
In Vitro Assays Here isolated leukocytes (3), platelets (5), or other cell suspensions are perfused through a chamber, where they interact with immobilized reagents or cells (2). Various modifications of the in vitro flow chamber [e.g., parallel plate flow chamber (40), Stamper-Woodruff assay (58), or cone and plate assay (62)] exist. In most scenarios physiological shear rates are applied to cells interacting with immobilized mediators of cell rolling, arrest, or firm adhesion. Commonly, syringe pumps generate the driving force for the fluid suspensions to pass once through the chambers (40). The cells used in these systems are also in most cases derived from in vitro cultures or human blood. Although cultured cells or human leukocytes may be conveniently generated in sufficient numbers, this is not as easily achieved with murine cells (15). For instance, the procedures to isolate murine leukocyte subtypes are time consuming and not as efficient, and sterile conditions may not be guaranteed because of the number of steps involved, which may unintentionally modify the cells, for example, through activation (49). Therefore, it is uncertain whether their performance would match that of native cells in the circulation.
Prevailing Shear Forces To have a proficient model, quantitative knowledge of the shear forces is necessary. Shear stresses in a system manifest themselves in two ways: they cause pressure loss and are related to the shear rates of the flow, which, in turn, are kinematically determined by the velocity distribution over the chamber's cross section. Deriving shear stresses from the pressure loss is based on balancing forces, which is a universal principle of mechanics and as such avoids dealing with specific fluid characteristics that are based on hypothetical fluid models and estimates of various model parameters. However, this approach can determine only those shear forces that actually enter that balance, that is, the average wall shear stress. Conversely, using the fluid's constitutional laws formally completes the set of equations that are needed to determine the entire spatial distribution of the flow variables, including shear rates and shear stresses at arbitrarily chosen points. However, this approach requires a realistic fluid model and accurate estimates of its parameters, such as viscosity, which is itself dependent on temperature and, in the specific case of blood, also on further properties such as hematocrit (17). More realistic fluid models for blood require the introduction of additional parameters, such as the viscosity's nonlinear dependence on the shear rate (63, 64) or effects of flow history (14). Because it is preferable to remain independent of particular models and parameter estimates wherever possible, the balance of forces approach was chosen for determining the average wall shear stress, consistent with Reference 1.
In summary, in vivo and in vitro systems are adept at their respective domains but also have limitations. In vivo systems offer the entire complexity of a living organism, which may hinder the ability to pinpoint the function of a single molecule or cell type. In vitro systems focus on a few elements and excel in reductionism, sacrificing physiological fidelity. This inspired us to develop the autoperfused ex vivo flow chamber and apply the balance of force approach to characterize its fluid forces. The ex vivo flow chamber aims to reduce the complexity of an in vivo system while offering a more physiological flow environment for the study of murine cells.
|
MATERIALS AND METHODS |
|---|
|
TOPABSTRACTEXISTING MODELSMATERIALS AND METHODSRESULTSDISCUSSIONAPPENDIX A: THEORETICAL...APPENDIX B: SOLUTION OF...APPENDIX C: ESTIMATING ERROR...REFERENCES |
|---|
Components Used to Assemble the Ex Vivo Flow Chamber Translucent solid phase. For the solid phase (Figs. 1, iv, and 2A), microslides from VitroCom (vitrocom.com) were obtained [inner diameters (ID): 0.02 x 0.2 (catalog no. 5002S), 0.03 x 0.3 (catalog no. 5003), 0.04 x 0.4 (catalog no. 5004), 0.05 x 0.5 (catalog no. 5005S), and 0.1 mm x 1.0 mm (catalog no. 5010S)]. Microslides were coated with recombinant murine P-selectin (R&D Systems) at 4°C overnight and were washed with PBS (containing 10% FBS) before the experiments. To block P-selectin the MAb RB40.34 [rat immunoglobulin (Ig) G1, 30 µg/mouse; no. 09482D, Pharmingen] was used.
|
|
Inlet and outlet means. For the in- and outlet means (Fig. 1, ii and iii) heparinized PE-10 (ID 0.28 mm; no. 427401, B&D), PE-20 (ID 0.38 mm; no. 427406), PE-50 (ID 0.58 mm; no. 427411), and PE-60 (ID 0.76 mm; no. 427416) tubing and elastic connection pieces [ID 0.51 mm (no. 508–002) and ID 0.64 mm (no. 508–003), Dow Corning] were used. Sterile heparin in vacutainers was obtained from Becton Dickinson (no. ref. 366480; Franklin Lakes, NJ) to pretreat the ex vivo chamber before perfusion with blood. Heparin activates antithrombin III, resulting in the inhibition of factor Xa (4). It also inhibits thrombin, which reduces fibrin formation and is used to reduce blood activation upon contact with artificial surfaces (4). The size of the tubing was chosen to accommodate convenient cannulation of the carotid artery (Fig. 1, v) and the jugular vein (Fig. 1, vi) of the mouse (Fig. 1, i) (29). PE-10 tubing was connected to the microslides via a small piece of elastic tubing with its ID corresponding to the outer dimensions of the PE-10 tubing and the chamber. To create a tight seal around the connections between the smaller microslides and the tubing, commercially available instant glue was used. Use of the elastic connection pieces provided a versatile interface between the inlet and outlet means and the microslide, which allowed convenient exchange of several coated microslides during an experiment with the same mouse.
Flow regulators. To regulate the velocity of the blood flowing through the chamber, adjustable screw valves (Figs. 1, vii, and 2A) were used to reduce the diameter of the inlet and outlet tubing (Fig. 1, ii and iii).
Side ports. To make infusions into and draw materials out of the ex vivo chamber, side ports were placed before and after the microslide in the inlet and outlet means with three-way connection pieces (Fisher Scientific; Figs. 1, viii, and 2A).
Pressure measurement. To measure the pressure of blood before and after the solid phase a blood pressure transducer with a range between –10 and 300 mmHg was used (no. 60-3002, Harvard Apparatus). Three-way stopcocks (no. 2C6302, Baxter) were used to connect tubing from both side ports to the two Luer-Lock fittings of the transducer's sensor dome. This allows convenient switching between the connecting tubing from the high- and low-pressure ends. To monitor the systemic blood pressure of the mouse the side port in the inlet tubing was connected to the transducer while the flow restrictor in the outlet means was in the closed position (Fig. 1).
Temperature control. To prevent a drop in the temperature of the extracorporal blood, the ex vivo flow chamber was kept submerged in a petri dish with constant inflow and outflow of water from a temperature-controlled bath. With this method the temperature of the extracorporal blood is kept at physiological or any other experimentally desired level.
Microscopy. The ex vivo flow chamber is designed to provide quantitative visual analysis of the interaction of leukocytes, platelets, or other blood components with immobilized reagents on the solid phase of the chamber by microscopy (Fig. 2A). An upright microscope with a saline immersion objective (SW x40, 0.75 numerical aperture) was used. Optionally, x25 to x100 oil-immersion objectives were used for better resolution. Real-time images were recorded with a charge-coupled device (CCD) camera (Hamamatsu) connected to an S-VHS. Gridlines were used to calibrate the spatial dimensions of the recorded material (Fig. 2A). Velocity of rolling leukocytes was quantified as described previously (29).
Calculating Shear Forces The theoretic fundamentals are given in APPENDIX A. Here the notations and requirements are summarized and their results are presented.
Geometry of the solid phase. We presume flow through a prismatic vessel with rectangular cross section, such as the microslides used in the experiments. Let a denote the length of the long (wide) side of the rectangle and b the length of the short side. We introduce Cartesian coordinates, x for the direction of flow and y and z for the cross section (Fig. 2B), such that y = ± a/2 and z = ± b/2 identify the boundaries of the flow, i.e., the chamber walls.
Requirements. The cross section must be constant over x, because the microslide must be prismatic to avoid Bernoulli-like pressure changes due to velocity changes enforced by flow continuity. Another Bernoulli contribution would be changes in gravitational potential; therefore, the relevant pressure difference is to be taken between points of equal height. We assume the fluid to be homogeneous and incompressible and suppose that the flow is laminar and that entry effects have diminished such that all velocity vectors point into the main direction of the flow and the fluid's extra stresses have become independent from x. Furthermore, the fluid is assumed to reflect the symmetries of the rectangular cross section in its flow-induced stress distribution. Finally, we assume that unsteady flow acceleration and deceleration, if any, employs negligible forces only (as in our case with low flow velocities and low oscillation frequencies) or gets averaged out during pressure measurement.
Average wall shear stress. Given measures 
of the pressure at two points x0 and x1 of the above flow, the balance of forces results in
 | (1) |
p/x is the measured pressure difference quotient (note that p/x is negative, hence –p/x is positive). This equation determines the average wall shear stress when given a pressure difference over a definite section length. Note that the derivation does not require use of a specific fluid model and that it holds even if the pressure measurement is influenced by other flow-induced stresses, e.g., extra axial stress. It is also valid for arbitrary shapes of the cross section, in contrast to the following result.
Maximum wall shear stress. Suppose that the point of greatest flow velocity is in the center of the cross section and that the points of maximal wall shear stress are those wall points closest to it, that is, the midpoints of the long sides of the cross section. Assume further that the side length ratio a/b of the rectangular cross section is large, e.g., a/b = 10 or greater. Then the flow behaves much like a parallel plate flow, and the following formula closely approximates the greatest wall shear stress
 | (2) |
Characterizing Fluid Dynamics of the Ex Vivo Chamber
Because of the rectangular cross section of the ex vivo flow chamber some rules governing the relations of key variables such as center line velocity, (volumetric) mean velocity, wall shear rates, wall shear stresses, and pressure drop may be different from those applying to a cylindrical pipe. For example, the typical ratio of maximum velocity to volumetric velocity (vmax/vvol) 
1.6 observed for the flow of blood in cylindrical glass tubes with internal diameters between 23 and 90 µm (7) need not hold for rectangular cross sections. To estimate the relations in the ex vivo chamber, exemplary numerical simulations were performed. The simulations require a definite constitutive equation for the fluid. A basic fluid model is the linear Newtonian model, which assumes a constant viscosity but does not represent the complex rheological properties of blood. To account for the effect known as shear thinning a nonlinear model with a viscosity that decreases with increasing shear rate was proposed (63). The linear Oldroyd-B fluid model (14) allows simulation of viscoelastic effects such as relaxation and retardation over time. The generalized Oldroyd-B model used in References 28, 48, and 64 combines both approaches and is therefore nonlinear and time dependent. For our steady-flow simulations, we chose a nonlinear isotropic shear thinning model (63), which fits experimental blood flow data exceedingly well (64). We have verified that when the generalized Oldroyd-B fluid model is used in a steady prismatic flow scenario the equation of motion remains the same as in the pure shear thinning case, thus creating the same velocity profile and shear stress distribution. Similar results for cylindrical pipes can be found in Reference 64. Note, however, that the generalized Oldroyd-B model would nevertheless generate an additional axial stress 
(64). As explained in APPENDIX A, additional axial stress does not affect the determination of the wall shear stress by the balance of forces approach; hence, the nonviscoelastic shear thinning model is sufficient for steady-flow scenarios as long as one is not interested in determining .
Additional notation. Let ex, ey, and ez be the orthonormal basis vectors accompanying x, y, z, and let v = vex denote the velocity profile over the cross section. Introduce the spatial nabla operator
 |
Shear rates. Because v is independent from x, which is a direct result from applying the continuity equation 
·v = 0 for homogenous incompressible flow to the velocity vector field v = vex, the velocity gradient v and the corresponding strain rate tensor C have only two independent components
 | (3) |
being the shear rates derived from the distribution of v over the cross section. Because v does not vary along x, neither do the strain rates.
Constitutive law. The chosen constitutive law, modeling an isotropic shear thinning fluid, largely resembles that of an incompressible Newtonian fluid
 | (4) |
representing the apparent viscosity. However, to model the shear-thinning properties of blood, is made dependent on the shear rate as follows
 | (5) |
0 and 
are the viscosity limits for shear rates approaching zero and infinity, respectively, and 
is a sensitivity parameter with the dimension of time. The argument 
is the magnitude of the shear rate, defined as a suitable norm of the strain rate tensor
 | (6) |
simplifies to
 | (7) |
= 0 or 0 = , remains constant and thus reduces the fluid model to the Newtonian case.) With C as of Eq. 3, the stress tensor T features the spherical pressure term plus only two symmetric shear stress components
 | (8) |
Equation of motion. Applying the local balance-of-forces equation ·T = 0 and successively substituting T, C, and v results in the equation of motion
 | (9) |
= · being the Laplace operator. Equation 9 also indicates that the spherical pressure p varies over x only and is therefore constant over the cross section. Furthermore, the right-hand side of Eq. 9 is determined by derivatives of v and is thus independent from x. Equality of both sides implies that both are constant over the entire domain of this particular flow and the pressure drop is linear in x. This justifies our writing of p/x instead of 
p/x.
Using the chain rule to decompose results in
 | (10) |
 | (11) |
Boundary conditions. In addition to the partial differential equation for the velocity profile, boundary conditions must be specified to disambiguate the solution. Applying the no-slip requirement to the rectangular cross section, we get
 | (12) |
Treatment of the Newtonian case. With = const., Eq. 10 reduces to the classic Poisson equation
 | (13) |
Treatment of the non-Newtonian case. Because of the nonlinearity of Eq. 10 in the shear thinning case, the solution of the boundary value problem cannot be given explicitly. To obtain the solutions a finite-difference method with a regularly spaced grid was used. Because of the symmetries of the velocity profile, the domain for the computation was restricted to the quarter y 
[0, a/2], z [0, b/2]. The symmetries imply v/y = 0 at the cut y = 0 and v/z = 0 at z = 0, which are Neumann-type boundary conditions for the restricted domain. Discretization was performed with difference quotients consistent up to second order, resulting in difference equations where the value of v at each grid point depends on its eight neighbors. Note that this dependence is nonlinear in the main four neighbors that are needed to approximate v but linear in the diagonal neighbors. To solve the system, an iterative method based on successive overrelaxation (SOR) and some concepts from multigrid techniques and wave front computations was chosen. After initialization, the algorithm sweeps the domain diagonally and, for each diagonal tuple of unknown values of v, builds and solves a simple three-band linear system based on the finite-difference equations and the discretized boundary conditions, regarding values from adjacent diagonals as being readily given instead of unknown. The computed changes to the current diagonal are relaxed or overrelaxed by a factor 
chosen from the range 0 < < 2. The sweeps are repeated until the maximum deviation from the difference equation (that is, the maximum change requested by the linear solver before overrelaxation) is found to be sufficiently low. Because the overrelaxation can destabilize the convergence in regions of high nonlinearity, it was combined with grid refinement by using bicubic interpolation.
Computations
Initial numerical computations were performed ad hoc with the Pari/GP interpreter on an i686-Linux-GNU platform. The computations involved evaluation of Fourier series using Pari's series functions to obtain flow velocities, strain rates, or shear stresses. To account for nonlinear effects such as shear thinning, an interactive program, RECTFLOW, was written in C99. The inputs are the chamber's cross-sectional dimensions, pressure drop per chamber length, viscosity parameters, grid resolution, and the overrelaxation factor. All parameters can be changed interactively, and the approximation process can be stopped, single-stepped, and continued. Screen outputs include maximum and average velocity, maximum shear rates and stresses at the walls, and the approximation status. File outputs include the detailed velocity profile and its volumetric average, shear rate distributions, local viscosity, and derived variables including shear stresses and dissipation power density. Convergence was monitored by the maximum change per sweep relative to the product yz x p/x and by testing the balance of force between the boundary integral of the computed wall shear stresses and the prescribed pressure drop. For the latter step, the AWK tool was applied to RECTFLOW's output files. The resulting fluid variable distributions were visualized with Gnuplot.
|
RESULTS |
|---|
|
TOPABSTRACTEXISTING MODELSMATERIALS AND METHODSRESULTSDISCUSSIONAPPENDIX A: THEORETICAL...APPENDIX B: SOLUTION OF...APPENDIX C: ESTIMATING ERROR...REFERENCES |
|---|
Description of the Ex Vivo Flow Chamber The ex vivo flow chamber comprises various components that allow the extracorporal flow of murine blood, originating from the carotid artery, passing through an inlet means, passing next through a translucent solid phase, proceeding through an outlet means, and subsequently reentering the animal's body through the jugular vein (Fig. 1). The solid phase is a hollow, translucent structure that serves as the observational area for microscopy and can vary in its length and cross-sectional dimensions (Figs. 1 and 2A). These dimensions influence the flow properties by creating different pressure gradients between the high- and low-pressure ends of the chamber. To assemble the ex vivo flow chamber, microslides of 1–5 cm in length and 0.004–0.1 mm2 in cross-sectional surface area were used as the solid phase. The force generated by the animal's own cardiac function perfuses the blood continuously through the ex vivo chamber, eliminating the need for external vacuum pumps as used in conventional in vitro flow chambers to generate flow. The inlet means is comprised of tubing connected on one end to the carotid artery (high-pressure end; Fig. 1, v) of the mouse and on the other end to the translucent solid phase (Fig. 1, iv). The outlet means is comprised of tubing connected on one end to the solid phase and on the other end to the jugular vein (low-pressure end; Fig. 1, vi). Three-way connections (Fig. 1, viii) in the inlet and outlet means provide external access to the ex vivo chamber for intake and outtake of fluids or reagents or measurement of fluid pressures.
To expand the capability of the ex vivo chamber, the solid phase and the inlet and outlet tubing were submerged in a petri dish with inflow and outflow from a water bath. With this improvement the ex vivo chamber can be kept under a range of temperatures without a significant change of the flow conditions, because the major parameters (i.e., the animal's systemic blood pressure) are not affected. If a temperature other than 37°C is chosen, on reentry of the blood from the chamber into the animal, it quickly resumes the core temperature of the animal without significantly affecting it, because the extracorporal blood volume— 47 µl when including the internal volume of the three-way connection pieces or <10 µl without them—is substantially lower than the body mass of the mouse.
Measurement of Pressure Drop Over the Ex Vivo Chamber Here the inlet and outlet means were attached to a pressure transducer via the three-way connections while the mouse perfused the chamber with blood (Figs. 1, viii, and 2A). However, to obtain accurate results it is imperative to keep losses before and after the microslide proper as low as possible. Such extra losses occur along the tubing, at the entry and exit points of the microslide, or at any other connection pieces. External pressure losses can be diminished by dimensioning the flow chamber such that the additional flow resistance of the external components is negligible. Therefore, to minimize these losses we used for those areas tubing (i.e., PE-50 or larger) with significantly larger cross-sectional areas than the microslide.
Estimating Pressure Measurement Errors APPENDIX C describes our approach to estimating the error in the drop of pressure introduced by the transport of blood through the PE-50 tubing used to connect the three-way pieces and the microslides (Fig. 2A). Table 1 suggests that the 1.0-mm x 0.1-mm microslide may be inappropriate for pressure drop measurements unless shear thinning can be supposed to be insignificant for other reasons, e.g., when shear rates are very low or because the change in cross-sectional area between the microslide and the connection pieces is rather moderate, which limits the multiplicative effect of shear thinning on the measurement error. Slight reductions in the internal dimensions of the microslide make the error estimate drop drastically. The 1.0-mm x 0.1-mm microslide should be avoided for yet another reason: when the 1.0-mm x 0.1-mm microslide was used, the entry pressure was noticeably lower than the animal's systemic blood pressure, indicating a significant shunt that may disturb the organism's circulation.
|
Between the three-way connections and the vessels, PE-10 tubing (ID = 0.28 mm) was used. These connection pieces also contribute to the total ex vivo flow resistance; however, their resistances do not contribute to the error, because these parts are outside of the measured section.
Calculated Shear Stresses
We used Eq. 2 for rectangular cross sections with large aspect ratio, resulting in
 | (14) |
x[slide]) can be used to calculate 
max from the measured pressure drop (–p)[slide]. Table 1 shows values of that factor for several microslides of length x[slide] = 50 mm, including the conversion factor from milligrams of mercury to dynes per square centimeter, whereas Table 2 shows examples of max calculations from measured pressure drops that demonstrate the range of shear stresses that can be obtained depending on the microslide dimensions. These max values are calculated for experimentally measured pressure drops in wild-type animals. Knockout mice such as endothelial nitric oxide synthase (eNOS)–/– mice that have a higher arterial blood pressure (55) would generate a larger pressure drop resulting in higher max values within the same experimental design compared with normotensive wild-type animals.
|
Numerical Simulations
Our simulations confirm that flow of a Newtonian or an isotropic shear-thinning fluid through a rectangular cross section with side length ratio a/b = 10 or greater well approximates the parallel plate flow case and thus supports the approximation formula for max.
Figure 3 shows characteristics of Newtonian flow, whereas Figs. 4 and 5 illustrate examples of non-Newtonian flow. In all plots, the cross section coordinates y and z (Fig. 2B) are both normalized with respect to b/2 (half the short side); therefore, the normalized ranges are [–a/b; +a/b] x [–1; +1] (with a/b = 10).
|
|
|
For comparison with the non-Newtonian flow simulation, we first present the Newtonian flow characteristics. 1) For larger aspect ratios in general and for a/b = 10 in particular, the velocity profile is practically constant throughout most of the y range except near the ends posed by the short sides and is almost perfectly parabolic along the z-axis (Fig. 3A). For a/b = 10, 
70% of the y range exhibits constant values. 2) The distributions of shear rate and shear stress inherit the broad constancy over y from the velocity profile (Fig. 3, B–D). As a consequence, 
and yx are mostly zero except near the short sides. Furthermore, the distributions of and zx are mostly linear over z because of the parabolic shape of the velocity profile. 3) The maximum wall shear stress at the short sides is smaller than that at the long sides by a factor of 0.74 (Fig. 3, B–D). 4) In the corners, all shear rates and shear stresses vanish together with the flow velocity, thus creating resting zones. 5) For a/b = 10, the ratio of the maximum velocity in the center of the microslide's cross section to the volumetric mean velocity is vmax/vvol = 1.60. For a/b 
, the influence of the short sides on the velocity distribution vanishes and the ratio vmax/vvol converges to 1.5, as characteristic for a profile that is constant over y and parabolic over z. 6) Given the viscosity of the fluid, the parabolic velocity profile over z allows calculation of max from the magnitude of the z component of the velocity gradient at the long side, resulting in
 | (15) |
Figures 4 and 5 show characteristics of three simulated flows of a shear-thinning fluid through a rectangular cross section of size 0.4 mm x 0.04 mm with viscosity parameters o = 73.8 cP, = 5 cP, = 13.42 s, according to data from page 34 of Reference 28 for whole human blood (hematocrit 40%) at room temperature rather than body temperature. The flow simulations for three different pressure drops were chosen to clearly demonstrate the effect of shear thinning on various parameters: –p/x = 20 (Figs. 4 and 5, top), 200 (Figs. 4 and 5, middle), and 10,000 (Figs. 4 and 5, bottom) dyn/cm3 (1,000 dyn/cm3 = 0.75 mmHg/cm). For body temperature, lower values of the viscosity parameters o and can be expected. However, these would result in the same phenomena at correspondingly lower pressure drops. Generally, at very small pressure drops, shear rates are so small that the viscosity remains close to o (Fig. 4B, top), thus staying nearly constant, which makes the velocity profile resemble a Newtonian one (Fig. 4A, top). At higher pressure drops, the viscosity changes drastically over the cross section (Fig. 4B, middle), particularly over z, while preserving constancy over the inner part of the y range, as expected for a scenario close to parallel plate flow. The velocity profile therefore looks blunted over z (Fig. 4A, middle). At very high pressure drops, shear rates are so high that the viscosity almost immediately drops to its lower limit when leaving the shear-free center (Fig. 4B, bottom). The viscosity is therefore constant over those parts of the cross section where it can be applied to nonzero shear rates. Consequently, the velocity profile again looks Newtonian (Fig. 4A, bottom). Although the viscosity near the center can rise steeply by more than one order of magnitude, this would not cause extra non-monotonicities in the shear stress distribution (Fig. 5, bottom). Therefore, high values of the viscosity over a very narrow area at the center and in the corners do not affect the flow significantly. Consequently, at very small and very high pressure drops, the distributions of shear rate and shear stress are again very similar and Newtonian-like (Fig. 5, top and bottom). In the range between the two Newtonian-like extremes, the distribution of the shear rate changes shape significantly (Fig. 5A, middle) because it is derived from the blunted velocity profile. However, the shear stress distribution remains quite similar to its Newtonian version, at least when linearity over z is considered (Fig. 5B, middle). This is because of the linear momentum balance ·T = 0, which imposes severe restrictions on any variation of the stress distribution. As in the Newtonian case, resting zones with vanishing velocity, shear rates, and shear stresses are created in the corners. This is a general consequence of the no-slip boundary condition applied to the rectangular shape of the cross section.
Figure 6 summarizes shear thinning characteristics over a wide range of pressure gradients. As Figs. 4 and 5 suggest, there is a transition region where the fluid behaves in a highly non-Newtonian manner, whereas outside that region the characteristics are similar to the Newtonian case, with the apparent viscosity approaching o for small pressure drops and for high drops. Figure 6A demonstrates this region of transition with a plot of velocities vvol and vmax vs. the pressure drop. Besides showing the asymptotic Newtonian-like behavior, Fig. 6A also indicates that the overall behavior is nonlinear and therefore non-Newtonian: whereas the pressure drop varies over less than four orders of magnitude, the velocities span nearly five.
|
Figure 6B shows how characteristics change in the non-Newtonian region. The ratio vvol/vmax rises from its Newtonian limit at 0.625 to nearly 0.675. As the velocity profile gets blunted, its slopes at the walls and thus the wall shear rates rise relative to the Newtonian value 4vmax/b suggested by Eq. 15. The maximal shear rate, which occurs at the middle of the long sides, exhibits the strongest change, followed by the shear rate at the middle of the short sides. The latter changes less drastically, and therefore its ratio to the former dips below the Newtonian value 0.74 in the non-Newtonian region. In contrast, the ratio of the main two wall shear stresses remains almost invariant because of the restrictions imposed by the linear momentum balance. Figure 6B also shows that the absolute value of the maximal wall shear stress is accurately determined by Eq. 2.
Laminarity
To account for the quality of flow we calculated Reynolds numbers (Re), with Re = 
vavgb/. As an example, for b 
0.01 cm, vavg < 3 cm/s, and = 1 g/cm3, = 5 cP, the resulting Re is less than 0.6, which is well below any turbulence threshold (48). In all chamber geometries used our Re calculations reaffirmed the rule that the microcirculation features low Re (48) and is thus laminar. Furthermore, in the ex vivo experiments, the laminarity of the blood flow was visually verified through the streaks of cells passing through the visual field.
Study of Leukocyte Rolling in the Ex Vivo Flow Chamber To demonstrate the capability of the ex vivo flow chamber for the study of leukocyte interactions we coated the solid phase (microslide) with recombinant murine P-selectin. P-selectin (8) is an endothelial ligand for P-selectin glycoprotein ligand-1 (PSGL-1; Ref. 46), expressed on leukocytes, and mediates leukocyte rolling (44). Whereas on uncoated microslides we did not observe any leukocyte rolling or significant numbers of firm adhesion, on P-selectin-coated microslides (50 µg/ml) leukocytes rolled with average velocities of 35 µm/s (Fig. 7). To test whether this interaction was specific for P-selectin we systemically applied a P-selectin-blocking MAb, RB40 (39), which abolished the rolling. Interacting leukocytes were visualized as round, distinct objects with individual rolling velocities ranging in the order of 5–80 µm/s. Noninteracting blood components appeared as blurred streaks in the standard camera frame speed.
|
Platelet Interactions in the Ex Vivo Flow Chamber Shear is an important factor for platelet function (47). Approximately 70% of the total surface area of the solid phase of the ex vivo chamber provides constant levels of the highest shear rates along the x- and y-axes (Fig. 5). These shear rates decrease when approaching either end of a (edges of the microslide) along the y-axes (Figs. 3D and 5). This pattern of shear allows the study of platelet function under a range of shear rates. Native murine platelets readily interacted with uncoated surfaces in the constant-shear areas of the ex vivo flow chamber in single fashion along x, the path of the blood flow (Fig. 8A). However, in areas closer to the junction of the chamber's walls a qualitative change in the pattern of the platelet interaction was observed. There, platelets rolled along the chamber surface in strands of grapelike conglomerates (Fig. 8B). As our simulations show, a rapid drop of shear rates along the y-axis occurs in these areas, which may be the cause of this mode of platelet function. These strands of grapelike conglomerates appeared to be an intermediate developmental stage, a prelude of individual platelets on their way to full thrombi, and appeared to be a result of the sustained blood flow over time at specific shear levels. At the actual junctions of the walls of uncoated slides, areas of lowest flow velocity and shear rates, formation of thrombi starting within the first 5 min of blood flow through the ex vivo chamber was observed (Fig. 8C).
|
|
DISCUSSION |
|---|
|
TOPABSTRACTEXISTING MODELSMATERIALS AND METHODSRESULTSDISCUSSIONAPPENDIX A: THEORETICAL...APPENDIX B: SOLUTION OF...APPENDIX C: ESTIMATING ERROR...REFERENCES |
|---|
Advantages of Recirculating Blood In the ex vivo model, the blood cells reside and circulate within their physiological habitat, the animal's vasculature, except during their brief extracorporal passage through the ex vivo flow chamber. This characteristic eliminates the need for isolation of cells before the experiment and minimizes the likelihood for contamination or autostimulation of the cells during the often time-consuming and laborious isolation procedures. Furthermore, exogenous cells are kept in media with a physical (viscosity, partial pressure of gasses, etc.) and chemical (glucose concentration, etc.) composition differing from that of blood and produce metabolic products that accumulate around them. Such changes are likely to affect the cells, for instance, by changing the expression pattern or conformation of adhesion molecules, which could result in artificial or uncontrolled experimental conditions (49). With the ex vivo model, crucial physiological parameters—such as oxygenation of the blood, pH, reduction of waste products, and blood pressure—are continuously maintained at homeostasis by the animal's body.
Also, soluble adhesion molecules occur at various concentrations in plasma and are thought to have physiological functions (54). For instance, soluble P-selectin is procoagulant (5) and soluble ICAM-1 can inhibit inflammation and adhesion of rhinovirus to ICAM-1 (13, 61). In vitro experiments often lack these important factors, which may alter the outcome of an experiment, whereas the cells in the recirculating ex vivo model are exposed to physiological levels.
The Driving Circulatory Force Conventional in vitro systems are driven by external vacuum pumps. Even though the flow rate can be regulated, in cases with limited cell suspension volume, which is usually passed through the chamber once, the duration of an experiment is restricted to minutes (5, 40). In vivo observations of the microcirculation do not have this limitation, because in intravital microscopy a microvessel is perfused by the animal's heart in a close circuit, which makes it possible to observe the microvessel virtually indefinitely. The ex vivo flow chamber adds this advantage of an in vivo system to a flow chamber system through its integration within the vascular circuitry of the mouse.
Flow and Shear Rates Shear forces are crucial for maintenance of the physiology of blood-borne cells and the endothelium (24, 25). For instance, shear above a critical threshold is required to promote and maintain rolling interactions through L-selectin (20). In the autoperfused ex vivo model the native cells are continuously under physiological shear conditions. In vivo an animal's systemic blood pressure and vascular dimensions mainly determine the flow rate in a vessel. Thus the rate of flow of observed vessels is in general not within the sphere of the experimenter's influence, whereas in an in vitro flow chamber system, specific flow rates can be chosen (40, 41). The flow restrictors in the inlet and outlet means are translations of this advantage of an in vitro system into the ex vivo model, allowing regulation of the flow rate and the prevailing shear forces. We measured drops of pressure over the lengths of the ex vivo chamber and obtained through a balance of forces approach the shear forces at the inner chamber surfaces for the various cross-sectional dimensions, determining that the ex vivo chamber offers a wide range of physiological shear stresses (up to 40 dyn/cm2). These values correspond to the physiological time-averaged shear stress levels in the human arterial and venous systems, which are 20–30 and 0.8–8 dyn/cm2, respectively (52). We estimated the errors for the loss of pressure apart from the flow through the microslides and found that, indeed, the substantial portion of the measured pressure gradient is due to the loss over the microslides, thus making corrections unnecessary, except in the case of the 1.0 x 0.1-mm microslide.
Study of Platelet Interaction and Thrombus Formation Thrombus formation is a complex process involving cellular and noncellular components, i.e., platelets and fibrin, respectively (18), and is linked to clinical scenarios such as atherosclerosis or myocardial infarction (43, 51). During this process, flow, neutrophil and platelet adhesion, and blood coagulation are intricately interrelated (26). This complexity poses a challenge to the study of thrombus formation in vitro, where only a few elements are present (50, 60). The ex vivo flow chamber offers a powerful model to study platelet function and its relation to thrombus formation or leukocyte recruitment (16), because all cellular and plasma components are present. For instance, as a result of the emergent hemodynamic properties of the ex vivo chamber we observed the previously unreported phenomena of grapelike platelet conglomerates moving along the surfaces of the chamber in areas that show a rapid drop of shear perpendicular to the direction of flow. Rapid changes in shear do occur in vivo, i.e., around sclerotic valves or in stenotic arteries (21). Furthermore, pathological shear changes promote platelet aggregation in vitro (33). Therefore, under pathological conditions such grapelike conglomerates of platelets may accompany rapid changes of shear in vivo. However, it is virtually impossible to have accurate quantification of the prevailing shear conditions in vivo, i.e., because of the structural complexity and inaccessible location of the vessels. Therefore, having an experimental model that contains all blood components and can generate a variety of well-defined shear rates is of value to the study of thrombus formation. For instance, Goel and Diamond (26) showed that adherent neutrophils can induce fibrin formation independent of platelets, a process that is inversely correlated to shear (occurring at 62.5 s–1, however, hardly so at 250 s–1), whereas addition of platelets dramatically increases the amount of fibrin deposition. The elucidation of this shear-dependent property of adherent neutrophils at lower flow conditions is important for the understanding of the pathophysiology of venous thrombosis or inflammation.
Analogous to the local catheter technique that was developed for insertion of reagents and cells directly into the microcirculation, the side ports of the ex vivo chamber are designed for direct and immediate access to a developing thrombus, for instance, to insert agents that would inhibit its growth or cause its dissolution. In case the entry of such agents into the mouse circulation is not desired, the blood containing the agents can be extracted from the ex vivo chamber after its passage through the solid phase from the side port in the outlet means, whereas closure of the ipsilateral flow restrictor would block the reentry of the mixture into the mouse's vein.
Thrombogenicity of Implants The ex vivo flow chamber may furthermore offer an opportunity to directly observe the long-term interaction of blood with coated materials. Contact between blood and a biomaterial surface takes place in many applications and is known to activate the coagulation and complement systems (4). Depositions of blood-borne cells and other materials around implants, catheters, or vascular stents pose a major problem in modern medicine (42). For instance, thrombus formation within stent implants can cause restenosis of treated coronary arteries (19). Therefore, it is of interest to examine the long-term performance of materials that are used for stents or prostheses under blood flow conditions and what might prevent depositions (53). In this respect it would be of interest to study the process of neoepithelialization at the inner surfaces of the chamber under visual control. To inhibit the contact pathway of coagulation that can be initiated when blood comes in contact with an artificial surface, heparin is coated on surfaces (4), as was done in our experiments. Alternatively, corn trypsin inhibitor (CTI), a specific inhibitor of coagulation factor XIIa, can be used to reduce fibrin formation (26, 27).
Working Temperature The ability to operate the ex vivo chamber under a range of temperatures offers an asset in the investigation of microcirculatory behavior of blood components in conditions such as fever or hypothermia. For instance, hypothermia primes platelets for activation and causes their rapid clearance from the circulation through hepatic macrophages (32); however, little is known about the effects of subtle changes in temperature on leukocyte function. Furthermore, temperature changes in vivo substantially impact the vascular tone and the hemodynamics of the observed vessels (17, 30), complicating the study of the isolated effect of temperature. In contrast, if given the viscosity parameters at any given temperature, our simulations offer accurate estimates of the important flow characteristics.
In summary, conventional in vitro and in vivo models for studying microcirculatory processes are two ends of a spectrum. Whereas in vivo systems impress with their physiological fidelity, in vitro systems excel in the amount of reduction that can be achieved. The autoperfused ex vivo flow chamber presented here combines strengths of conventional in vitro and in vivo systems.
|
APPENDIX A: THEORETICAL FOUNDATIONS FOR THE BALANCE OF FORCES APPROACH |
|---|
|
TOPABSTRACTEXISTING MODELSMATERIALS AND METHODSRESULTSDISCUSSIONAPPENDIX A: THEORETICAL...APPENDIX B: SOLUTION OF...APPENDIX C: ESTIMATING ERROR...REFERENCES |
|---|
Geometry
Assume flow through a prismatic vessel with rectangular cross section as in the microslides used in our experiments. Let a denote the length of the long (wide) side of the rectangle and b the length of the short side. Introduce Cartesian coordinates: x for the direction of flow and y and z for the cross section (Fig. 2B) such that y = ±a/2 and z = ±b/2 identify the boundaries of the flow, i.e., the chamber walls. Let ex, ey, and ez be the orthonormal basis vectors accompanying x, y, z. Introduce the spatial nabla operator
|
|
v = grad v, ·v = div v, and x v = rot v. To transpose v, write v. Requirements The cross section must be constant over x, that is, the microslide must be prismatic to avoid Bernoulli-like pressure changes due to velocity changes enforced by flow continuity. Another Bernoulli contribution would be changes in gravitational potential; therefore, the relevant pressure difference is to be taken between points of equal height. Assume the fluid to be homogenous and incompressible and suppose that the flow is laminar and that entry effects have diminished such that the velocity vectors v take the form v = vex. Furthermore, the fluid is assumed to reflect the symmetries of the rectangular cross section in its flow-induced stress distribution. Finally, assume that unsteady flow acceleration and deceleration, if any, employs negligible force only (as in our case with low flow velocities and low oscillation frequencies) or is averaged out during pressure measurement.
Shear Rates
Applying the continuity equation ·v = 0 for homogenous incompressible flow to the velocity vector field v = vex results in v/x = 0, meaning that the v is independent from x. Consequently, the velocity gradient v and the corresponding strain rate tensor C have only two independent components
 | (A1) |
being the shear rates derived from the distribution of v over the cross section. Because v does not vary along x, neither do the strain rates.
Stresses
Let T denote the fluid's (symmetric) Cauchy stress tensor, partitioned as
 | (A2) |
S/x = 0.
Global Balance
To balance linear momentum (and thus forces) in a section of the microslide over x [x0,x1], let D denote the cross section, parameterized by y [–a/2, +a/2], z [–b/2, +b/2], and let V be the domain enclosed by the section. In its general form, the balance is
 | (A3) |
is the fluid's mass density, t is the time coordinate, V is the boundary of V, that is, the cross sections at x0 and x1 and the side walls in between, n is the outer surface normal vector of each face of V, and k = –
is the gradient of a (presumably gravitational) potential , which is height dependent. The integrals are next considered in order. The first integral sums forces for unsteady acceleration/deceleration and is negligible or averaged out by hypothesis. The surface integral over vv·n has its only contributions from the areas where fluid enters and leaves, i.e., from the cross sections at x0 and x1 where n = ±ex; hence it totals the changes in v2 between x0 and x1. But v does not change over x, and therefore the second integral becomes zero. As for the third integral, k can be neutralized by adding to the spherical pressure p hidden in T. This hydrostatic pressure component makes p height dependent, but significant height differences have already been ruled out, and therefore it can be assumed without further loss of generality that k = 0; thus the pressure component not related to the fluid dynamics is discarded. Hence the essential term is the remaining integral 
VT·ndA. Because of the assumed symmetries in the stress distribution, this integral must yield a vector parallel to ex, taking the form
 | (A4) |
is the xx-coefficient of the extra stress tensor S and gets cancelled because it is independent from x. Assume that at x0 and x1 an average pressure
 | (A5) |
 | (A6) |
 | (A7) |
|
| (A8) |
p/x is a short expression for the measured pressure difference quotient. Note that p/x is negative, hence –p/x is positive, despite the suggestive minus sign. This is the equation used for determining the average wall shear stress when given a pressure difference over a definite section length. Note that the derivation did not require use of a specific fluid model and that it holds even if the pressure measurement is influenced by parts of S.
Local Balance
The integral form of the linear momentum balance has an analog differential form that can be used to examine the flow in the neighborhood of individual points
 | (A9) |
·T = 0, and after substituting T from Eq. A2, we get
 | (A10) |
 | (A11) |
p/x is constant over the cross section. Therefore, p/x can be identified with the measured pressure difference quotient p/x.
More precisely, the x-component of the local force balance reveals
 | (A12) |
yx and zx are the respective shear stresses from S. Note that the right-hand side is constant. The local force balance can help to determine maximal shear stresses if the side length ratio a/b is large. Supposedly, the points at the wall exposed to the greatest shear stress max are those closest to the center of the cross section, namely, the midpoints of the long sides: y = 0, z = ± b/2. Because at y = 0 the short sides are far away, one can expect that the flow there behaves much like a parallel plate flow, that is, the y-derivatives vanish. Given that, the local force balance Eq. A12 approaches
 | (A13) |
 | (A14) |
max, leading to |
| (A15) |
avg/max approximates
 | (A16) |
avg approaches max with increasing side length ratio a/b. This means that the wall shear stress at the long sides tends to be constant, as must be expected for an approximation to parallel plate flow. |
APPENDIX B: SOLUTION OF THE POISSON EQUATION WITH FOURIER SERIES |
|---|
|
TOPABSTRACTEXISTING MODELSMATERIALS AND METHODSRESULTSDISCUSSIONAPPENDIX A: THEORETICAL...APPENDIX B: SOLUTION OF...APPENDIX C: ESTIMATING ERROR...REFERENCES |
|---|
 | (B1) |
 | (B2) |
 | (B3) |
 | (B4) |
 | (B5) |
 | (B6) |
|
APPENDIX C: ESTIMATING ERROR IN THE PRESSURE MEASUREMENT |
|---|
|
TOPABSTRACTEXISTING MODELSMATERIALS AND METHODSRESULTSDISCUSSIONAPPENDIX A: THEORETICAL...APPENDIX B: SOLUTION OF...APPENDIX C: ESTIMATING ERROR...REFERENCES |
|---|
x[PE-50] = 2 x 2 cm, i.e., 2 cm on each side] used to connect the three-way pieces and the microslides (Fig. 2A), we partitioned the measured pressure difference as
 | (C1) |
p[junctions] denotes pressure changes between the microslide and the PE-50 parts due to abrupt changes in geometry. These are typically quadratic in the flow velocity v and given as fractions of v2/2. For the range of velocities observed in our system, v < 3 cm/s, such fractions can be estimated to not exceed v2/2 = 4.5 dyn/cm2 = 0.0034 mmHg (using a mass density of 1 g/cm3), which is negligible. Consequently, the measured pressure difference can be identified as
 | (C2) |
p[PE-50]/p[slide] is the relative error of the pressure difference measurement. To estimate this error, we supposed a worst-case shear thinning scenario where shear rates in the PE-50 tubing are low enough to keep the apparent viscosity at its thick limit o but in the microslide are so high that the apparent viscosity drops to its thin limit over most parts of the microslide's cross section. Thus the viscosities of the blood are assumed to be different between the PE-50 parts and the microslide but can be treated as constants within either part and therefore allow calculations as for a Newtonian fluid. The choice of this approach was motivated by our numerical simulations using the shear-thinning fluid model as proposed in Reference 63. Hence, the flow resistance of the PE-50 parts, idealized to cylindrical pipes, can be given as
 | (C3) |
max, yielding
 | (C4) |
 | (C5) |
p[PE-50]/p[slide] becomes
 | (C6) |
o/ 15 (63), which is greater than the ratio at 37°C because o grows significantly when temperature decreases, whereas the accompanying relative change of tends to be smaller. Yeleswarapu et al. (64) provide data for porcine blood at 23°C with o/ 30. For the purpose of estimating worst-case error bounds, we assumed o/ = 30, yielding the error estimates summarized in Table 1, realizing that the actual errors are likely much smaller. However, this depends on temperature and flow rate; hence, there is no easy way to determine the actual errors, and the only way to obtain reliable pressure drop data is to keep the error estimate for p[PE-50]/ p[slide] below the desired accuracy level. To that end, Eq. C6 suggests that the choice of internal dimensions has much greater influence on the error than the lengths of the parts. |
ACKNOWLEDGMENTS |
|---|
This work was supported by National Institute of Allergy and Infectious Diseases Grant KO8-AI-50775 (to A. Hafezi-Moghadam).
|
FOOTNOTES |
|---|
The costs of publication of this article were defrayed in part by the payment of page charges. The article must therefore be hereby marked "advertisement" in accordance with 18 U.S.C. Section 1734 solely to indicate this fact.
* A. Hafezi-Moghadam, K. L. Thomas, and C. Cornelssen contributed equally to this work.
|
REFERENCES |
|---|
|
TOPABSTRACTEXISTING MODELSMATERIALS AND METHODSRESULTSDISCUSSIONAPPENDIX A: THEORETICAL...APPENDIX B: SOLUTION OF...APPENDIX C: ESTIMATING ERROR...REFERENCES |
|---|
2. Allport JR, Ding H, Collins T, Gerritsen ME, and Luscinskas FW. Endothelial-dependent mechanisms regulate leukocyte transmigration: a process involving the proteasome and disruption of the vascular endothelial-cadherin complex at endothelial cell-to-cell junctions. J Exp Med 186: 517–527, 1997.
3. Alon R, Chen S, Fuhlbrigge R, Puri KD, and Springer TA. The kinetics and shear threshold of transient and rolling interactions of L-selectin with its ligand on leukocytes. Proc Natl Acad Sci USA 95: 11631–11636, 1998.
4. Andersson J, Sanchez J, Ekdahl KN, Elgue G, Nilsson B, and Larsson R. Optimal heparin surface concentration and antithrombin binding capacity as evaluated with human non-anticoagulated blood in vitro. J Biomed Mater Res 67A: 458–466, 2003.
5. Andre P, Hartwell D, Hrachovinova I, Saffaripour S, and Wagner DD. Pro-coagulant state resulting from high levels of soluble P-selectin in blood. Proc Natl Acad Sci USA 97: 13835–13840, 2000.
6. Baez S. An open cremaster muscle preparation for the study of blood vessels by in vivo microscopy. Microvasc Res 5: 384–394, 1973.[CrossRef][Web of Science][Medline]
7. Baker M and Wayland H. On-line volume flow rate and velocity profile measurement for blood in microvessels. Microvasc Res 7: 131–143, 1974.[CrossRef][Web of Science][Medline]
8. Berman CL, Yeo EL, Wencel-Drake JD, Furie BC, Ginsberg MH, and Furie B. A platelet alpha granule membrane protein that is associated with the plasma membrane after activation. Characterization and subcellular localization of platelet activation-dependent granule-external membrane protein. J Clin Invest 78: 130–137, 1986.[Web of Science][Medline]
9. Breslow JL. Mouse models of atherosclerosis. Science 272: 685–688, 1996.[Abstract]
10. Brown EB, Campbell RB, Tsuzuki Y, Xu L, Carmeliet P, Fukumura D, and Jain RK. In vivo measurement of gene expression, angiogenesis and physiological function in tumors using multiphoton laser scanning microscopy. Nat Med 7: 864–868, 2001. [Corrigenda. Nat Med 7: September 2001, p. 1069.][CrossRef][Web of Science][Medline]
11. Butcher EC. Leukocyte-endothelial cell recognition: three (or more) steps to specificity and diversity. Cell 67: 1033–1036, 1991.[CrossRef][Web of Science][Medline]
12. Butcher EC and Picker LJ. Lymphocyte homing and homeostasis. Science 272: 60–66, 1996.[Abstract]
13. Casasnovas JM and Springer TA. Kinetics and thermodynamics of virus binding to receptor. Studies with rhinovirus, intercellular adhesion molecule-1 (ICAM-1), and surface plasmon resonance. J Biol Chem 270: 13216–13224, 1995.
14. Chmiel H, Anadere I, and Walitza E. The determination of blood viscoelasticity in clinical hemorheology. Biorheology 27: 883–894, 1990.[Web of Science][Medline]
15. Cotter MJ, Norman KE, Hellewell PG, and Ridger VC. A novel method for isolation of neutrophils from murine blood using negative immunomagnetic separation. Am J Pathol 159: 473–481, 2001.
16. Diacovo TG, Puri KD, Warnock RA, Springer TA, and von Andrian UH. Platelet-mediated lymphocyte delivery to high endothelial venules. Science 273: 252–255, 1996.[Abstract]
17. Eckmann DM, Bowers S, Stecker M, and Cheung AT. Hematocrit, volume expander, temperature, and shear rate effects on blood viscosity. Anesth Analg 91: 539–545, 2000.
18. Falati S, Gross P, Merrill-Skoloff G, Furie BC, and Furie B. Real-time in vivo imaging of platelets, tissue factor and fibrin during arterial thrombus formation in the mouse. Nat Med 8: 1175–1181, 2002.[CrossRef][Web of Science][Medline]
19. Ferns GA, Raines EW, Sprugel KH, Motani AS, Reidy MA, and Ross R. Inhibition of neointimal smooth muscle accumulation after angioplasty by an antibody to PDGF. Science 253: 1129–1132, 1991.
20. Finger EB, Puri KD, Alon R, Lawrence MB, von Andrian UH, and Springer TA. Adhesion through L-selectin requires a threshold hydrodynamic shear. Nature 379: 266–269, 1996.[CrossRef][Medline]
21. Freis ED. Studies in hemodynamics and hypertension. Hypertension 38: 1–5, 2001.
22. Frenette PS, Moyna C, Hartwell DW, Lowe JB, Hynes RO, and Wagner DD. Platelet-endothelial interactions in inflamed mesenteric venules. Blood 91: 1318–1324, 1998.
23. Gautam N, Herwald H, Hedqvist P, and Lindbom L. Signaling via 
2 integrins triggers neutrophil-dependent alteration in endothelial barrier function. J Exp Med 191: 1829–1839, 2000.
24. Gimbrone MA Jr, Nagel T, and Topper JN. Biomechanical activation: an emerging paradigm in endothelial adhesion biology. J Clin Invest 100: S61–S65, 1997.[Web of Science][Medline]
25. Gimbrone, MA Jr, Topper JN, Nagel T, Anderson KR, and Garcia-Cardena G. Endothelial dysfunction, hemodynamic forces, and atherogenesis. Ann NY Acad Sci 902: 230–240, 2000.[Web of Science][Medline]
26. Goel MS and Diamond SL. Neutrophil enhancement of fibrin deposition under flow through platelet-dependent and -independent mechanisms. Arterioscler Thromb Vasc Biol 21: 2093–2098, 2001.
27. Goel MS and Diamond SL. Neutrophil cathepsin G promotes prothrombinase and fibrin formation under flow conditions by activating fibrinogen-adherent platelets. J Biol Chem 278: 9458–9463, 2003.
28. Grigioni M, Daniele C, and D'Avenio G. The role of wall shear stress in unsteady vascular dynamics. Prog Biomed Res 7: 204–212, 2002.
29. Hafezi-Moghadam A and Ley K. Relevance of L-selectin shedding for leukocyte rolling in vivo. J Exp Med 189: 939–948, 1999.
30. Hafezi-Moghadam A, Simoncini T, Yang E, Limbourg FP, Plumier JC, Rebsamen MC, Hsieh CM, Chui DS, Thomas KL, Prorock AJ, Laubach VE, Moskowitz MA, French BA, Ley K, and Liao JK. Acute cardiovascular protective effects of corticosteroids are mediated by non-transcriptional activation of endothelial nitric oxide synthase. Nat Med 8: 473–479, 2002.[CrossRef][Web of Science][Medline]
31. Hafezi-Moghadam A, Thomas KL, Prorock AJ, Huo Y, and Ley K. L-selectin shedding regulates leukocyte recruitment. J Exp Med 193: 863–872, 2001.
32. Hoffmeister KM, Felbinger TW, Falet H, Denis CV, Bergmeier W, Mayadas TN, von Andrian UH, Wagner DD, Stossel TP, and Hartwig JH. The clearance mechanism of chilled blood platelets. Cell 112: 87–97, 2003.[CrossRef][Web of Science][Medline]
33. Holme PA, Orvim U, Hamers MJ, Solum NO, Brosstad FR, Barstad RM, and Sakariassen KS. Shear-induced platelet activation and platelet microparticle formation at blood flow conditions as in arteries with a severe stenosis. Arteriosclerosis Thrombosis Vasc Biol 17: 646–653, 1997.
34. House SD and Lipowsky HH. Leukocyte-endothelium adhesion: microhemodynamics in mesentery of the cat. Microvasc Res 34: 363–379, 1987.[CrossRef][Web of Science][Medline]
35. Huo Y, Hafezi-Moghadam A, and Ley K. Role of vascular cell adhesion molecule-1 and fibronectin connecting segment-1 in monocyte rolling and adhesion on early atherosclerotic lesions. Circ Res 87: 153–159, 2000.
36. Hynes RO and Wagner DD. Genetic manipulation of vascular adhesion molecules in mice. J Clin Invest 100: S11–S13, 1997.[Web of Science][Medline]
37. Ishii H, Yoshida M, Rosenzweig A, Gimbrone MA Jr, Yasukochi Y, and Numano F. Adenoviral transduction of human E-selectin into isolated, perfused, rat aortic segments: an ex vivo model for studying leukocyte-endothelial interactions. J Leukoc Biol 68: 687–692, 2000.
38. Kansas GS. Selectins and their ligands: current concepts and controversies. Blood 88: 3259–3287, 1996.
39. Kanwar S, Smith CW, and Kubes P. An absolute requirement for P-selectin in ischemia/reperfusion-induced leukocyte recruitment in cremaster muscle. Microcirculation 5: 281–287, 1998.[CrossRef][Web of Science][Medline]
40. Lawrence MB, McIntire LV, and Eskin SG. Effect of flow on polymorphonuclear leukocyte/endothelial cell adhesion. Blood 70: 1284–1290, 1987.
41. Lawrence MB and Springer TA. Leukocytes roll on a selectin at physiologic flow rates: distinction from and prerequisite for adhesion through integrins. Cell 65: 859–873, 1991.[CrossRef][Web of Science][Medline]
42. Levy RJ, Schoen FJ, Anderson HC, Harasaki H, Koch TH, Brown W, Lian JB, Cumming R, and Gavin JB. Cardiovascular implant calcification: a survey and update. Biomaterials 12: 707–714, 1991.[CrossRef][Web of Science][Medline]
43. Libby P. Inflammation in atherosclerosis. Nature 420: 868–874, 2002.[CrossRef][Medline]
44. Mayadas TN, Johnson RC, Rayburn H, Hynes RO, and Wagner DD. Leukocyte rolling and extravasation are severely compromised in P selectin-deficient mice. Cell 74: 541–554, 1993.[CrossRef][Web of Science][Medline]
45. McEver RP. Adhesive interactions of leukocytes, platelets, and the vessel wall during hemostasis and inflammation. Thrombosis Haemostasis 86: 746–756, 2001.[Web of Science][Medline]
46. McEver RP and Cummings RD. Role of PSGL-1 binding to selectins in leukocyte recruitment. J Clin Invest 100: S97–S103, 1997.[Web of Science][Medline]
47. Merten M, Chow T, Hellums JD, and Thiagarajan P. A new role for P-selectin in shear-induced platelet aggregation. Circulation 102: 2045–2050, 2000.
48. Pontrelli G. Blood flow through an axisymmetric stenosis. Proc Inst Mech Eng [H] 215: 1–10, 2001.[Web of Science][Medline]
49. Preece G, Murphy G, and Ager A. Metalloproteinase-mediated regulation of L-selectin levels on leucocytes. J Biol Chem 271: 11634–11640, 1996.
50. Robbie L and Libby P. Inflammation and atherothrombosis. Ann NY Acad Sci 947: 167–180, 2001.[Web of Science][Medline]
51. Ross R. Atherosclerosis—an inflammatory disease. N Engl J Med 340: 115–126, 1999.
52. Ruggeri ZM. Mechanisms of shear-induced platelet adhesion and aggregation. Thrombosis Haemostasis 70: 119–123, 1993.[Web of Science][Medline]
53. Sakakibara M, Goto S, Eto K, Tamura N, Isshiki T, and Handa S. Application of ex vivo flow chamber system for assessment of stent thrombosis. Arteriosclerosis Thrombosis Vasc Biol 22: 1360–1364, 2002.
54. Schleiffenbaum B, Spertini O, and Tedder TF. Soluble L-selectin is present in human plasma at high levels and retains functional activity. J Cell Biol 119: 229–38, 1992.
55. Shesely EG, Maeda N, Kim HS, Desai KM, Krege JH, Laubach VE, Sherman PA, Sessa WC, and Smithies O. Elevated blood pressures in mice lacking endothelial nitric oxide synthase. Proc Natl Acad Sci USA 93: 13176–13181, 1996.
56. Springer TA. Adhesion receptors of the immune system. Nature 346: 425–434, 1990.[CrossRef][Medline]
57. Springer TA. Traffic signals for lymphocyte recirculation and leukocyte emigration: the multistep paradigm. Cell 76: 301–314, 1994.[CrossRef][Web of Science][Medline]
58. Stamper HB Jr and Woodruff JJ. Lymphocyte homing into lymph nodes: in vitro demonstration of the selective affinity of recirculating lymphocytes for high-endothelial venules. J Exp Med 144: 828–833, 1976.
59. Tangelder GJ, Slaaf DW, Tierlinck HC, Alewijnse R, and Reneman RS. Localization within a thin optical section of fluorescent blood platelets flowing in a microvessel. Microvasc Res 23: 214–230, 1982.[CrossRef][Web of Science][Medline]
60. Turitto VT and Weiss HJ. Red blood cells: their dual role in thrombus formation. Science 207: 541–543, 1980.
61. Van de Stolpe A and van der Saag PT. Intercellular adhesion molecule-1. J Mol Med 74: 13–33, 1996.[Web of Science][Medline]
62. Xia Z and Frojmovic MM. Aggregation efficiency of activated normal or fixed platelets in a simple shear field: effect of shear and fibrinogen occupancy. Biophys J 66: 2190–2201, 1994.[Web of Science][Medline]
63. Yeleswarapu KK. Evaluation of Continuum Models for Characterizing the Constitutive Behavior of Blood. University of Pittsburgh, 1996, p. 169.
64. Yeleswarapu KK, Kameneva MV, Rajagopal KR, and Antaki JF. The flow of blood in tubes: theory and experiment. Mech Res Commun 25: 257–262, 1998.[CrossRef]
This article has been cited by other articles:
|
|
 |
|
  S. Miyahara, L. Almulki, K. Noda, T. Nakazawa, T. Hisatomi, S. Nakao, K. L. Thomas, A. Schering, S. Zandi, S. Frimmel, et al. In vivo imaging of endothelial injury in choriocapillaris during endotoxin-induced uveitis FASEB J, June 1, 2008; 22(6): 1973 - 1980. [Abstract] [Full Text] [PDF]
|
|
| ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| HOME | HELP | FEEDBACK | SUBSCRIPTIONS | ARCHIVE | SEARCH | TABLE OF CONTENTS |
| Visit Other APS Journals Online |
with unit (vmax/b) of the Newtonian flow through a rectangular cross section with a/b = 10. This is equivalent to the normalized z-shear stress 
, relative to vmax/b. Because in the Newtonian model the shear stress is directly proportional to the shear rate, this plot also shows the distribution of the magnitude of the shear stress 
relative to 
