## Abstract

Various in vitro and in vivo techniques exist for study of the microcirculation. Whereas in vivo systems impress with their physiological fidelity, in vitro systems excel in the amount of reduction that can be achieved. Here we introduce the autoperfused ex vivo flow chamber designed to study murine leukocytes and platelets under well-defined hemodynamic conditions. In our model, the murine heart continuously drives the blood flow through the chamber, providing a wide range of physiological shear rates. We used a balance of force approach to quantify the prevailing forces at the chamber walls. Numerical simulations show the flow characteristics in the chamber based on a shear-thinning fluid model. We demonstrate specific rolling of wild-type leukocytes on immobilized P-selectin, abolished by a blocking MAb. When uncoated, the surfaces having a constant shear rate supported individual platelet rolling, whereas on areas showing a rapid drop in shear platelets interacted in previously unreported grapelike conglomerates, suggesting an influence of shear rate on the type of platelet interaction. In summary, the ex vivo chamber amounts to an external vessel connecting the arterial and venous systems of a live mouse. This method combines the strengths of existing in vivo and in vitro systems in the study of leukocyte and platelet function.

- autoperfused flow chamber
- intravital microscopy
- inflammation
- thrombus formation

proper immune function requires recruitment of leukocytes to various sites of an organism (56). Leukocyte recruitment is a complex process, orchestrated by various signals, chemicals, and specialized molecules (i.e., adhesion molecules) (11, 57). Analogously, the growth of a thrombus requires the recruitment of platelets and plasma components at sites of vascular injury (22, 27, 45). To understand the details of these events cell interactions are studied in vivo and in vitro under flow conditions (12, 41).

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

### In Vivo Study of Microcirculation

In vivo models closely mimic the physiology and pathology of an organism (9, 10). Here, the microcirculation of various organs is traditionally studied by microscopic visualization. Intravital microscopy predominantly makes use of transparent tissues such as mesentery (34), cheek pouch (23), cremaster muscle (6), or equivalently thin tissues. In such tissues interactions of leukocytes, platelets, or inserted cells with the endothelial layer are observed with transluminescence or epifluorescence microscopy (29–31). Despite the physiological relevance of the in vivo models, the complexity and the simultaneous presence of most elements of interest (i.e., immune cells, endothelium, adhesion molecules, ECM proteins, and other blood components) limit the amount of reduction achievable in vivo. For instance, leukocytes and the endothelium express numerous, different adhesion molecules, some of which may have overlapping functions (38). This makes, for instance, pinpointing the role of a single molecule in vivo difficult or sometimes impossible. Similarly, ex vivo models that use isolated, perfused organs show the full complexity of the endothelial layer (35, 37).

### 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

### Anesthesia and Surgery

Eight- to ten-week-old wild-type C57BL/6 mice (Jackson Labs) weighing 22–24 g were anesthetized and kept at 37°C as described previously (29). Cannulation of the vessels and intubation of the trachea to secure respiration were performed microsurgically (29). All animal experiments adhered to the “Guiding Principles for Research Involving Animals and Human Beings” of the American Physiological Society.

### Components Used to Assemble the Ex Vivo Flow Chamber

*Translucent solid phase.* For the solid phase (Figs. 1, *iv*, and 2*A*), microslides from VitroCom (vitrocom.com) were obtained [inner diameters (ID): 0.02 × 0.2 (catalog no. 5002S), 0.03 × 0.3 (catalog no. 5003), 0.04 × 0.4 (catalog no. 5004), 0.05 × 0.5 (catalog no. 5005S), and 0.1 mm × 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 2*A*) 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 2*A*).

*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. 2*A*). An upright microscope with a saline immersion objective (SW ×40, 0.75 numerical aperture) was used. Optionally, ×25 to ×100 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. 2*A*). 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. 2*B*), 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 p̄ of the pressure at two points *x*_{0} and *x*_{1} of the above flow, the balance of forces results in (1) where Δp/Δ*x* is the measured pressure difference quotient (note that Δp/Δ*x* is negative, hence –Δp/Δ*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 (*v*_{max}/*v*_{vol}) ≈ 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 **e*** _{x}*,

**e**

*, and*

_{y}**e**

*be the orthonormal basis vectors accompanying*

_{z}*x, y, z*, and let

**v**=

*v*

**e**

*denote the velocity profile over the cross section. Introduce the spatial nabla operator and use · to denote the dot product.*

_{x}*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** = *v***e**_{x}, the velocity gradient ∇**v** and the corresponding strain rate tensor **C** have only two independent components (3) with 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) with **T** denoting the fluid's (symmetric) Cauchy stress tensor, *p* meaning the spherical pressure, **1** being the identity tensor, and η representing the apparent viscosity. However, to model the shear-thinning properties of blood, η is made dependent on the shear rate as follows (5) Here η_{0} and η_{∞} are the viscosity limits for shear rates approaching zero and infinity, respectively, and Λ 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) where ·· denotes the double dot product, i.e., the trace of the dot product. In our case, simplifies to (7) (Note that for Λ = 0 or η_{0} = η_{∞}, η 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) with Δ = ∇·∇ 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 Δp/Δ*x* instead of ∂p/∂*x*.

Using the chain rule to decompose ∇η results in (10) with (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 η = const., *Eq. 10* reduces to the classic Poisson equation (13) which can be solved by using Fourier series (appendix b). For given coordinates *y* and *z*, the corresponding value of any flow variable can be determined as a doubly infinite series whose coefficients depend on the side ratio *a*/*b*, using the series computing functions of modern mathematical software packages. To obtain the values of any flow variable for all points of a regular grid, fast Fourier transform (FFT) techniques have been used.

*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 ω chosen from the range 0 < ω < 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 Δ*y*Δ*z* × Δp/Δ*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

To study the interaction of murine blood with coated surfaces in a model that offers a higher level of reduction than intravital microscopy and is closer to the physiology than conventional in vitro systems, we developed the autoperfused ex vivo flow chamber for the mouse (Fig. 1). In the following we describe its assembly, characterize and simulate its flow properties, and show examples of its capabilities for blood research.

### 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 2*A*). 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 mm^{2} 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 2*A*). 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. 2*A*). Table 1 suggests that the 1.0-mm × 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 × 0.1-mm microslide should be avoided for yet another reason: when the 1.0-mm × 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) Thus a nondimensional factor *b*/(2Δ*x*_{[slide]}) can be used to calculate τ_{max} from the measured pressure drop (–Δp)_{[slide]}. Table 1 shows values of that factor for several microslides of length Δ*x*_{[slide]} = 50 mm, including the conversion factor from milligrams of mercury to dynes per square centimeter, whereas Table 2 shows examples of τ_{max} calculations from measured pressure drops that demonstrate the range of shear stresses that can be obtained depending on the microslide dimensions. These τ_{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 τ_{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 τ_{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. 2*B*) are both normalized with respect to *b*/2 (half the short side); therefore, the normalized ranges are [–*a*/*b*; +*a*/*b*] × [–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. 3*A*). 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 τ* _{yx}* are mostly zero except near the short sides. Furthermore, the distributions of and τ

*are mostly linear over*

_{zx}*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

*v*

_{max}/

*v*

_{vol}= 1.60. For

*a*/

*b*→ ∞, the influence of the short sides on the velocity distribution vanishes and the ratio

*v*

_{max}/

*v*

_{vol}converges to 1.5, as characteristic for a profile that is constant over

*y*and parabolic over

*z. 6*) Given the viscosity η of the fluid, the parabolic velocity profile over

*z*allows calculation of τ

_{max}from the magnitude of the

*z*component of the velocity gradient at the long side, resulting in (15) The constancy of the velocity profile over most of the

*y*domain is a general effect of a large aspect ratio

*a*/

*b*(i.e., 10 in the microslides we used), whereas the parabolic distribution over

*z*is tightly bound to the additional assumption of laminar Newtonian flow (Fig. 3

*A*). Therefore,

*Eq. 2*generally holds whereas

*Eq. 15*depends on fluid viscosity and thus on a particular fluid model.

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 × 0.04 mm with viscosity parameters η_{o} = 73.8 cP, η_{∞} = 5 cP, Λ = 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: –Δ*p/*Δ*x* = 20 (Figs. 4 and 5, *top*), 200 (Figs. 4 and 5, *middle*), and 10,000 (Figs. 4 and 5, *bottom*) dyn/cm^{3} (1,000 dyn/cm^{3} = 0.75 mmHg/cm). For body temperature, lower values of the viscosity parameters η_{o} and η_{∞} 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 η_{o} (Fig. 4*B*, *top*), thus staying nearly constant, which makes the velocity profile resemble a Newtonian one (Fig. 4*A*, *top*). At higher pressure drops, the viscosity changes drastically over the cross section (Fig. 4*B*, *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. 4*A*, *middle*). At very high pressure drops, shear rates are so high that the viscosity almost immediately drops to its lower limit η_{∞} when leaving the shear-free center (Fig. 4*B*, *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. 4*A*, *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. 5*A*, *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. 5*B*, *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 η_{o} for small pressure drops and η_{∞} for high drops. Figure 6*A* demonstrates this region of transition with a plot of velocities *v*_{vol} and *v*_{max} vs. the pressure drop. Besides showing the asymptotic Newtonian-like behavior, Fig. 6*A* 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 6*B* shows how characteristics change in the non-Newtonian region. The ratio *v*_{vol}/*v*_{max} 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 4*v*_{max}/*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 6*B* 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 = ρ*v*_{avg}*b*/η_{∞}. As an example, for *b* ≤ 0.01 cm, *v*_{avg} < 3 cm/s, and ρ = 1 g/cm^{3}, η_{∞} = 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. 3*D* 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. 8*A*). 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. 8*B*). 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. 8*C*).

## DISCUSSION

For functional studies of blood cells, genetically engineered mice provide an excellent resource (36). To optimize the use of this resource for the study of the microcirculation we previously reported (29) the development of a model of intravital microscopy allowing direct insertion of exogenous cells or compounds into the microcirculation. In this model native murine blood cells or inserted transfectants are faced with the full biological complexity of the vascular wall (29). Despite the high physiological fidelity of this system, there remained a need to study the interaction of murine blood or its components with single molecular layers. Therefore, we developed the autoperfused ex vivo flow chamber for the study of murine blood and its components.

### 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/cm^{2}). 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/cm^{2}, 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 × 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

To determine essential flow characteristics, we introduce some necessary theoretical preliminaries valid for any homogenous incompressible fluid and derive the equations for average and maximum wall shear stress based on pressure drop measurements.

### 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. 2*B*) such that *y* = ±*a*/2 and z = ±*b*/2 identify the boundaries of the flow, i.e., the chamber walls. Let **e*** _{x}*,

**e**

*, and*

_{y}**e**

*be the orthonormal basis vectors accompanying*

_{z}*x, y, z*. Introduce the spatial nabla operator and use · to denote the dot product. Take simple concatenation of vector symbols for the tensor product. Symbolic calculus will be done in tensor style, not matrix style; that is, mere vector symbols will never be subjected to matrix transposition to obtain the right product with adjacent vectors. Instead, the multiplication operators themselves are distinguished as above. Thus for a vector field

**v**, ∇

**v**= grad

**v**, ∇·

**v**= div

**v**, and ∇ ×

**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** = *v***e**_{x}. 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** = *v***e**_{x} 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) with 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) where p is the spherical pressure, **1** is the identity tensor, and **S** is the extra stress tensor containing flow-induced stresses. We claim that **S** is independent from *x*. At first this may seem trivial because, by definition, **S** is determined by the distribution of *v*, which does not vary over *x*, but the fluid may feature nonlocal properties that delay or anticipate stress reactions over space. However, as a consequence of the principle that the influence from any point's neighbors should not produce singularities in that point, the effects of finite local variations must vanish in the distance; hence, such nonlocal effects can be assumed to decay over *x*, actually representing a (prolonged) entry effect. Given that entry effects have diminished, we can indeed assume that ∂**S**/∂*x* = 0.

### Global Balance

To balance linear momentum (and thus forces) in a section of the microslide over *x* ∈ [*x*_{0},*x*_{1}], 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) where ρ is the fluid's mass density, *t* is the time coordinate, ∂*V* is the boundary of *V*, that is, the cross sections at *x*_{0} and *x*_{1} and the side walls in between, **n** is the outer surface normal vector of each face of ∂*V*, and **k** = –∇ϕ is the gradient of a (presumably gravitational) potential ϕ, 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 ρ**vv**·**n** has its only contributions from the areas where fluid enters and leaves, i.e., from the cross sections at *x*_{0} and *x*_{1} where **n** = ±**e*** _{x}*; hence it totals the changes in ρ

*v*

^{2}between

*x*

_{0}and

*x*

_{1}. 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 ρϕ 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 ∫

_{∂}

_{V}**T**·

**n**d

*A*. Because of the assumed symmetries in the stress distribution, this integral must yield a vector parallel to

**e**

*, taking the form (A4) The first right-hand side integral collects the forces on the cross sections at*

_{x}*x*

_{0}and

*x*

_{1}(with different signs due to the opposite normal vectors), whereas the second represents the total shear force from the walls to the fluid. is the

*xx*-coefficient of the extra stress tensor

**S**and gets cancelled because it is independent from

*x*. Assume that at

*x*

_{0}and

*x*

_{1}an average pressure (A5) is measured with some function

*q*representing the influence of

**S**on the measurement;

*q*cannot disturb the difference of two measured pressures. Let us also define the average wall shear stress (A6) (with minus sign because it is a stress from the fluid to the wall, not vice versa). Then the total surface integral of the stresses amounts to (A7) Because the global balance of linear momentum requires this integral to vanish, we get (A8) where Δp/Δ

*x*is a short expression for the measured pressure difference quotient. Note that Δp/Δ

*x*is negative, hence –Δp/Δ

*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) As in the global case, the acceleration forces on the left-hand side vanish and **k** can be easily dealt with by adding a height-dependent hydrostatic pressure term to **T**, but as this is irrelevant here, we simply assume **k** = **0**. Thus we only have to deal with ∇·**T** = 0, and after substituting **T** from *Eq. A2*, we get (A10) By the “no entry effect” assumption, the right-hand side is independent from *x*, implying (A11) which in turn means that the pressure p changes linearly over *x* and that its slope ∂p/∂*x* is constant over the cross section. Therefore, ∂p/∂*x* can be identified with the measured pressure difference quotient Δp/Δ*x*.

More precisely, the *x*-component of the local force balance reveals (A12) Here, τ* _{yx}* and τ

*are the respective shear stresses from*

_{zx}**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 τ

_{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) Integrating over the entire

*z*range, we obtain (A14) Now the left-hand side is two times the shear stress from the middle of the long wall, i.e., –2τ

_{max}, leading to (A15) Note that the ratio τ

_{avg}/τ

_{max}approximates (A16) that is, τ

_{avg}approaches τ

_{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

We are interested in the solution of the boundary value problem (B1) with *c* denoting an arbitrary constant. For the given rectangular domain, the solution should take the form (B2) which can represent any continuous function that is even in *y* and *z* and vanishes at the rectangle boundaries. With the pairwise orthogonality of the above summand functions over the rectangular domain, the inverse formula can be generated (B3) Substituting *v*(*y,z*) in the PDE with its Fourier series and transforming back yields (B4) Hence (B5) And therefore (B6) This representation allows integration and differentiation. Values of the resulting functions can be represented for any given *y* and *z* as limits of a doubly infinite series and approximated as such. For specific values of *a*/*b* such as *a*/*b* = 1, some closed-form results are achievable.

## APPENDIX C: ESTIMATING ERROR IN THE PRESSURE MEASUREMENT

To estimate the error in the drop of pressure introduced by the transport of blood through the PE-50 tubing [ID (*d*) = 0.58 mm, maximal total length Δ*x*_{[PE-50]} = 2 × 2 cm, i.e., 2 cm on each side] used to connect the three-way pieces and the microslides (Fig. 2*A*), we partitioned the measured pressure difference as (C1) Where Δp_{[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 ρ*v*^{2}/2. For the range of velocities observed in our system, *v* < 3 cm/s, such fractions can be estimated to not exceed ρ*v*^{2}/2 = 4.5 dyn/cm^{2} = 0.0034 mmHg (using a mass density ρ of ∼1 g/cm^{3}), which is negligible. Consequently, the measured pressure difference can be identified as (C2) Thus Δp_{[PE-50]}/Δp_{[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 η_{o} but in the microslide are so high that the apparent viscosity drops to its thin limit η_{∞} 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) where *Q* is the (volumetric) flow rate. For the microslides themselves, we can calculate the flow resistance from *Eqs. 14* and *15* by eliminating τ_{max}, yielding (C4) and with *v*_{max} = 1.6 *Q*/(*ab*), as for the Newtonian velocity profile on a 10:1 rectangle (Figs. 3*A* and 6*B*), we get (C5) Consequently, the ratio Δp_{[PE-50]}/Δp_{[slide]} becomes (C6) Thus the error depends on the ratio of the apparent viscosities. For human blood with hematocrit 40 at room temperature, the ratio is approximately η_{o}/η_{∞} ≈ 15 (63), which is greater than the ratio at 37°C because η_{o} grows significantly when temperature decreases, whereas the accompanying relative change of η_{∞} tends to be smaller. Yeleswarapu et al. (64) provide data for porcine blood at 23°C with η_{o}/η_{∞} ≈ 30. For the purpose of estimating worst-case error bounds, we assumed η_{o}/η_{∞} = 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 Δp_{[PE-50]}/ Δp_{[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

GRANTS

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.

- Copyright © 2004 the American Physiological Society