3-D modeling
Computational fluid models were created for three commercial cell flow/perfusion chambers (FCS, Oligene GmbH; FCS2, Bioptechs; RC-30 HV, Warner Instruments) to elucidate the effect of their specific design parameters on flow fields and resulting stress regimes that are imparted to cells seeded within the chambers (Figure 1). First, dimensions of all surfaces that define the fluid geometry (inlets, outlets, and chamber walls) were measured using a precision caliper and micrometer. Then the fluid was mapped to track the flow from inlet to outlet. Thereafter, for each commercial chamber, flow regimes were analyzed and compared for two target fluid shear stress magnitudes representative of those typically imparted to an osteoblastic monolayer (1, 10 dyn/cm2).
Fluid meshing
The creation of the fluid mesh is critical to the computational analysis, as it delineates interfaces between fluid cavities as well as node locations where each calculation is made by the solver. Care was taken to place node locations in the critical areas throughout the model, in particular at flow transition areas such as at inlets, outlets, and points of flow expansion or contraction. This procedure not only ensures the accurate description of flow through the channels, but it also reduces the computational requirements of the simulation. Hence, the mesh includes only fluid volumes within the chamber itself and not flow volumes within inlet/outlet tubing or volumes outside of the device. Two sets of models were created, accounting for (i) the chamber geometries without cells seeded on the bottom surface (for all chambers), and (ii) the chamber geometries with an array of cells modeled on the lower surface of the flow chamber. Similar to previous studies on flow over cell-shaped protrusions, the cells were modeled as rigid spheroid protrusions on the chamber surface, with dimensions typical of osteocytes (height = 10 μ m, radius = 15 μ m) [43, 45, 50]. The number of nodes used in each chamber was 64000, 480000, and 89000 for chambers 1, 2, 3 respectively, where the average finite volume modeled was on the order of 10-13 m3. Finally, the mesh, which provides a visual map of the flow geometry, was imported into a computational fluid dynamics (CFD) package (CFD-ACE, CFDRC), to allow for the definition of boundary conditions and simulation of flow regimes for targeted stress magnitudes.
Computational fluid dynamics
For each chamber, the velocity profile and pressure variation were determined at the inlet and outlet, for a corresponding maximum target shear stress 1 and 10 dyn/cm2, at the location where cells are placed within the chamber (i.e. bottom surface of chamber). These were then applied as boundary conditions, to focus simulations on the interior of the chamber cavity where the cells are cultured. Using a discretization convective-upwind scheme, velocity profiles were calculated from the continuity equation and Navier-Stokes equations in three dimensions (3D),
∇·u = 0 (1)
ρ (u·∇u) = -∇p + μ∇2
u (2)
where u is velocity vector, μ is the fluid viscosity, p is pressure, and ρ is density. Pressure and velocity at the center of each finite volume are decoupled by linear interpolation, where instabilities are avoided by averaging the Navier-Stokes equations for each volume face and relating the face velocity to the pressure gradient. Reynolds number, Re,
was also calculated for each case to further characterize the flow. Values calculated based on the mean velocity, u
m
, and hydraulic diameter, D
h
, at the midplane of the chambers are estimated to be on the order of 1 – 4. As this Re number falls well within the laminar region (laminar flow, Re<1400), viscous-dominated flows are anticipated. Using velocity components and pressure from above, the fluid shear stress, τ, at the surface of the chamber was calculated from the viscosity and rate of strain,
,
τ
wall
= μ
(4)
The perfusion medium was idealized as water with appropriate constant fluid properties: μ = 0.001 kg/m-s and ρ = 1000 kg/m3. A no-slip boundary condition was used for all chamber walls, and the inlet/outlet conditions were determined for standard pipe (tubing) flow with a laminar parabolic velocity profile and corresponding pressure gradient. Simulations were carried out using a finite-volume numerical method under steady flow conditions, with a convergence criterion of 0.0001, for the solution of each velocity component and pressure gradient per finite volume. The resulting calculations included 3D spatial resolution of the velocity profiles, pressure gradients along the flow direction (axial), and the shear stress at the bottom surface. These data were recorded for each chamber.
Node densities were increased at the center of the chamber in order to track flow and stress fields at higher resolution in the area where cells are seeded in mechanotransduction studies. The volume of fluid directly above this center section was isolated for each case and the velocity profile and pressure gradient were magnified in this section to increase resolution and to extract maximum and minimum values. Shear stresses experienced at the surface were then determined for each flow chamber. Thus, accurate comparisons could be made between global flow regimes in the commercial perfusion chambers as well as local flow regimes that impart stresses to cells within the chamber.
μ-PIV validation
In order to validate the velocity and shear stress components found in the computational models, microparticle image velocimetry (μ PIV) techniques were performed to measure the rate of flow found within each chamber design. A Leica DMIRE-2 (Leica Microsystems, Inc, Bannockburn, IL) inverted epifluorescent microscope with integrated (hardware and software) Scan IM 100 × 120 automated stage (Marzhauser GmbH & CO, Wetzlar-Steindorf, Germany) and Retiga EXi camera (Q-Imaging, Burnaby, BC, Canada) were used to image TetraSpeck fluorescent microparticles (4 μ m diameter; excitation wavelengths 365/505/560/660 nm; emission wavelengths 430/515/580/680 nm; T-7283, Molecular Probes, Eugene, OR) as they traveled through the chamber in a 2.8 × 104 microspheres per ml DH2O suspension. As the suspension moved through the flow channel, an automated imaging routine (implemented in OpenLab 4.0.3, Improvision Inc, Lexington, MA) captured images of a grid containing the entire flow field. This procedure was then repeated 5 times consecutively to capture the maximum number of particles, and to minimize sampling error. This process was then repeated at several planes spaced 50–100 μ m apart through the depth the flow channel. The microspheres appeared in the images as streaked lines of varying length, where the length of the streak was equal to the distance traveled during the exposure time interval.
The same set of procedures was also used to perform another μ PIV study to determine any effects that seeded cells might have on flow fields within the chamber. The Oligene chamber (chamber 1) was implemented for this set of experiments. Degreased silica glass coverslips were etched with sodium hydroxide for 1 hour, and then covered with a 0.15 mg/mL solution of a collagen/acetic acid solution for 1 hour. After rinsing, MLO-Y4 osteocyte-like cells (a generous gift from Lynda Bonewald, University of Missouri-Kansas City) were seeded onto the coverslips at a density of approximately 5500 cells/cm2. The cells were then incubated for 48 hours before being fixed in a 3.7% solution of formaldehyde for 10 minutes.
The particle velocities within each chamber were calculated using a combination of image processing and symbolic mathematical manipulation software. After conversion to gray-scale, the images were auto-leveled in Adobe Photoshop CS (Adobe Systems, Inc.) to enhance contrast between the particle streaks and background noise. Image thresholding and particle analysis was completed using ImageJ 1.34 (NIH, Bethesda, MD). After exclusion parameters were applied to remove any artifacts (too large or small to be particles), the output data file for each image was processed using a Mathematica (Wolfram Research, Inc.) notebook file. Velocity was calculated as the distance traveled per duration of imaging (i.e. shutter speed). The sequential data at each point was combined to create an array of sample-averaged velocities, and used to generate a vector field depicting particle velocity through the entire flow field. The measured profiles were then compared to the calculated velocity components obtained from the CFD models for flow rates equivalent to the target shear stresses in order to validate the computational results.
Particular care was taken to ensure repeatability of trials as well as to minimize random error. Using the automated stage and OpenLab, the exact position of the flow channel (with respect to the stage adapter) was recorded in the software for each chamber. This allowed for the automations to be repeated using the same image coordinates each time. Any random errors that were introduced when capturing the particle streaks were minimized by running the automation five times consecutively for each focal depth, in order to capture the maximum number of particle streaks possible. Two-times binning, which acquires 2 × 2 adjacent pixels as one large pixel, was used to increase the speed of image (and particle streak) capture. Pixel size is 0.5 × 0.5 μ m for the 20× objective and 1.0 × 1.0 μ m for the 10× objective. Hence, binning, which is implemented to minimize any lag time in real-time imaging, could potentially introduce an error of 1–2% in measurement of microsphere displacement, e.g. considering a 100 μ m total displacement. During image processing, particle streaks attributable to background noise or particles not moving with the rest of the fluid flow were identified as being outside of the range of lengths possible for the given flow regime and were removed.