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Research Papers

# Mechanical Stress Analysis of Microfluidic Environments Designed for Isolated Biological Cell Investigations

[+] Author and Article Information
Sean S. Kohles1

Department of Mechanical and Materials Engineering, Portland State University, Portland, OR 97207; Department of Surgery, Oregon Health and Science University, Portland, OR 97239kohles@cecs.pdx.edu

Nathalie Nève, Jeremiah D. Zimmerman, Derek C. Tretheway

Department of Mechanical and Materials Engineering, Portland State University, Portland, OR 97207

1

Corresponding author.

J Biomech Eng 131(12), 121006 (Nov 10, 2009) (10 pages) doi:10.1115/1.4000121 History: Received January 09, 2009; Revised June 03, 2009; Posted September 01, 2009; Published November 10, 2009; Online November 10, 2009

## Abstract

Advancements in technologies for assessing biomechanics at the cellular level have led to discoveries in mechanotransduction and the investigation of cell mechanics as a biomarker for disease. With the recent development of an integrated optical tweezer with micron resolution particle image velocimetry, the opportunity to apply controlled multiaxial stresses to suspended single cells is available (Nève, N., Lingwood, J. K., Zimmerman, J., Kohles, S. S., and Tretheway, D. C., 2008, “The $μPIVOT$: An Integrated Particle Image Velocimetry and Optical Tweezers Instrument for Microenvironment Investigations,” Meas. Sci. Technol., 19(9), pp. 095403). A stress analysis was applied to experimental and theoretical flow velocity gradients of suspended cell-sized polystyrene microspheres demonstrating the relevant geometry of nonadhered spherical cells, as observed for osteoblasts, chondrocytes, and fibroblasts. Three flow conditions were assessed: a uniform flow field generated by moving the fluid sample with an automated translation stage, a gravity driven flow through a straight microchannel, and a gravity driven flow through a microchannel cross junction. The analysis showed that fluid-induced stresses on suspended cells (hydrodynamic shear, normal, and principal stresses in the range of 0.02–0.04 Pa) are generally at least an order of magnitude lower than adhered single cell studies for uniform and straight microchannel flows (0.5–1.0 Pa). In addition, hydrostatic pressures dominate (1–100 Pa) over hydrodynamic stresses. However, in a cross junction configuration, orders of magnitude larger hydrodynamic stresses are possible without the influence of physical contact and with minimal laser trapping power.

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## Figures

Figure 1

Schematics of applied microenvironments including (a) a uniform flow field generated by moving the fluid sample with an automated translation stage, (b) a gravity driven flow through a straight microchannel, and (c) a gravity driven flow through a microchannel cross junction. The Cartesian coordinate system was established at the center of the fixed cell position and converted to (d) spherical polar coordinates during stress analysis (here ϕ=0 deg).

Figure 2

(a) Representative time-lapse image sequence of a living rat osteoblast suspended by an optical trap within the geometric center of the cross-junctional channel design during flow (Fig. 1). The dashed arrows follow the streamline path of a particle within the culture media, as indicated by the white arrow. (b) Velocity flow field surrounding an analogous cell (20.6 μm diameter polystyrene microsphere), as measured with micron resolution particle image velocimetry.

Figure 3

Localized theoretical elastic elements representing applied stresses due to (a) unidirectional and (b) bidirectional flow fields (only hydrodynamic state shown with mild deformation into ellipsoid shape). Mohr’s Circle characterizations of two-dimensional normal (σr with or without σp) and shear (τrθ) stresses with conversion into planar principal stresses (σ1 and σ2) and maximum shear stress (τmax) for (c) straight and (d) cross-junctional channel flow conditions (axes not drawn to scale). The rotation of the element required to minimize or maximize shear stress is also shown (φ).

Figure 4

(a) Shear and (b) normal hydrodynamic stresses applied around the central perimeter (θ=0 to ±180 deg at z=0) of a stationary microsphere (a=10.9 μm radius) in uniform flow (nonchannelized, stage motion). Coincident experimental and theoretical data were compared in the region near the microsphere surface. These experimental and theoretical values were strongly correlated for both shear (R2=0.938) and normal (R2=0.964) stresses. Continuous theoretical data were produced to explore surface stresses (Eqs. 14,15). The total normal stress (static plus dynamic) would have an additional pressure, p=−1.469 Pa, superimposed over the entire surface (Eq. 4).

Figure 5

Experimental and theoretical (a) shear and (b) normal stresses applied around the central perimeter (θ=0 to ±180 deg at z=0) of a stationary microsphere (a=14.0 μm radius) in gravity driven unidirectional flow (straight channel). Coincident data were compared in the region near the microsphere surface. These experimental and theoretical values were strongly correlated for both shear (R2=0.916) and normal (R2=0.974) stresses. Continuous theoretical data were produced to explore surface stresses (Eqs. 14,15). The total normal stress (static plus dynamic) would have an additional pressure, p=−73.45 Pa, superimposed over the entire surface (Eq. 4).

Figure 6

Experimental, theoretical, and numerical (a) shear and (b) normal stresses, as applied around the central perimeter (θ=0 to ±180 deg at z=0) of a stationary microsphere (a=10.3 μm radius) in gravity driven bidirectional flow (cross junction channel). Coincident experimental and theoretical values were strongly correlated for both shear (R2=0.964) and normal (R2=0.927) stresses. Continuous theoretical data were produced to explore surface stresses (Eqs. 24,25). The total normal stress (static plus dynamic) would have an additional pressure, p=−119.75 Pa, superimposed over the entire surface (Eq. 4).

Figure 7

Normalized and thereby dimensionless shear stress (τrθ/μγ̇) plotted as a function of dimensionless radial position (r/a) for a quarter section of a sphere suspended in planar extensional flow. Shear stress is at maximum at the sphere surface, at minimum for r/a=1.67, and approximately equal to the freestream value for r/a>3.

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