Abstract
We present a two-dimensional model that describes the concentration profile of a class of previously reported microfluidic devices which are of particular interest in cellular taxis research. The devices generate stable concentration gradients by mixing and dividing two or more external inputs into a large number of discrete streams. This study focuses specifically on modeling the confluence of the discrete streams in a long chamber. We derive a closed-form solution for gradient generators with any arbitrary number of sampling streams. By relating the physical dimensions to the Péclet number, we create a model independent of flow rate and therefore dependent only on the specific nature of the boundary condition provided by the upstream network. As a result, the modeling method we propose may help evaluate the effectiveness of competing gradient generation schemes. Finally, our analytical work introduces a framework for developing simple design rules of interest to experimentalists working with these devices.
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Notes
We implemented the DFT solution algorithm in MATLAB (Mathworks, Inc., Natick, MA). Further details are available from the authors upon request.
Further details are available from the authors upon request.
Typically, the target function, η(x *), and the impulse response function, h(x *), are equivalent. However, it may be necessary to approximate a complicated target function with a simpler impulse response function, and thus it would be appropriate to minimize the RMS deviation with respect to the target function.
Other norms may be considered. However, the general principle of calculating a single constant representing all flow conditions Pe ≫ 1 remains.
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Acknowledgements
We thank Michael Hwang for programming assistance. We are further indebted to Drs. G. Kane Jennings, Dmitry Markov, and Robert Roselli for reviewing early drafts of this manuscript. The Advanced Computing Center for Research and Education (ACCRE) at Vanderbilt graciously provided computational resources. We acknowledge support from the NSF-sponsored VaNTH ERC, the Systems Biology/Bioengineering Undergraduate Research Experience (SyBBURE), the Vanderbilt Institute for Integrative Biosystems Research and Education (VIIBRE), NIH Grant 5U01AI061223, and a Whitaker Foundation Special Opportunity Award.
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Gorman, B.R., Wikswo, J.P. Characterization of transport in microfluidic gradient generators. Microfluid Nanofluid 4, 273–285 (2008). https://doi.org/10.1007/s10404-007-0169-0
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DOI: https://doi.org/10.1007/s10404-007-0169-0