Nonlinear Gradients - Greg Schneider

Background

In order to generate solutions which do not mix linearly along a concentration gradient, nonlinear (e.g., exponential, logarithmic, or sigmoidal) mixing gradients can be creating using microfluidic mixing techniques. This is primarily achieved by varying fluid channel length or by designing asymmetrical microfluidic channels. As opposed to traditional gradient generation devices, such as Microwell plates, the gradient across a microfluidic device can be quantified and adjusted based on flow rate and channel design. By varying these parameters in a nonlinear gradient generator, microfluidic networks can be significantly less complex and smaller than traditional, linear designs.

Figure 1 Microfluidic channel design capable of producing logarithmic concentration gradients. The horizontal channel width determines the degree of nonlinearity. ^[1]

Figure 2 Microfluidic channel design which forms a linear concentration gradient. The gradient is generated by maintaining equal flow rates in the two liquids and the limited fluid to fluid contact allowed by the "contact space". ^[1]

Uses

Nonlinear gradient generation devices are particularly useful for biological applications. For example, studying chemotaxis. A nonlinear microfluidic gradient which flows continuously can show how a cell vresponds to a gradient changing through time as well as how shear stress (i.e., changes in the flow behavior) influences cell motility. Being able to study cell motility in continuous, temporally varying conditions more closely mimics real biological systems. Flow-based nonlinear gradients in particular are used for chemotaxis studies on cancer cells.

Mathematical Model

Assuming that a nonlinear gradient can be generated in a time-independent manner, the Navier-Stokes Equation can be simplified to describe the nonlinear gradient.

Figure 2 Selimovic et al. represent a 2-D model of a nonlinear gradient via this simplified form of the Navier-Stokes equation. ^[3]

Selimovic et al. showed close agreement between their mathematical model and experimental results, as depicted in Figure 3.

Figure 3 Selimovic et al. represent a 2-D model of a nonlinear gradient via this simplified form of the Navier-Stokes equation. ^[4]

where,

• [math]\displaystyle{ }[/math] is the velocity vector
• [math]\displaystyle{ }[/math] is pressure
• [math]\displaystyle{ }[/math] is the Reynolds Number
• [math]\displaystyle{ }[/math] is the Peclet Number
• [math]\displaystyle{ }[/math] is the normalized concentration

References

1. Abe, Y., Kamiya, K., Osaki, T., Sasaki, H., Kawano, R., Mikia, N., Takeuchi, S. Nonlinear concentration gradients regulated by the width of channels for observation of half maximal inhibitory concentration (IC50) of transporter protein. Royal Society of Chemistry. 2015,140. DOI: https://doi.org/10.1039/C4AN02201G

2. Li, B., Qi, J., Fu, L., Han, J., Choo, J., deMello, A. J., Lin, B., Chen, L. Integrated hand-powered centrifugation and paper-based diagnosis with blood-in/answer-out capabilities. Biosensors and Bioelectronics. 2020, 165. DOI: https://doi.org/10.1016/j.bios.2020.112282

3. Pan, Z., Nong X., Xie, Y., Zeng, H., Liang, Y., Zhang, M. Field Determination of Phosphate in Environmental Water by Using a Hand-Powered Paper Centrifuge for Preconcentration and Digital Image Colorimetric Sensing. Journal of Analytical Methods in Chemistry. 2022. DOI: https://doi.org/10.1155/2022/7359197

4. This Simple Paper Centrifuge Could Revolutionize Global Health | WIRED. WIRED. YouTube, January 10, 2017. https://www.youtube.com/watch?v=L5ppD07DMKQ

5. Raju, S. P. and Chu, X. Rapid Low-Cost Microfluidic Detection in Point of Care Diagnostics. Mobile and Wireless Health. 2018, 42. DOI: https://doi.org/10.1007/s10916-018-1043-1

6. Yang, S., Lv, S., Zhang, W., Cue, Y. Microfluidic Point-of-Care (POC) Devices in Early Diagnosis: A Review of Opportunities and Challenges. Sensors. 2022, 22 (4). DOI: https://doi.org/10.3390/s22041620