Microfluidic Gradient Generators - Greg Schneider
Microfluidic Gradients Background
Simple microfluidic devices can allow for extremely good control over fluid mixing using laminar flow. Due to laminar flow, mixing in microfluidic devices is largely controlled by diffusion (see also Péclet number (Pe)) between different fluids at the interface. Microfluidic gradient devices can be precisely programmed to mix fluids by choosing how long two or more fluids contact one another and diffuse.[1] Diffusion can be tuned by adjusting the respective flow rates of fluids, relative channel sizes and lengths, and the total contact area. Gradient generation is useful for a broad range of fluid-related research applications. Biological studies and analytical research dominate.
Typically, equipment such as microtiter/microwell plate arrays with fluid wells have been used to create concentration gradients via serial dilution. A microwell plate is shown in Figure 1. Microfluidic gradient generators can significantly reduce preparation time compared to microwell plates, as well as the amount of liquid required to achieve similar results to microwell plates. Walker et al., compared the performance of a microfluidic gradient generator to a 96-well plate in the context of linear dilution. The microfluidic device used 100 times less fluid than the 96-well plate to achieve the same linearity. Furthermore, a microfluidic gradient generator can be used in a continuous fashion, and by changing flow rates the gradient concentrations can be changed in real time.[2]
Mathematical Model
Slow diffusion at the liquid-liquid interface controls passive mixing within microfluidic devices, which is behavior occuring in the low Péclet number (Pe) region. Fick's law determines the overall mixing time for diffusive mixing. The mixing time [math]\displaystyle{ \mathrm\tau_{mix} }[/math] in a microfluidic channel is described by:[9]
[math]\displaystyle{ \mathrm\tau_{mix} = \frac{d^2}{2D}\ }[/math]
Where:
- [math]\displaystyle{ \mathrm\tau_{mix} }[/math] is the mixing time
- d is the diffusive distance
- D is the diffusion coefficient (literature value)
The mixing time influences channel length. For a given microfluidic device, the channel length must be adequate for the fluids in solution to mix completely. In gradient trees, for example, long channels must be compacted into a small area for adequate mixing to occur in a small device.
Gradient Trees
A gradient tree is a hierarchical microfluidic structure that can create highly complex and precise concentration gradients. The gradient resulting from a gradient tree is dependent on channel length, total contact area, and the number of contact areas.[5] In Figure 2, the gradient tree features a multitude of squiggly channels. The channels are intended to be as long as possible in a compact area to allow for the fluids to fully diffuse and mix completely before the next contact area is reached. Figure 2 represents a gradient tree which results in 7 linear dilutions based on 2 input streams. The leftmost outlet is a pure stream of the left inlet fluid. The rightmost outlet is a pure stream of the right inlet fluid. The centermost outlet is an equal mixture of the right and left inlet fluids. Linearly formed concentration steps are formed in between. Each additional branch or layer added to the gradient tree introduces a new, stepwise change in the concentration between the two lines which come into contact in the horizontal "contact zone" channels. Linearity between each step can be maintained in this device by preserving equal volumetric flow rates between the two inlets. A nonlinear gradient could theoretically be achieved using this device by simply changing one of the flow rates of the two inlet fluids (i.e., making the volumetric flow rates unbalanced).
Nonlinear Gradients
In order to generate solutions which do not mix in linear steps 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. In a gradient tree, nonlinear gradients are formed using unequal flow rates of the inlet fluids. Microwell plate gradients are typically nonlinear (logarithmic). A microfluidic nonlinear gradient can be easily programmed for different types of nonlinearity.[2] Channel width variation is considered more precise for forming nonlinear gradients, so equal flows can be maintained between inlets, and etched channel widths control the degree of nonlinearity. However, this requires more device planning and preparation.[2]
Uses
Gradient trees are regularly used in biological studies. Walker et al., proposed the use of a microfluidic gradient generator to perform cytotoxicity studies.[2] In the case of cytotoxicity studies, these miniaturized studies reduced cell waste and reagent use. After creating a desirable gradient profile, the authors were able to dose each gradient outlet with various treatments in a precise manner with continuous flow. Materials are a further advantage for microfluidic devices in this type of study. Since microfluidic devices are often made of optically transparent materials such as glass or PDMS, Walker et al., were able to record data in situ using florescence.[2]
Another example of gradient application is in drug resistance research; microfluidic devices were used to form concentration gradients of antimalarial drugs to test their efficacy across a range of concentrations. In the case of Rhomphosri et al., a linear gradient generation device was used.[8]
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 responds 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. Nonlinear gradients based on flow rate changes are often used for chemotaxis studies on cancer cells. [3]
MIMIC devices, which utilize capillary action to form precise patterns, often incorporate microfluidic concentration gradients. MIMIC devices themselves can be used to form concentration gradients via surface patterning.
References
1. Gao, Y., Sun, J., Lin, W., Webb, J., Li, D. A compact microfluidic gradient generator using passive pumping. Microfludics and Nanofluidics. 2012, 12. DOI: https://doi.org/10.1007/s10404-011-0908-0
2. 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
3. Walker, G., Monteiro, N., Rouse, J., O'Neill, A. A linear dilution microfluidic device for cytotoxicity assays. Lab on a Chip. 2006, 7. DOI: https://doi.org/10.1039/b608990a
4. Selimovic, S., Sim, W., Sang Bok Kim, Yun Ho Jang, Won Gu Lee, Masoud Khabiry, Hojae Bae, Sachin Jambovane, Jong Wook Hong, Ali Khademhosseini. Generating Nonlinear Concentration Gradients in Microfluidic Devices for Cell Studies. Analytical Chemistry. 2020, 83. DOI: https://doi.org/10.1021/ac2001737
5. Wang, Y.-H., Ping, C.-H., Sun, Y.-S. Design of Christmas-Tree-like Microfluidic Gradient Generators for Cell-Based Studies. Chemosensors. 2023, 11. DOI: https://doi.org/10.3390/chemosensors11010002
6. Shen, Q., Zhou, Q., Lu, Z., Zhang, N. Generation of Linear and Parabolic Concentration Gradients by Using a Christmas Tree-Shaped Microfluidic Network. Wuhan University Journal of Natural Sciences. 2018, 23. DOI: https://doi.org/10.1007/s11859-018-1317-y
7. Corning 96 Well Plate. Wyatt Technology Corporation. Product number P8802-09602. https://store.wyatt.com/shop/plate-reader/wpr3/corning-96-well-plate/
8. Rhomphosri, S., Changruenngam, S., Chookajorn, T., Modchang, C. Role of a Concentration Gradient in Malaria Drug Resistance Evolution: A Combined within- and between-Hosts Modelling Approach. Scientific Reports. 2020, 10. DOI: https://doi.org/10.1038/s41598-020-63283-2
9. Perry, S.L. ChE 535, Lecture 04, Fluid Flow and Mixing. UMass Amherst Department of Chemical Engineering. 2023.
