# Towards optimal strategies for moving droplets in digital microfluidic systems

## Summary (3 min read)

### INTRODUCTION

- Advances in microfabrication and microelectromechanical systems (MEMS) over the past decades have lead to a rapidly expanding collection of techniques to build systems for the handling and analyzing of very small quantities of liquids (see, e.g., [1] ).
- These microfluidic systems typically consist of submillimeter scale components such as channels, valves, pumps, and reservoirs, as well as application-specific sensors and actuators.
- Microfluidic devices hold great promise, for example for novel fast, low-cost, portable, and disposable diagnostic tools.
- Applications include the massively parallel testing of new drugs, the on-site, real-time detection of toxins and pathogens, and PCR (polymerase chain reaction) for DNA sequence analysis.
- They usually operate with continuous flows of liquids, in analogy to traditional macro-scale laboratory setups, and integrate all functionality into complete "bio-systemson-a-chip ".

### A. Digital Microfluidic Systems

- More recently, there has been increased interest in microfluidic devices that handle discrete droplets, with volumes usually in the sub-microliter range.
- In these "digital microfluidic systems" (DMFS), droplets are generated, transported, merged, analyzed, and disposed on planar arrays of addressable cells.
- This architecture for microfluidic systems is attractive because of (1) greater flexibility -analyte handling may be reconfigured simply by re-programming rather than by changing the physical layout of the microfluidic components; (2) high droplet speeds -reportedly up to 25cm/s [2] ; (3) no dilution and cross-contamination due to diffusion and shearflow; and (4) the possibility for massively parallel microfluidic circuits.

### D. Paper Overview

- The goal of this paper is to generate optimal sequences of control signals to move droplets from start to goal positions in the shortest number of steps.
- With growing array size and number of droplets, this becomes increasingly challenging: closely related optimizations are the traveling salesman problem, VLSI circuit routing, factory floor plan layout, resource scheduling, and motion planning with multiple moving robots, which are known to be computationally expensive (i.e., NP-hard [11] ).
- Section III gives a more formal problem definition.

### A. DMFS Design Specifications

- These specifications provide a physical framework within which a DMFS can operate.
- Once a sufficiently general DMFS model exists, the authors can investigate algorithmic solutions at an abstract level, without worrying about the specific details of varying hardware implementations.

### B. Problem Definition

- Various kinds of transitions exist, including droplet generation, moving, disposing, merging, and splitting.
- In addition, to avoid accidental merging of droplets, at least one empty cell is required between two occupied cells at all times.
- Transitions are further restricted by the addressing circuitry and cells with specialized functions.

### IV. DMFS CONTROL STRATEGIES

- The authors will first give a simple, complete algorithm based on A* search, but find that its computational complexity is very high (exponential in number of droplets).
- The authors then present a more efficient algorithm that trades off completeness for faster execution times.

### A. Basic Algorithm Outline

- This algorithm maintains a graph data structure to represent the array (inclusive special cells and obstacles) and to keep track of droplet locations.
- Transitions between states define edges in this graph, and finding an optimal control strategy to transform start state A s into goal state A g becomes a standard graph search problem, which can be solved, for example, using an A* algorithm known from artificial intelligence programming [19] :.
- This estimate provides a heuristic that gives preference to the more promising paths.
- The downside of this approach is its high asymptotic complexity.
- One might hope that in practice, most of these choices need not be explored.

### B. Prioritized Droplet Control

- The discussion above has shown that droplet motion planning for DMFS has two main aspects: generating efficient droplet motion plans, and finding efficient algorithms to generate these plans.
- This section applies ideas from Erdmann and Lozano-Pérez [12] to DMFS control.
- (2) For each droplet, starting with the highest priority, generate an optimal motion plan.
- Droplets with higher priorities are considered time-dependent obstacles.
- Figures 4b-e depict the individual traces for each of the four droplets.

### V. OTHER SAMPLE DROPLET MANIPULATION STRATEGIES

- In this section the authors show two additional examples of optimal control strategies.
- This strategy assumes that the electrode in each cell can be activated independently from all other cells.
- The two droplets are always separated by at least one empty cell, such that accidental merging is avoided.
- Note that the darker droplet moves more than necessary (gratuitous steps 4 and 5), but this does not affect the overall number of 8 steps in the control strategy.

### A. Limited Row-Column Addressing

- The previous examples assumed that each cell in the array is individually addressable.
- Two droplets trade places as in Figure 5 above, but here droplets move only to cells whose row and column address has been activated (indicated by triangular arrows).
- Instead of making full use of these advantages, the computational complexity may limit DMFS to much more constraint applications.
- The authors have shown one possible answer to this challenge: Instead of insisting on optimal strategies, an algorithm that trades off completeness and optimality for polynomial run-time was presented.
- Thus, even if the hardware allows simultaneous motion of droplets (e.g., with individually addressable cells), it may be more effective to first generate a motion plan consisting of single droplet moves, and then perform a post-processing step that "parallelizes" the plan as much as possible.

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##### Related Papers (5)

##### Frequently Asked Questions (15)

###### Q2. What are the main techniques used to move droplets across a planar surface?

Small droplets can be moved across a planar surface effectively with a variety of techniques, for example with electric fields (e.g., [2-6]), the thermocapillary effect (e.g., [7]), electrochemical surface modulation (e.g., [8]), or conformational changes in molecular surface layers (e.g., [9]).

###### Q3. What is the way to solve the graph search problem?

As into goal state Ag becomes a standard graph search problem, which can be solved, for example, using an A* algorithm known from artificial intelligence programming [19]:

###### Q4. What is the way to control droplets?

VI. CONCLUSIONS AND FUTURE WORK Digital microfluidic systems (DMFS) based on droplet manipulation are promising because of their flexibility and reconfigurability: they shift complexity from microfluidics hardware to control software.

###### Q5. What is the main technique used for dielectrophoresis?

More information on dielectrophoresis can be found, e.g., at [10].2) Electrowetting Electrowetting on dielectric (EWOD) exploits the decrease of contact angle that an aqueous droplet on a dielectric surface experiences when exposed to an electric field.

###### Q6. How many steps can be found with the A* algorithm?

The authors conclude that the search graph explored with the A* algorithm has O((mn)!) nodes and a branching factor of O(4s), leading to prohibitive complexity for any non-trivial array size with more than a few droplets.

###### Q7. What is the basic structure of a DMFS?

Layout: Typically, a DMFS consists of a rectangular arrayA with m×n cells (but, e.g., an arrangement of hexagonal cells is also possible).•

###### Q8. Why is the architecture for microfluidic systems attractive?

This architecture for microfluidic systems is attractive because of (1) greater flexibility – analyte handling may be reconfigured simply by re-programming rather than by changing the physical layout of the microfluidic components; (2) high droplet speeds – reportedly up to 25cm/s [2]; (3) no dilution and cross-contamination due to diffusion and shearflow; and (4) the possibility for massively parallel microfluidic circuits.

###### Q9. what is the simplest way to generate a droplet?

Droplet generation: For (x,y) ∈ {1…m}×{1…n} and some i ∈ {1…t}, a droplet is generated at coordinate (x,y) ifA(x,y) = ∅ at time t and A(x,y) = Ti at time t+1.•

###### Q10. What is the main idea of the paper?

In this paper, the authors have shown one possible answer to this challenge: Instead of insisting on optimal strategies, an algorithm that trades off completeness and optimality for polynomial run-time was presented.

###### Q11. How many different placements of droplets on the array?

Then the number of different placements of droplets on the array is )(mnd , which for modest numbers m=n=10 and d=10 yields more than 1.7×1013 possibilities.

###### Q12. What is the way to generate a control strategy for a DMFS?

e.g., a control strategy for a complex DMFS can be generated in polynomial time that is guaranteed to be at most twice as long as an optimal solution then this might be sufficient for most practical purposes.

###### Q13. What is the recent interest in microfluidic devices?

More recently, there has been increased interest in microfluidic devices that handle discrete droplets, with volumes usually in the sub-microliter range.

###### Q14. What is the main aspect of a DMFS plan?

The discussion above has shown that droplet motion planning for DMFS has two main aspects: generating efficient droplet motion plans, and finding efficient algorithms to generate these plans.

###### Q15. What is the way to avoid droplets?

Droplet 3 (Figure 4d) has to evade droplets 1 and 2 and therefore turns left in steps 10 and 13, instead of choosing the shorter path towards the right.