sample sample
buffer
reagent
reagent
waste
Figure 5.13: The routing result of the 2D plane of the diagnostics 1 benchmark obtained by the 2D routing algorithm. There are total five nets in this 2D plane.
The arrows represent droplets movement directions.
Now we show the result of the minimization of the number of cells used for routing. Table 5.4 shows our experimental results. We report the total number of cells used for routing on all 2D planes (#Tcell) and the CPU time to route all 2D planes. As shown in this table, our routing algorithm can route all benchmarks while previous works cannot. For example, for the diagnostics 2 benchmark, neither of the two-stage routing and the prioritized A∗-search algorithms is able to gener-ate a routing solution, while our schemes have successfully routed this benchmark with reasonable CPU time. Furthermore, for those benchmarks where previous ap-proaches can generate a feasible solution, e.g., diagnostics 1, our routing algorithm provides solutions with fewer cells used for routing in less CPU time compared with the two-stage routing algorithm and the prioritized A∗-search algorithm. This result demonstrates the robustness and efficiency of our routing algorithm. Figure 5.13 shows the routing result of the 2D plane of the diagnostics 1 benchmark obtained by the 2D routing algorithm. This 2D planes has 5 nets and 45 cells used for routing.
Arrows represent the droplets’ moving directions.
Here we discuss why our approach is more robust and effective than the other
Table 5.5: Timing-aware routing result of the two bioassays. N/A denotes that some 2D planes are failed for routing.
Circuit [50] [6] Ours
Rtd CPU Rtd CPU Rtd CPU
time time time
(sec) (sec) (sec)
Diagnostics 1 2.22 0.17 1.17 45.26 1.16 0.05 Diagnostics 2 N/A N/A N/A N/A 1.33 0.05 Protein 1 2.85 1.35 N/A N/A 1.44 0.24 Protein 2 N/A N/A N/A N/A 1.12 0.17
two routing algorithms. For the two-stage algorithm, droplet routing and scheduling are performed in separate stages without considering the interaction among them.
Moreover, the alternative routing path generation stage only finds the shortest path without explicitly minimizing the number of cells used for routing. In contrast, our algorithm can concurrently route a set of independent nets in global routing and simultaneously perform droplet routing and scheduling in detailed routing. For the prioritized A∗-search algorithm, the possible routing path and schedules of low-priority droplets are not considered while routing high-low-priority droplets. Therefore, droplets with lower priorities may be blocked by droplets with higher priorities, making this approach harder to find a feasible solution. In contrast, our algorithm adopts a negotiation based routing scheme. We iteratively rip up and reroute a set of nets to modify the routing solution. Therefore, our algorithm is more robust for various bioassays.
Finally we show the timing-aware droplet routing results in Table 5.5. Note that in this experiment, for fair comparison, we adopt the original prioritized A∗ -search algorithm without any modification. Let Rdt be the ratio of the maximum droplet transportation time (in cycles) over the maximum Manhattan distance of one 2D plane. The maximum Manhattan distance is the minimum time to route all droplets from their sources to sinks. Therefore, a smaller Rtd indicates shorter time to route all droplets. We report the maximum Rtd of all 2D planes and CPU time. Compared with the two-stage routing algorithm, our algorithm obtains a better solution; i.e., smaller Rtd (1.16 vs. 2.22), in less CPU time (0.05 sec vs. 0.17
sec) for the diagnostics 1 benchmark. The experimental results also show that our routing algorithm outperforms the prioritized A∗-search algorithm. For the same benchmark, our algorithm obtains a routing solution with shorter routing time (1.16 vs. 1.17) in much less CPU time (0.05 sec vs. 45.26 sec).
Chapter 6
Droplet Routing on Cross-referencing Biochips
In this chapter, we handle the droplet routing problem on cross-referencing biochips, which uses row/column addressing scheme to activate electrodes. The main chal-lenge of this type of biochips is the electrode interference which prevents simultane-ous movement of multiple droplets. We first present a basic integer linear program-ming (ILP) formulation to optimally solve the droplet routing problem. Due to its complexity, we also propose a progressive-ILP scheme to determine the locations of droplets at each time step. Experimental results demonstrate the efficiency and effectiveness of the proposed progressive-ILP scheme on a set of practical bioassays.
6.1 Introduction
In previous chapters, we solve the placement and routing problems on gen-eral biochips, which allow each electrode to be individually activated. Due to this property, the general biochips are also referred to as the direct-addressing biochips in this chapter. While this architecture provides the flexibility for droplet move-ment, it suffers from the major drawback that the number of control pins rapidly increases as the system complexity (i.e., the size of the array) increases. To overcome these limitations, recently a new digital microfluidic biochip, referred to as cross-referencing architecture has been proposed [18], as presented in Chapter 1. This architecture uses a row/column addressing scheme, where a set of electrodes in one row/column is connected to a control pin. Therefore, the number of control pins is greatly reduced: this number is now proportional to the perimeter of the chip rather than the area of the chip. However, this architecture also introduces a new set of limitations. Since an electrode can potentially control the movement of all droplets
111
in a row/column at the same time, this architecture incurs higher droplet movement complexity than that of direct-addressing biochips. Moreover, the manipulation of more than two droplets causes electrode interference among droplets, which prevents multiple droplets to move at the same time. This performance limitation is a major drawback to high-performance applications, such as large-scale protein analysis.
In this chapter, we tackle the problem of droplet routing on cross-referencing biochips [62]. The main challenge of this routing problem is to ensure the correctness of droplet movement; the fluidic property which avoids unexpected mixing among droplets needs to be satisfied, and electrode interference patterns that prevent mul-tiple droplets from moving at the same time must be avoided. The goal of droplet routing is to minimize the maximum droplet transportation time, and has several motivations, such as the the benefit for real-time applications and the maintenance of bioassay execution integrity, as detailed in the previous chapter.
6.1.1 The Contribution
In this chapter, we propose an integer linear programming (ILP) based droplet routing algorithm for cross-referencing biochips. We derive the basic ILP formulation to simultaneously perform droplet routing and assign voltages to the cross-referencing electrodes, while minimizing the maximum droplet transportation time. Moreover, we also model multi-pin nets in the proposed ILP formulation for practical bioassays, where multiple droplets are merged during their transportation.
To overcome the computational cost of the ILP, we also propose a progressive-ILP routing scheme, which is used to find the min-cost droplet locations at each time step using an ILP. Unlike the progressive-ILP scheme proposed in [10], which divides the original problem spatially, the proposed algorithm divides the original routing problem temporally. In this way, the original problem is reduced to a manageable size, and we can practically apply an ILP-based method to find a good solution within reasonable CPU time. The major contributions of this chapter include the following:
• We propose the first routing algorithm that directly solves the routing problem in cross-referencing biochips. In contrast with previous works that start with
an initial routing solution, the proposed algorithm has higher flexibility and can obtain better solutions for droplet routing on cross-referencing biochips.
• To tackle the complexity of the basic ILP formulation, we propose the progres-sive-ILP routing scheme. We iteratively determine the locations of droplets at each time step by ILP formulation. Therefore, the proposed algorithm can obtain a high-quality solution within reasonable CPU time.
• Unlike previous works that only move a subset of droplets at each time (for example, the algorithm proposed in [57] only moves droplets whose destination cells are in the same row/column), the proposed algorithm maximizes the number of droplets that can simultaneously move at the same time, even the destination cells are not in the same row/column. Therefore, the proposd algorithm can obtain a routing solution with lower droplet transportation time.
This minimization is especially important for cross-referencing biochips due to the electrode interference problem.
Experimental results demonstrate the efficiency of the progressive routing scheme compared with the basic ILP formulation. The ILP formulation needs more than five days while the progressive-ILP routing scheme needs at most 15.36 sec-onds for one bioassay. Experimental results also demonstrate the effectiveness of the algorithm compared with previous work. For example, for the protein assay, the progressive-ILP routing scheme obtains 45.83% smaller maximum droplet trans-portation time than the network-based method proposed in [65], plus the clique partitioning based algorithm proposed in [57].
The reminder of this chapter is organized as follows. Section 6.2 details routing on cross-referencing biochips and formulates the droplet routing problem.
Section 6.3 presents the basic ILP formulation for droplet routing problem the com-plexity analysis. Section 6.4 details the progressive-ILP routing scheme, while Sec-tion 6.5 shows the experimental results.