Our algorithm was implemented in the C++ programming language and run on a 1.066 GHz SUN Blade 1500 machine with 4 GB memory. We implemented the unified synthesis and placement algorithm proposed in Su and Chakrabarty [2005b] and the subTCG representation on the same machine. For 3D-subTCG, we used the same SA engine as T-tree. We modified the operations of 3D-subTCG to satisfy the storage constraint at each perturbation. We also applied the clustering algorithm proposed in Section 5.1 to 3D-subTCG for fair comparison. For all experiments, we setα = 21.51 ,β = 21.50.5, andγ = 21.520 . We also assumed that there exists one segregation cell between any two operations. All experimental results are the best result obtained by simulated annealing.
We evaluated our placement algorithm with two bioassays: the colorimetric protein assay [Srinivasan et al. 2004] and the multiplexed in-vitro diagnos-tics [Su and Chakrabarty 2004]. Figure 20 shows the sequencing graph of the
… … … … … … …
S11 S1q Sp1 Spq R11 R1q Rp1 Rpq
M11 M1q Mp1 Mpq
Fig. 21. The sequencing graph of multiplexed in-virto diagnostics [Su and Chakrabarty 2004].
colorimetric protein assay while Figure 21 shows the multiplexed in-virto di-agnostics with p samples (Plasma, Serum, Urine, and Saliva) and q reagents (glucose, lactate, pyruvate, and glutamate). For the colorimetric protein as-say, we applied the same design specification (resource constraint) and used the same microfluidic module library as Su and Chakrabarty [2005b]. We as-sumed that there is only one reservoir/dispensing port for sample fluid, two such ports for buffer fluid, two ports for reagent fluid, and one port for waste fluid. We also assumed that there are at most four optical detectors integrated on the biochip [Su and Chakrabarty 2005b]. For the multiplexed in-virto diag-nostics, we used the same design specifications (resource constraint) as Su and Chakrabarty [2004]. We assumed that there is one reservoir/dispensing port for each type of samples and reagents and one optical detector for each enzy-matic assay [Su and Chakrabarty 2004]. However, since [Su and Chakrabarty 2004] did not specify the width, height, and duration of each reconfigurable operation, we generated the areas/durations of each type of the mix operations based on the ratio of areas/durations of each reconfigurable operation in Su and Chakrabarty [2005b]. Table I shows the microfluidic module library used for the multiplexed in-vitro diagnostics.
First, we assumed that no defective cells exist. Table II summarizes the re-sult of the colorimetric protein bioassay. Column 2 lists four different design specifications (fixed-cube constraints). We report the resulting volume (area times assay completion time) and CPU time (in seconds). As shown in this ta-ble, our algorithm can meet all design specifications (fixed-cube constraints) while both [Su and Chakrabarty 2005b] and 3D-subTCG cannot. More impor-tantly, [Su and Chakrabarty 2005b] (3D-subTCG) requires, on average, 1.68X (1.61X) larger volume and 4.32X (39.96X) longer CPU time than our algorithm.
Table III shows the result of the multiplexed in-vitro diagnostics. In this ex-periment, we used three examples for evaluation. Column 1 shows the num-ber of types of samples and reagents, and column 2 lists the type of samples and reagents used in each example. For each example, we applied three dif-ferent design specifications, as listed in column 3. We also report the volume and CPU time in this experiment. As shown in this table, our algorithm can meet all design specifications (fixed-cube constraints) while both 3D-subTCG and Su and Chakrabarty [2005b] cannot. Su and Chakrabarty [2005b] obtains larger volumes in all three examples (3.48X, 4.90X, and 3.84X) with longer CPU times (9.71X, 9.81X, and 19.19X, respectively); 3D-subTCG also obtains larger
Operation Resource Duration (sec.)
Table II. The Experimental Result of the Colorimetric Protein Bioassay [Su and Chakrabarty 2005b] T-tree
Design CPU CPU
Bioassay Spec. Volume Time (sec.) Volume Time (sec.)
Protein 10× 10 × 400 10× 10 × 349 275.05 9× 9 × 241 78.03
Bioassay Spec. Volume Time (sec.) Volume Time (sec.)
Protein 10× 10 × 400 10× 10 × 239 2497.94 9× 9 × 241 78.03 10× 10 × 360 10× 10 × 331 2226.71 10× 9 × 211 57.27 11× 11 × 320 11× 11 × 272 4036.13 10× 10 × 221 68.32 9× 9 × 400 (11× 9 × 398)* 1984.04 9× 9 × 240 65.21
Average 1.61 39.96 1.00 1.00
Volume= Area × Completion Time. ()*: the result cannot meet the design specification.
volumes in all three examples (2.50X, 2.05X, and 1.86X, respectively) with longer CPU times (38.52X, 23.89X, and 41.39X, respectively). The two exper-imental results clearly show the efficiency and effectiveness of our algorithm with different bioassays and design specifications. The results of 3D-subTCG also support our claim in Section 3.5 that the T-tree is a more suitable 3D repre-sentation for the placement problem of biochips. Figure 23 shows the placement
Table III. The Experimental Result of the Multiplexed in-vitro Diagnostics [Su and Chakrabarty 2005b] T-tree
CPU CPU
Design Time Time
Bioassay Description Spec. Volume (sec.) Volume (sec.)
in vitro S1, S2, S3, and S4 9× 9 × 100 9× 9 × 98 85.28 6× 9 × 67 9.12
Bioassay Description Spec. Volume (sec.) Volume (sec.)
in vitro S1, S2, S3, and S4 9× 9 × 100 9× 9 × 97 474.43 6× 9 × 67 9.12
()*: the result cannot meet the design specification. (S1: Plasma, S2: Serum, S3: Urine, S4: Saliva, A1: Glucose, A2: Lactate, A3: Pyruvate, A4: Glutamate).
result of the colorimetric protein assay with the 10× 10 × 400 design specifica-tion. For simplicity, we only show the reconfigurable and detection operations.
Now we demonstrate the effectiveness of our algorithm for handling the de-fective cells. Assume that the biochip of Figure 23 is fabricated. Similarly to Su and Chakrabarty [2005b], we assumed that one optical detector is rendered defective due to fabrication. Therefore, the detection operations that were orig-inally mapped to the defective detector must be re-mapped to other detectors.
In this experiment, we set the fixed architecture as 9× 9 and the limit of as-say completion time as infinity. In this way, our algorithm can minimize the assay completion time while satisfying the design specification. Table IV lists the result of defect tolerance. Column 2 lists the locations of the defective cells.
We considered four different cases with different number and location of de-fective cells. We report the assay completion time (in seconds) and the CPU time (in seconds). As shown in this table, our algorithm can obtain 16% longer
Fig. 22. The 3D view of the placement result of the protein bioassay with the 10× 10 × 400 design specification.
average assay completion time (280 vs. 241) with 42% longer average CPU time (111.46 vs. 78.03) than defect-free placement. This experimental result demon-strates that our defect-tolerance algorithm can operate a bioassay on a defective biochip with reasonable CPU time. Figure 23 shows the two placement results with three and four defective cells.
For the last experiment, we demonstrate the effect of our clustering method proposed in Section 5.1 on the protein bioassay. Table V shows the result of the protein bioassay with and without clustering. Columns 3 and 4 show the volume and CPU time without clustering and columns 5 and 6 show the volume and CPU time with clustering. As shown in this table, we can observe that the T-tree with clustering achieves 27% smaller volume and 6% less CPU time compared with the T-tree without clustering. The reduction on volume comes from the elimination of unnecessary storage units between generation operations and reconfigurable operations. Therefore, the SA engine can obtain a more compact 3D floorplan. The saving in CPU time is not as significant as the reduction on volume, because we do not actually cluster two operations into one operation.
Moreover, we need extra CPU time to ensure that a clustered node is the left child of another clustered node during the tree reconstruction process. This result shows the effectiveness of the proposed clustering algorithm. The result also shows that it is important to make use of the properties of a bioassay during floorplanning.
Fig. 23. (a) The 3D view of the placement result with three defective cells located at (4, 2), (1. 6), (5, 8). (b) The 3D view of the placement result with four defective cells located at (0, 4), (4, 0), (2, 7), (7, 5).
Bioassay cells Time (seconds) Time (seconds)
Table V. The Experimental Result of the Colorimetric Protein Bioassay with and Without Clustering. Volume= area × completion time
w/o Clustering w/ Clustering
Design CPU CPU
Bioassay Spec. Volume Time (sec.) Volume Time (sec.)
Protein 10× 10 × 400 10× 10 × 249 59.73 9× 9 × 241 78.03
In this article, we have applied the temporal floorplanning technique to the placement problem of digital microfluidic biochips. The motivation is that the physical placement of operations can be handled by 2D floorplanning tech-niques. Moreover, previous works show that floorplanning techniques are ap-plicable to some scheduling problems, such as Xia et al. [2003] and Wuu et al.
[2004]. Therefore, to simultaneously perform scheduling and physical place-ment, we model the placement problem of biochips as the temporal (3D) floor-planning problem. The advantage of this approach is that we have a high flex-ibility to optimize both the assay completion time and the biochip area (and other constraints, such as the defect tolerance requirement, as well).
To our best knowledge, our work is the first to adopt a topological representa-tion (the T-tree representarepresenta-tion) for the placement problem of digital microfluidic biochips. We have also proposed a clustering algorithm to cluster a generation operation and a reconfigurable operation to obtain a smaller volume and to re-duce the CPU time. Due to the need to perform a bioassay on a biochip with the existence of defects, the proposed placement algorithm handles the defect tolerant issue by modeling each defective cell as an obstacle and not allowing overlaps among operations and obstacles. We have shown the efficiency and the effectiveness of our algorithm over previous works.
Future work lies in finding more sophisticated methods for handling the storage units as well as considering the fault tolerance issue and the design-for-defect/fault tolerance requirement during floorplanning. Another potential research direction lies in mapping the placement problem of biochips to other problems, instead of the floorplanning one. For example, the placement problem can be mapped to the unified high-level synthesis and physical design problem ([Dougherty and Thomas 2000; Gu et al. 2005]). A bioassay is represented as