1. Optimal Testing of Digital Microfluidic Biochips: A Multiple Traveling Salesman Problem R. Garfinkel 1, I.I. Măndoiu 2, B. Paşaniuc 2 and A. Zelikovsky 3 1 Operations and Information Management, University of Connecticut 2 Computer Science and Engineering, University of Connecticut 3 Computer Science, Georgia State University

2. Outline Introduction Problem definition ILP Formulation Bounds and Heuristic Experimental results Conclusions

3. Introduction Lab-on-chip  Systems for performing biomedical analyses of very small quantities of liquids Advantages  Fast reaction times  Low-cost, portable and disposable  Compactness  massive parallelization  high-throughput 2 Types:  Continuous-flow: enclosed, interconnecting, micron- dimension channels  Digital: discrete droplets of fluid across the surface of an array of electrodes.

4. Digital Microfluidic Biochips [Srinivasan et al. 04] [Su&Chakrabarty 06] I/O Cell Electrodes typically arranged in rectangular grid Droplets moved by applying voltage to adjacent cell Can be used for analyses of DNA, proteins, metabolites…

5. Optimization Challenges Module placement  Assay operations (mixing, amplification, etc.) can be mapped to overlapping areas of the chip if performed at different times Droplet routing  When multiple droplets are routed simultaneously must prevent accidental droplet merging or interference Testing  High electrode failure rate, but can re-configure around  Performed both after manufacturing and concurrent with chip operation  Main objective is minimization of completion time

6. Concurrent Testing Problem GIVEN:  Input/Output cells  Position of obstacles (cells in use by ongoing reactions) FIND:  Trajectories for test droplets such that  Every non-blocked cell is visited by at least one test droplet  Droplet trajectories meet non-merging and non- interference constraints  Completion time is minimized Defect model: test droplet gets stuck at defective electrode

7. Concurrent Testing Problem [Su et al. 04] ILP-based solution for single test droplet case & heuristic for multiple input-output pairs with single test droplet/pair Our problem formulation allows an unbounded number of droplets out of each input cell  additional droplets can be used at no extra cost  completion time can be reduced substantially by splitting the work among multiple droplets  however, too many droplets may interfere with each other Test problem for multiple droplets is NP-hard by reduction from the Hamiltonian path problem in grid graphs [Itai et. al. 82]  we seek approximation algorithms and heuristics with good practical performance

8. Merging region Set of cells to be kept empty when (i,j) is occupied by a droplet Merging region:

9. Interference region Set of cells to be kept empty when a droplet moves away from (i,j) Interference region:

10. ILP formulation 0/1 variable for each pair of neighbor cells: is set to 1 iff a droplet that occupies cell (i,j) at time t-1 occupies cell (k,l) at time t j: i: l: k: Time t-1: Time t:

11. ILP Formulation for Unconstrained Number of Droplets Each cell (i,j) visited at least once: Droplet conservation: No droplet merging: No droplet interference: Minimize completion time:

12. Special Case NxN Chip I/O cells in Opposite Corners No Obstacles  Single droplet solution needs N 2 cycles

13. Stripe Algorithm with N/3 Droplets Completion time:

14. Lower Bound Lemma 1: Completion time is at least when k droplets are used Proof: In each cycle, each of the k droplets places 1 dollar in current cell  3k(k-1)/2 dollars paid waiting to depart  3k(k-1)/2 dollars paid waiting for last droplet  k dollars in each diagonal  1 dollar in each cell

15. Approximation guarantee Lemma 2: Completion time for any #droplets is at least Proof: Minimum for is achieved when Theorem: Stripe algorithm with N/3 droplets has approximation factor of

16. Stripe Algorithm with Obstacles of width ≤ Q Divide array into vertical stripes of width Q+1 Use one droplet per stripe All droplets visit cells in assigned stripes in parallel  In case of interference droplet on left stripe waits for droplet in right stripe

17. Results for 120x120 Chip, 2x2 Obstacles ~20x decrease in completion time by using multiple droplets Obstacle Area Average completion time (cycles) k=40 vs. k=1 speed-up k=1k=12k=20k=30k=40 0%144001412944710593 24x 1%142561420953.4715.2598.8 24x 5%136801473982.8725596.2 23x 10%1296014901010.8734.8592.6 22x 15%1224015011025.8730.8588.2 21x 20%1152015011046.8738.4580.8 20x 25%1080015011071736.6570 19x

18. Conclusions Presented ILP formulation, approximation algorithm and heuristic for microfluidic biochip testing problem Combinatorial optimization techniques can yield significant improvements