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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

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

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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.

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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…

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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

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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

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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

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

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

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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:

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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:

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

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Stripe Algorithm with N/3 Droplets Completion time:

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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

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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

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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

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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

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

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