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Optimal Testing of Digital Microfluidic Biochips: A Multiple Traveling
Salesman Problem
R. Garfinkel1, I.I. Măndoiu2, B. Paşaniuc2 and A. Zelikovsky3
1Operations and Information Management, University of Connecticut2Computer Science and Engineering, University of Connecticut
3Computer 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 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:
)}1,1(),,1(),1,1(
),1,(),,(),1,(
),1,1(),,1(),1,1{(),(
jijiji
jijiji
jijijijiMR
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Interference region
Set of cells to be kept empty when a droplet moves away from (i,j)
Interference region:
),(),( jiMRjiIR
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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
tlkjix ),)(,(
tlkjix ),)(,(
j:
i:
l:
k:
Time t-1: Time t:
)},(),1,(),1,(),,1(),,1{(),( jijijijijijiN
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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:
1),(),(
),)(,( t jiNlk
tjilkx
0),(),(
1),)(,(
),(),(),)(,(
jiNlk
tlkji
jiNlk
tjilk xx
1),(),( ),()'',(
)'',)(,(),(),(
),)(,( jiNlk lkNlk
tlklk
jiNlk
tjilk xx
1 ),()'',( )'',()'',(
)'',)('',(),(),(
),)(,( jiNji jiNlk
tjilk
jiNlk
tlkji xx
Ojizxt
zt
jilk ),(every for ,0
Minimize
),)(,(
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Special Case
• NxN Chip
• I/O cells in Opposite Corners
• No Obstacles
Single droplet solution needs N2 cycles
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Stripe Algorithm with N/3 Droplets
65)2(3 NNNNCompletion 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
44 1213
T222
opt
kk
N
k
)k(kN k ) k(k-
442
kk
N
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Approximation guarantee
Lemma 2: Completion time for any #droplets is at least
Proof: Minimum for is achieved when
44 N
2/Nk
Theorem: Stripe algorithm with N/3 droplets has
approximation factor of 4
5
44
65
N
N
442
kk
N
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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-upk=1 k=12 k=20 k=30 k=40
0% 14400 1412 944 710 593 24x1% 14256 1420 953.4 715.2 598.8 24x5% 13680 1473 982.8 725 596.2 23x
10% 12960 1490 1010.8 734.8 592.6 22x15% 12240 1501 1025.8 730.8 588.2 21x20% 11520 1501 1046.8 738.4 580.8 20x25% 10800 1501 1071 736.6 570 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