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Optimal Testing of Digital Microfluidic Biochips: A Multiple Traveling Salesman Problem

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Title: Optimal Testing of Digital Microfluidic Biochips: A Multiple Traveling Salesman Problem


1
Optimal Testing of Digital Microfluidic Biochips
A Multiple Traveling Salesman Problem
R. Garfinkel1, I.I. Mandoiu2, B. Pasaniuc2 and A.
Zelikovsky3
1Operations and Information Management,
University of Connecticut 2Computer Science and
Engineering, University of Connecticut 3Computer
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
  • 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

i
k










j
l
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 N2 cycles

13
Stripe Algorithm with N/3 Droplets









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









? 1 dollar in each cell
? k dollars in each diagonal
? 3k(k-1)/2 dollars paid waiting for last droplet
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 Q1
  • 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
Obstacle Area Average completion time (cycles) Average completion time (cycles) Average completion time (cycles) Average completion time (cycles) Average completion time (cycles) k40 vs. k1 speed-up
k1 k12 k20 k30 k40 k40 vs. k1 speed-up
0 14400 1412 944 710 593 24x
1 14256 1420 953.4 715.2 598.8 24x
5 13680 1473 982.8 725 596.2 23x
10 12960 1490 1010.8 734.8 592.6 22x
15 12240 1501 1025.8 730.8 588.2 21x
20 11520 1501 1046.8 738.4 580.8 20x
25 10800 1501 1071 736.6 570 19x
20x decrease in completion time by using
multiple droplets
18
Conclusions
  • Presented ILP formulation, approximation
    algorithm and heuristic for microfluidic biochip
    testing problem
  • Combinatorial optimization techniques can yield
    significant improvements
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