Title: Optimal Testing of Digital Microfluidic Biochips: A Multiple Traveling Salesman Problem
1Optimal 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
2Outline
- Introduction
- Problem definition
- ILP Formulation
- Bounds and Heuristic
- Experimental results
- Conclusions
3Introduction
- 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.
4Digital 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
5Optimization 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
6Concurrent 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
7Concurrent 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
8Merging region
- Set of cells to be kept empty when (i,j) is
occupied by a droplet - Merging region
9Interference region
- Set of cells to be kept empty when a droplet
moves away from (i,j) - Interference region
10ILP 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
11ILP 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
12Special Case
- NxN Chip
- I/O cells in Opposite Corners
- No Obstacles
- ? Single droplet solution needs N2 cycles
13Stripe Algorithm with N/3 Droplets
14Lower 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
15Approximation 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
16Stripe 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
17Results 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
18Conclusions
- Presented ILP formulation, approximation
algorithm and heuristic for microfluidic biochip
testing problem - Combinatorial optimization techniques can yield
significant improvements