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

Outline

Introduction

Problem definition

ILP Formulation

Bounds and Heuristic

Experimental results

Conclusions

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.

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…

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

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

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

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

Interference region

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

Interference region:

),(),( jiMRjiIR

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

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

),)(,(

Special Case

• NxN Chip

• I/O cells in Opposite Corners

• No Obstacles

Single droplet solution needs N2 cycles

Stripe Algorithm with N/3 Droplets

65)2(3 NNNNCompletion time:

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

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

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

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

Conclusions

Presented ILP formulation, approximation algorithm and heuristic for microfluidic biochip testing problem

Combinatorial optimization techniques can yield significant improvements