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Proceedings ArticleDOI

Towards optimal strategies for moving droplets in digital microfluidic systems

27 Sep 2004-Vol. 2, pp 1468-1474
TL;DR: In this article, a high-level approach to optimally control digital microfluidic systems is presented, i.e., to generate a sequence of control signals for moving one or many droplets from start to goal positions in the shortest number of steps, subject to constraints such as minimum required separation between droplets, obstacles on the array surface, and limitations in the control circuitry.
Abstract: In digital microfluidic systems, analyte droplets (volume typically less than 1 /spl mu/l) are transported across a planar electrode array by dielectrophoretic or electrowetting effects. This paper outlines a high-level approach to optimally control digital microfluidic systems, i.e., to develop efficient algorithms that generate a sequence of control signals for moving one or many droplets from start to goal positions in the shortest number of steps, subject to constraints such as minimum required separation between droplets, obstacles on the array surface, and limitations in the control circuitry. However, optimality may be prohibitive for large-scale configurations because of the high asymptotic complexity. Alternative solutions include (1) an investigation of still useful but more limited system configurations; and (2) approximation algorithms that trade off optimality of the control sequences with higher efficiency of the algorithms that generate these control sequences.

Summary (3 min read)

INTRODUCTION

  • Advances in microfabrication and microelectromechanical systems (MEMS) over the past decades have lead to a rapidly expanding collection of techniques to build systems for the handling and analyzing of very small quantities of liquids (see, e.g., [1] ).
  • These microfluidic systems typically consist of submillimeter scale components such as channels, valves, pumps, and reservoirs, as well as application-specific sensors and actuators.
  • Microfluidic devices hold great promise, for example for novel fast, low-cost, portable, and disposable diagnostic tools.
  • Applications include the massively parallel testing of new drugs, the on-site, real-time detection of toxins and pathogens, and PCR (polymerase chain reaction) for DNA sequence analysis.
  • They usually operate with continuous flows of liquids, in analogy to traditional macro-scale laboratory setups, and integrate all functionality into complete "bio-systemson-a-chip ".

A. Digital Microfluidic Systems

  • More recently, there has been increased interest in microfluidic devices that handle discrete droplets, with volumes usually in the sub-microliter range.
  • In these "digital microfluidic systems" (DMFS), droplets are generated, transported, merged, analyzed, and disposed on planar arrays of addressable cells.
  • This architecture for microfluidic systems is attractive because of (1) greater flexibility -analyte handling may be reconfigured simply by re-programming rather than by changing the physical layout of the microfluidic components; (2) high droplet speeds -reportedly up to 25cm/s [2] ; (3) no dilution and cross-contamination due to diffusion and shearflow; and (4) the possibility for massively parallel microfluidic circuits.

D. Paper Overview

  • The goal of this paper is to generate optimal sequences of control signals to move droplets from start to goal positions in the shortest number of steps.
  • With growing array size and number of droplets, this becomes increasingly challenging: closely related optimizations are the traveling salesman problem, VLSI circuit routing, factory floor plan layout, resource scheduling, and motion planning with multiple moving robots, which are known to be computationally expensive (i.e., NP-hard [11] ).
  • Section III gives a more formal problem definition.

A. DMFS Design Specifications

  • These specifications provide a physical framework within which a DMFS can operate.
  • Once a sufficiently general DMFS model exists, the authors can investigate algorithmic solutions at an abstract level, without worrying about the specific details of varying hardware implementations.

B. Problem Definition

  • Various kinds of transitions exist, including droplet generation, moving, disposing, merging, and splitting.
  • In addition, to avoid accidental merging of droplets, at least one empty cell is required between two occupied cells at all times.
  • Transitions are further restricted by the addressing circuitry and cells with specialized functions.

IV. DMFS CONTROL STRATEGIES

  • The authors will first give a simple, complete algorithm based on A* search, but find that its computational complexity is very high (exponential in number of droplets).
  • The authors then present a more efficient algorithm that trades off completeness for faster execution times.

A. Basic Algorithm Outline

  • This algorithm maintains a graph data structure to represent the array (inclusive special cells and obstacles) and to keep track of droplet locations.
  • Transitions between states define edges in this graph, and finding an optimal control strategy to transform start state A s into goal state A g becomes a standard graph search problem, which can be solved, for example, using an A* algorithm known from artificial intelligence programming [19] :.
  • This estimate provides a heuristic that gives preference to the more promising paths.
  • The downside of this approach is its high asymptotic complexity.
  • One might hope that in practice, most of these choices need not be explored.

B. Prioritized Droplet Control

  • The discussion above has shown that droplet motion planning for DMFS has two main aspects: generating efficient droplet motion plans, and finding efficient algorithms to generate these plans.
  • This section applies ideas from Erdmann and Lozano-Pérez [12] to DMFS control.
  • (2) For each droplet, starting with the highest priority, generate an optimal motion plan.
  • Droplets with higher priorities are considered time-dependent obstacles.
  • Figures 4b-e depict the individual traces for each of the four droplets.

V. OTHER SAMPLE DROPLET MANIPULATION STRATEGIES

  • In this section the authors show two additional examples of optimal control strategies.
  • This strategy assumes that the electrode in each cell can be activated independently from all other cells.
  • The two droplets are always separated by at least one empty cell, such that accidental merging is avoided.
  • Note that the darker droplet moves more than necessary (gratuitous steps 4 and 5), but this does not affect the overall number of 8 steps in the control strategy.

A. Limited Row-Column Addressing

  • The previous examples assumed that each cell in the array is individually addressable.
  • Two droplets trade places as in Figure 5 above, but here droplets move only to cells whose row and column address has been activated (indicated by triangular arrows).
  • Instead of making full use of these advantages, the computational complexity may limit DMFS to much more constraint applications.
  • The authors have shown one possible answer to this challenge: Instead of insisting on optimal strategies, an algorithm that trades off completeness and optimality for polynomial run-time was presented.
  • Thus, even if the hardware allows simultaneous motion of droplets (e.g., with individually addressable cells), it may be more effective to first generate a motion plan consisting of single droplet moves, and then perform a post-processing step that "parallelizes" the plan as much as possible.

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Towards Optimal Strategies for Moving Droplets
in Digital Microfluidic Systems
Karl F. Böhringer
Department of Electrical Engineering
University of Washington, Seattle
Seattle, WA 98195, USA
karl@ee.washington.edu
Abstract - In digital microfluidic systems, analyte droplets
(volume typically less than 1µl) are transported across a planar
electrode array by dielectrophoretic or electrowetting effects.
This paper outlines a high-level approach to optimally control
digital microfluidic systems, i.e., to develop efficient algorithms
that generate a sequence of control signals for moving one or
many droplets from start to goal positions in the shortest number
of steps, subject to constraints such as minimum required
separation between droplets, obstacles on the array surface, and
limitations in the control circuitry. However, optimality may be
prohibitive for large-scale configurations because of the high
asymptotic complexity. Alternative solutions include (1) an
investigation of still useful but more limited system
configurations; and (2) approximation algorithms that trade off
optimality of the control sequences with higher efficiency of the
algorithms that generate these control sequences.
Keywords - digital microfluidics; droplet manipulation; control
strategy; lab on a chip.
I. INTRODUCTION
Advances in microfabrication and microelectromechanical
systems (MEMS) over the past decades have lead to a rapidly
expanding collection of techniques to build systems for the
handling and analyzing of very small quantities of liquids (see,
e.g., [1]). These microfluidic systems typically consist of sub-
millimeter scale components such as channels, valves, pumps,
and reservoirs, as well as application-specific sensors and
actuators. Microfluidic devices hold great promise, for example
for novel fast, low-cost, portable, and disposable diagnostic
tools. Applications include the massively parallel testing of
new drugs, the on-site, real-time detection of toxins and
pathogens, and PCR (polymerase chain reaction) for DNA
sequence analysis. They usually operate with continuous flows
of liquids, in analogy to traditional macro-scale laboratory set-
ups, and integrate all functionality into complete “bio-systems-
on-a-chip (bioSOCs)”.
A. Digital Microfluidic Systems
More recently, there has been increased interest in
microfluidic devices that handle discrete droplets, with
volumes usually in the sub-microliter range. In these “digital
microfluidic systems” (DMFS), droplets are generated,
transported, merged, analyzed, and disposed on planar arrays of
addressable cells. This architecture for microfluidic systems is
attractive because of (1) greater flexibility – analyte handling
may be reconfigured simply by re-programming rather than by
changing the physical layout of the microfluidic components;
(2) high droplet speeds – reportedly up to 25cm/s [2]; (3) no
dilution and cross-contamination due to diffusion and shear-
flow; and (4) the possibility for massively parallel microfluidic
circuits.
B. Droplet Transport
Small droplets can be moved across a planar surface
effectively with a variety of techniques, for example with
electric fields (e.g., [2-6]), the thermocapillary effect (e.g., [7]),
electrochemical surface modulation (e.g., [8]), or
conformational changes in molecular surface layers (e.g., [9]).
For the work in this paper, droplet transport with electric fields
is most suitable; hence we briefly discuss the two main
techniques in this realm.
1) Dielectrophoresis
In dielectrophoresis (DEP), neutrally charged objects are
first polarized by an electric field, and then experience a net
force due to the field. This force can only be non-zero if a field
gradient exists, i.e., the positively and negatively polarized
regions of the object occupy areas of different field strengths. If
the object has stronger polarization than the surrounding
medium then it is pulled towards the areas of higher field
strength (this is called positive DEP), but if the surrounding
medium has higher polarization, then the object is pushed
towards areas of lower field strength (negative DEP). DEP can
be considered the electrostatic analogy of induced magnetism.
Common examples for DEP are charged clothes that attract
(neutral) lint particles. More information on dielectrophoresis
can be found, e.g., at [10].
2) Electrowetting
Electrowetting on dielectric (EWOD) exploits the decrease
of contact angle that an aqueous droplet on a dielectric surface
experiences when exposed to an electric field. If the field is
localized at only one side of the droplet, then the difference in
contact angle causes a pressure differential in the droplet,
which drives it towards the region of higher field strength.
Electrowetting and its applications in microfluidics have been
investigated by several groups, including [2-4, 6].
Support was provided from the National Science Foundation by grant
CCR-0342632, Sankar Basu, program director.

C. Examples
Figure 1 presents examples of control strategies for a
simple digital microfluidic system. On a 10×10 array, a single
droplet must be moved from cell (2,2) to cell (9,9). Figure 1a
shows an optimal strategy consisting of 14 steps. In the system
in Figure 1b, “forbidden” cells marked as black squares must
be circumnavigated, resulting in a slightly longer solution
sequence. In this paper, we describe how these solutions can be
generated automatically, and generalize the approach to more
complex scenarios with multiple moving droplets and
additional constraints stemming from the specific physical
implementation of the DMFS.
D. Paper Overview
The goal of this paper is to generate optimal sequences of
control signals to move droplets from start to goal positions in
the shortest number of steps. With growing array size and
number of droplets, this becomes increasingly challenging:
closely related optimizations are the traveling salesman
problem, VLSI circuit routing, factory floor plan layout,
resource scheduling, and motion planning with multiple
moving robots, which are known to be computationally
expensive (i.e., NP-hard [11]). Section II summarizes related
work. Section III gives a more formal problem definition.
Algorithms to control DMFS are discussed in Section IV, and
examples applicable to different DMFS hardware
configurations are presented in Section V. Section VI
summarizes the paper, and gives conclusions and an outlook on
future work.
II. R
ELATED WORK
Finding the optimal plan to generate, store, move, merge,
split, and dispose multiple droplets on a digital microfluidic
array is a complex problem, which combines general path
planning and scheduling tasks with the more application-
specific tasks of droplet generation, merging, and splitting.
Various researchers have studied parts of the overall problem
and have shown important results on algorithmic solutions and
their computational complexity.
Each droplet can be interpreted as a point robot moving in a
discrete two-dimensional configuration space. Under this
assumption, path planning of the droplets becomes a robot
motion planning problem with multiple moving robots.
Erdmann and Lozano-Pérez showed in 1987 that this problem
is NP-hard, but presented an algorithm that may find a good
solution in polynomial time [12]. This approach assigns
priorities to each robot (droplet) and generates paths
successively, starting with the highest priority robot. Lower
priority robots consider higher priority robots as time-varying
obstacles that must be avoided. The algorithm is not complete,
and generated solutions depend on the priority ranking of the
robots and may not be optimal.
A rather different approach to this problem can be taken
when the paths of the droplets are considered given a priori.
Under this assumption, we obtain a scheduling problem, where
the array cells en route are the limited resource that must be
shared among different droplets. Recently, Akella et al.
attacked this problem, again from the point of view of multiple
coordinated robots. The problem is formulated as an integer
programming problem, which can be solved with standard
optimization tools [13, 14].
A similar technique was used by Ding, Zhang, et al. [15-17]
who attack the problem from the VLSI design perspective.
Again, the problem leads to an integer programming
formulation, which is essentially equivalent to Akella’s
approach. Both groups show NP-hardness of the scheduling
problem even for fixed robot (droplet) routes.
VLSI circuit routing techniques could also be employed,
which address the path planning problem but do not apply
directly to the inherently two-dimensional layout of the digital
microfluidic platform.
In [18], this author described the problem as a graph search,
and suggested search techniques such as A*. Even though this
brute-force approach, unlike the other work mentioned above,
guarantees optimality and completeness, it is not practical for
larger scale problems because of its computational complexity,
which is exponential in the number of moving droplets.
While it is not within the scope of this paper to develop a
comprehensive algorithmic solution for the general problem of
droplet manipulation on massively parallel microfluidic
systems, we will attempt to present a formal problem definition
and algorithms for partial solutions, and point in the direction
of more general solutions for future work.
(a)
(b)
Figure 1. (a) Droplet moving on a 10×10 array from cell (2,2) to cell (9,9).
The trace of the droplet is shown, with darker color indicating earlier steps.
(b) Droplet moving from cell (2,2) to (9,9) while avoiding obstacles
(“forbidden” cells shown in black). Here, an optimal strategy requires 16
steps, two more than in (a).
star
t
goal
star
t
goal

III. DMFS HARDWARE SPECIFICATION AND
FORMAL PROBLEM DEFINITION
Let us first specify the important physical properties and
design parameters of a digital microfluidic system. Then we
can move on to a more abstract DMFS model that is
independent of specific implementation details.
A. DMFS Design Specifications
Layout:
Typically, a DMFS consists of a rectangular array
A with m×n cells (but, e.g., an arrangement of hexagonal
cells is also possible).
Control circuitry:
Various addressing schemes are possible
to activate individual cells in a DMFS. We can distinguish,
e.g., individually addressable electrodes for each cell, or
simpler row/column addressing. For the latter, entire rows
and columns are activated, and the droplet is attracted to a
neighboring cell A(x,y) only if it lies at the intersection of
active column x and row y.
Parallelism:
Does the DMFS controller allow simultaneous
activation of more than one cell, and is the total number of
active cells limited by a number significantly smaller than
m×n?
Location of cells with special functions:
Droplet
generators, reservoirs, cells for merging and splitting of
droplets, sensors, waste, etc. may require dedicated cells
with special embedded hardware.
These specifications provide a physical framework within
which a DMFS can operate. Based on this framework, we can
establish a formal description of the problem of controlling
droplets in a DMFS. Once a sufficiently general DMFS model
exists, we can investigate algorithmic solutions at an abstract
level, without worrying about the specific details of varying
hardware implementations.
B. Problem Definition
A digital microfluidic system is given by an array A with d
droplets, their start locations, and their goal locations. Our aim
is to automatically generate a strategy to move the droplets
from start to goal (as shown, e.g., in Figure 1). More
specifically, droplets can be of several types T
i
(e.g., T
1
= “DI
water”, T
2
= “buffer solution”, etc.), with i {1…t}, and t the
total number of different droplet types.
Each cell A(x,y) in the array can be either occupied by a
droplet (denoted as “T
i
”), empty (“”), or blocked by an
obstacle (“X”). Thus, at any given time the system can be
described by A(x,y) = c
xy
for (x,y) {1…m}×{1…n} and c
xy
C = {T
1
,…,T
t
,,X}. In particular, given a start placement A
s
C
m×n
and a goal placement A
g
C
m×n
, we need to find a
sequence of valid transitions that results in the desired droplet
motion from A
s
to A
g
.
Various kinds of transitions exist, including droplet
generation, moving, disposing, merging, and splitting.
Droplet generation:
For (x,y) {1…m}×{1…n} and some
i {1…t}, a droplet is generated at coordinate (x,y) if
A(x,y) = at time t and A(x,y) = T
i
at time t+1.
Moving:
Let (x,y) and (x',y') {1…m}×{1…n} and |xx'|
+ |yy'| = 1 (i.e., A(x,y) and A(x',y') are directly adjacent).
At time t, A(x,y) = T
i
and A(x',y') = and at time t+1,
A(x,y) = and A(x',y') = T
i
.
Merging:
Let again (x,y) and (x',y') {1…m}×{1…n} and
|xx'| + |yy'| = 1. At time t, A(x,y) = T
i
and A(x',y') = T
j
,
and at time t+1, A(x,y) = T
k
and A(x',y') = , where T
k
is
the droplet type that results in merging types T
i
and T
j
.
Splitting:
Definition similar to merging.
Disposing:
Definition similar to droplet generation.
In addition, to avoid accidental merging of droplets, at least one
empty cell is required between two occupied cells at all times.
Transitions are further restricted by the addressing circuitry and
cells with specialized functions.
IV. DMFS C
ONTROL STRATEGIES
This section focuses on a limited but important subproblem
in the control of DMFS: generating efficient paths for multiple
droplets that move from a given start configuration A
s
to a
desired goal configuration A
g
. We will first give a simple,
complete algorithm based on A* search, but find that its
computational complexity is very high (exponential in number
of droplets). We then present a more efficient algorithm that
trades off completeness for faster execution times.
A. Basic Algorithm Outline
This algorithm maintains a graph data structure to represent
the array (inclusive special cells and obstacles) and to keep
track of droplet locations. At any given time t
i
, the state of the
DMFS is described by A
i
C
m×n
, representing a node in the
graph. Transitions between states define edges in this graph,
and finding an optimal control strategy to transform start state
A
s
into goal state A
g
becomes a standard graph search problem,
which can be solved, for example, using an A* algorithm
known from artificial intelligence programming [19]: A* graph
search employs a metric that estimates the expected cost of a
Figure 2. Three droplets with respective start and goal positions (indicated by
S and G). The number of choices grows exponentially with the number of
droplets. At any time there are up to 4
3
choices for the next step, and at least 12
steps are required to move all droplets simultaneously from start to goal.
Hence, straightforward programming could produce software attempting to
explore (4
3
)
12
> 10
28
choices.
y
x
S
G
G
S
S
G

partial solution path in the directed graph. This estimate
provides a heuristic that gives preference to the more promising
paths. It can be shown that if certain “admissible” metrics are
used, then A* is guaranteed to find an optimal solution if one
exists, and indicates failure otherwise.
The downside of this approach is its high asymptotic
complexity. Suppose the number of droplets is d. In the
simplest case, all are of the same type T
0
. Then the number of
different placements of droplets on the array is
)(
mn
d
, which for
modest numbers m=n=10 and d=10 yields more than 1.7×10
13
possibilities. If all droplets are of distinct type T
1
T
d
, this
number increases by d! (to
3×10
19
). One might hope that in
practice, most of these choices need not be explored. However,
at each step, d droplets offer up to 4d choices to be moved,
assuming 4 neighbor cells per droplet. Thus, finding a strategy
with s steps could mean checking up to (4
d
)
s
choices or risk
missing the solution, resulting again in astronomical numbers
even for s<10. This is illustrated in Figure 2.
We conclude that the search graph explored with the A*
algorithm has O((mn)!) nodes and a branching factor of O(4s),
leading to prohibitive complexity for any non-trivial array size
with more than a few droplets.
B. Prioritized Droplet Control
The discussion above has shown that droplet motion
planning for DMFS has two main aspects: generating efficient
droplet motion plans, and finding efficient algorithms to
generate these plans. Because of the inherent complexity of the
problem, compromises need to be made to obtain practical
solutions, and completeness or optimality in motion plans has
to be traded off with efficiency in plan generation.
This section applies ideas from Erdmann and Lozano-Pérez
[12] to DMFS control. The algorithm proceeds as follows:
(1) Assign priorities to each droplet in the DMFS. This can be
done at random, or based on application-specific
guidelines (e.g., water may have lower priority than
droplets containing expensive or volatile compounds).
Figure 3. Optimal solution to the setup in Figure 2 by the prioritizing
algorithm. The blue droplet was assigned highest priority and an optimal
motion (12 steps) was generated. The yellow droplet requires 9 steps and
moves over cell (2,2) previously occupied by the blue droplet. The red droplet
does not interfere with the other droplets in this case.
(a)
(b)
(c)
(d)
(e)
Figure 4. (a) Four droplets moving simultaneously from start S
1
=(1,1),
S
2
=(16,1), S
3
=(8,16), S
4
=(16,8) to goal G
1
=(16,16), G
2
=(1,16), G
3
=(8,1),
G
4
=(1,8). (b-e) Individual paths (with time stamps) for droplets 1 through 4 in
decreasing order of priority. Solutions generated with sequential prioritized A*
algorithm.
S
1
y
x
S
G
G
S
S
G
S
2
S
3
S
4
G
1
G
2
G
3
G
4
Dro
p
let 1
Dro
p
let 2
Dro
p
let 3
Dro
p
let 4
Dro
p
lets 1-4

(2) For each droplet, starting with the highest priority,
generate an optimal motion plan. Droplets with higher
priorities are considered time-dependent obstacles.
Droplets with lower priorities are ignored.
This algorithm eliminates the exponential complexity in d,
where d is the number of droplets in the DMFS. Instead, as the
complexity of the A* algorithm for path planning of a single
droplet is O(nmlog(nm)), the complexity to determine d droplet
paths with this sequential prioritized approach is simply
O(dnmlog(nm)). However, as stated above, this algorithm is
neither complete, nor are the generated paths necessarily
optimal. Figure 3 gives an example of this algorithm for the
start and goal configurations of Figure 2.
Figure 4 shows a more extensive example of this algorithm.
On a 16×16 array with randomly distributed obstacles, four
droplets are initially placed at (1,1), (16,1), (8,16), and (16,8).
Their respective goals are at (16,16), (1,16), (8,1), and (1,8).
Figure 4a shows the simultaneous trace of all droplets. Figures
4b-e depict the individual traces for each of the four droplets.
We can observe that the two droplets with the highest priorities
(Figures 4b and 4c) achieve an optimal path with 31 steps each.
Droplet 3 (Figure 4d) has to evade droplets 1 and 2 and
therefore turns left in steps 10 and 13, instead of choosing the
shorter path towards the right. Similarly, droplet 4 (Figure 4e)
would interfere with higher priority droplets, were it to travel
on a more direct path towards its goal.
The solution in Figure 4 was generated in a few seconds by
a simple MATLAB implementation
of this algorithm.
In the following section we
show more examples performed
with variations of this algorithm.
They include multiple droplets,
obstacles, and constraints on the
control circuitry. Even though
rather simple, these examples
should summarize the basic
principles of DMFS control
strategies, and motivate ideas for
improved algorithms, which will be
summarized in Section VI.
V. O
THER SAMPLE DROPLET
MANIPULATION STRATEGIES
In this section we show two
additional examples of optimal
control strategies. In Figure 5 two
droplets of different types require 8
steps to switch their positions while
circumnavigating an obstacle.
This strategy assumes that the
electrode in each cell can be
activated independently from all
other cells. The two droplets are
always separated by at least one
empty cell, such that accidental
merging is avoided. Note that the
darker droplet moves more than necessary (gratuitous steps 4
and 5), but this does not affect the overall number of 8 steps in
the control strategy. Future software improvements will
eliminate this programming artifact.
A. Limited Row-Column Addressing
The previous examples (Figure 1-5) assumed that each cell
in the array is individually addressable. However, [20]
introduced a simpler addressing scheme for DMFS based on a
top layer of row electrodes and a bottom layer of column
electrodes. Droplets move to a neighboring cell whose row and
column address has been activated. This scheme creates
additional constraints on the droplet motion. Two droplets trade
places as in Figure 5 above, but here droplets move only to
cells whose row and column address
has been activated
(indicated by triangular arrows). An optimal strategy now
requires 9 steps, one more step than in Figure 5. Figure 6 shows
the same task as Figure 5 but performed only with row-column
addressing, resulting in a longer sequence.
Note that here we assumed that we can activate an arbitrary
number of rows and columns simultaneously (for d droplets, up
to d active rows and columns are useful). Further hardware
constraints could limit this number, possibly to a single row
and column. If so, longer control sequences could result, but
the branching factor at each step would drop from O(4
d
) to
O(d).
Figure 5. Two droplets moving simultaneously on a 6×6 array while avoiding an obstacle (black cells). The two
droplets start at cells (5,2) and (4,5), and require 8 steps to trade places. Solution generated with complete multi-
droplet A* algorithm.
1 2
4 5
7 8
3
6

Citations
More filters
Proceedings ArticleDOI
06 Mar 2006
TL;DR: This work develops the first systematic droplet routing method that can be integrated with biochip synthesis, which minimizes the number of cells used fordroplet routing, while satisfying constraints imposed by throughput considerations and fluidic properties.
Abstract: Recent advances in microfluidics are expected to lead to sensor systems for high-throughput biochemical analysis. CAD tools are needed to handle increased design complexity for such systems. Analogous to classical VLSI synthesis, a top-down design automation approach can shorten the design cycle and reduce human effort. We focus here on the droplet routing problem, which is a key issue in biochip physical design automation. We develop the first systematic droplet routing method that can be integrated with biochip synthesis. The proposed approach minimizes the number of cells used for droplet routing, while satisfying constraints imposed by throughput considerations and fluidic properties. A real-life biochemical application is used to evaluate the proposed method.

228 citations

Journal ArticleDOI
TL;DR: This work proposes a system design methodology that attempts to apply classical high-level synthesis techniques to the design of digital microfluidic biochips and develops an optimal scheduling strategy based on integer linear programming and two heuristic techniques that scale well for large problem instances.
Abstract: Microfluidic biochips offer a promising platform for massively parallel DNA analysis, automated drug discovery, and real-time biomolecular recognition. Current techniques for full-custom design of droplet-based “digital” biochips do not scale well for concurrent assays and for next-generation system-on-chip (SOC) designs that are expected to include microfluidic components. We propose a system design methodology that attempts to apply classical high-level synthesis techniques to the design of digital microfluidic biochips. We focus here on the problem of scheduling bioassay functions under resource constraints. We first develop an optimal scheduling strategy based on integer linear programming. However, because the scheduling problem is NP-complete, we also develop two heuristic techniques that scale well for large problem instances. A clinical diagnostic procedure, namely multiplexed in-vitro diagnostics on human physiological fluids, is first used to illustrate and evaluate the proposed method. Next, the synthesis approach is applied to a protein assay, which serves as a more complex bioassay application. The proposed synthesis approach is expected to reduce human effort and design cycle time, and it will facilitate the integration of microfluidic components with microelectronic components in next-generation SOCs.

172 citations

Journal ArticleDOI
TL;DR: A polynomial-time algorithm for coordinating droplet movement under such hardware limitations is developed and described, and a layout-based system that can be rapidly reconfigured for new biochemical analyses is introduced.
Abstract: This paper describes a computational approach to designing a digital microfluidic system (DMFS) that can be rapidly reconfigured for new biochemical analyses. Such a “lab-on-a-chip” system for biochemical analysis, based on electrowetting or dielectrophoresis, must coordinate the motions of discrete droplets or biological cells using a planar array of electrodes. The authors have earlier introduced a layout-based system and demonstrated its flexibility through simulation, including the system's ability to perform multiple assays simultaneously. Since array-layout design and droplet-routing strategies are closely related in such a DMFS, their goal is to provide designers with algorithms that enable rapid simulation and control of these DMFS devices. In this paper, the effects of variations in the basic array-layout design, droplet-routing control algorithms, and droplet spacing on system performance are characterized. DMFS arrays with hardware limited row-column addressing are considered, and a polynomial-time algorithm for coordinating droplet movement under such hardware limitations is developed. To demonstrate the capabilities of our system, we describe example scenarios, including dilution control and minimalist layouts, in which our system can be successfully applied.

164 citations

Journal ArticleDOI
TL;DR: This paper presents general hardware-independent models and algorithms to automate the operation of droplet-based microfluidic systems and an approach toward automatic mapping of a biochemical analysis task onto a DMFS is investigated.
Abstract: This paper presents general hardware-independent models and algorithms to automate the operation of droplet-based microfluidic systems. In these systems, discrete liquid volumes of typically less than 1 $muhboxl$ are transported across a planar array by dielectrophoretic or electrowetting effects for biochemical analysis. Unlike in systems based on continuous flow through channels, valves, and pumps, the droplet paths can be reconfigured on demand and even in real time. Algorithms that generate efficient sequences of control signals for moving one or many droplets from start to goal positions, subject to constraints such as specific features and obstacles on the array surface or limitations in the control circuitry, are developed. In addition, an approach toward automatic mapping of a biochemical analysis task onto a DMFS is investigated. Achieving optimality in these algorithms can be prohibitive for large-scale configurations because of the high asymptotic complexity of coordinating multiple moving droplets. Instead, these algorithms achieve a compromise between high runtime efficiency and a more limited nonglobal optimality in the generated control sequences.

149 citations

Journal ArticleDOI
TL;DR: A general-purpose system that uses simple algorithms and yet is versatile is described that has been able to successfully coordinate hundreds of droplets simultaneously and perform one or more chemical analyses in parallel.
Abstract: In this paper we present an approach to coordinate the motions of droplets in digital microfluidic systems, a new class of lab-on-a-chip systems for biochemical analysis. A digital microfluidic system typically consists of a planar array of cells with electrodes that control the droplets. The primary challenge in using droplet-based systems is that they require the simultaneous coordination of a potentially large number of droplets on the array as the droplets move, mix, and split. In this paper we describe a general-purpose system that uses simple algorithms and yet is versatile. First, we present a semi-automated approach to generate the array layout in terms of components. Next, we discuss simple algorithms to select destination components for the droplets and a decentralized scheme for components to route the droplets on the array. These are then combined into a reconfigurable system that has been simulated in software to perform analyses such as the DNA polymerase chain reaction. The algorithms have ...

77 citations

References
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Book
01 Jan 1979
TL;DR: The second edition of a quarterly column as discussed by the authors provides a continuing update to the list of problems (NP-complete and harder) presented by M. R. Garey and myself in our book "Computers and Intractability: A Guide to the Theory of NP-Completeness,” W. H. Freeman & Co., San Francisco, 1979.
Abstract: This is the second edition of a quarterly column the purpose of which is to provide a continuing update to the list of problems (NP-complete and harder) presented by M. R. Garey and myself in our book ‘‘Computers and Intractability: A Guide to the Theory of NP-Completeness,’’ W. H. Freeman & Co., San Francisco, 1979 (hereinafter referred to as ‘‘[G&J]’’; previous columns will be referred to by their dates). A background equivalent to that provided by [G&J] is assumed. Readers having results they would like mentioned (NP-hardness, PSPACE-hardness, polynomial-time-solvability, etc.), or open problems they would like publicized, should send them to David S. Johnson, Room 2C355, Bell Laboratories, Murray Hill, NJ 07974, including details, or at least sketches, of any new proofs (full papers are preferred). In the case of unpublished results, please state explicitly that you would like the results mentioned in the column. Comments and corrections are also welcome. For more details on the nature of the column and the form of desired submissions, see the December 1981 issue of this journal.

40,020 citations

Book
01 Jan 1980
TL;DR: This classic introduction to artificial intelligence describes fundamental AI ideas that underlie applications such as natural language processing, automatic programming, robotics, machine vision, automatic theorem proving, and intelligent data retrieval.
Abstract: A classic introduction to artificial intelligence intended to bridge the gap between theory and practice, "Principles of Artificial Intelligence" describes fundamental AI ideas that underlie applications such as natural language processing, automatic programming, robotics, machine vision, automatic theorem proving, and intelligent data retrieval. Rather than focusing on the subject matter of the applications, the book is organized around general computational concepts involving the kinds of data structures used, the types of operations performed on the data structures, and the properties of the control strategies used. "Principles of Artificial Intelligence"evolved from the author's courses and seminars at Stanford University and University of Massachusetts, Amherst, and is suitable for text use in a senior or graduate AI course, or for individual study.

3,754 citations

Journal ArticleDOI
TL;DR: In this article, a microactuator for rapid manipulation of discrete microdroplets is presented, which is accomplished by direct electrical control of the surface tension through two sets of opposing planar electrodes fabricated on glass.
Abstract: A microactuator for rapid manipulation of discrete microdroplets is presented. Microactuation is accomplished by direct electrical control of the surface tension through two sets of opposing planar electrodes fabricated on glass. A prototype device consisting of a linear array of seven electrodes at 1.5 mm pitch was fabricated and tested. Droplets (0.7–1.0 μl) of 100 mM KCl solution were successfully transferred between adjacent electrodes at voltages of 40–80 V. Repeatable transport of droplets at electrode switching rates of up to 20 Hz and average velocities of 30 mm/s have been demonstrated. This speed represents a nearly 100-fold increase over previously demonstrated electrical methods for the transport of droplets on solid surfaces.

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Book
01 Feb 1998
TL;DR: In this paper, the authors present an overview of Micromachining Techniques, Mechanical Transducers, Optical Transducers and Ionizing Radiation Transducers for Microfluidic Devices.
Abstract: 1 Introduction and Overview2 Micromachining Techniques3 Mechanical Transducers4 Optical Transducers5 Ionizing Radiation Transducers6 Thermal Transducers7 Magnetic & Electromagnetic Transducers8 Chemical & Biological Transducers9 Microfluidic Devices

1,212 citations

Journal ArticleDOI
17 Jan 2003-Science
TL;DR: The design of surfaces that exhibit dynamic changes in interfacial properties, such as wettability, in response to an electrical potential are reported, which enables amplification of molecular-level conformational transitions to macroscopic changes in surface properties without altering the chemical identity of the surface.
Abstract: We report the design of surfaces that exhibit dynamic changes in interfacial properties, such as wettability, in response to an electrical potential. The change in wetting behavior was caused by surface-confined, single-layered molecules undergoing conformational transitions between a hydrophilic and a moderately hydrophobic state. Reversible conformational transitions were confirmed at a molecular level with the use of sum-frequency generation spectroscopy and at a macroscopic level with the use of contact angle measurements. This type of surface design enables amplification of molecular-level conformational transitions to macroscopic changes in surface properties without altering the chemical identity of the surface. Such reversibly switching surfaces may open previously unknown opportunities in interfacial engineering.

1,055 citations

Frequently Asked Questions (15)
Q1. What are the contributions in "Towards optimal strategies for moving droplets in digital microfluidic systems" ?

This paper outlines a high-level approach to optimally control digital microfluidic systems, i. e., to develop efficient algorithms that generate a sequence of control signals for moving one or many droplets from start to goal positions in the shortest number of steps, subject to constraints such as minimum required separation between droplets, obstacles on the array surface, and limitations in the control circuitry. 

Small droplets can be moved across a planar surface effectively with a variety of techniques, for example with electric fields (e.g., [2-6]), the thermocapillary effect (e.g., [7]), electrochemical surface modulation (e.g., [8]), or conformational changes in molecular surface layers (e.g., [9]). 

As into goal state Ag becomes a standard graph search problem, which can be solved, for example, using an A* algorithm known from artificial intelligence programming [19]: 

VI. CONCLUSIONS AND FUTURE WORK Digital microfluidic systems (DMFS) based on droplet manipulation are promising because of their flexibility and reconfigurability: they shift complexity from microfluidics hardware to control software. 

More information on dielectrophoresis can be found, e.g., at [10].2) Electrowetting Electrowetting on dielectric (EWOD) exploits the decrease of contact angle that an aqueous droplet on a dielectric surface experiences when exposed to an electric field. 

The authors conclude that the search graph explored with the A* algorithm has O((mn)!) nodes and a branching factor of O(4s), leading to prohibitive complexity for any non-trivial array size with more than a few droplets. 

Layout: Typically, a DMFS consists of a rectangular arrayA with m×n cells (but, e.g., an arrangement of hexagonal cells is also possible).• 

This architecture for microfluidic systems is attractive because of (1) greater flexibility – analyte handling may be reconfigured simply by re-programming rather than by changing the physical layout of the microfluidic components; (2) high droplet speeds – reportedly up to 25cm/s [2]; (3) no dilution and cross-contamination due to diffusion and shearflow; and (4) the possibility for massively parallel microfluidic circuits. 

Droplet generation: For (x,y) ∈ {1…m}×{1…n} and some i ∈ {1…t}, a droplet is generated at coordinate (x,y) ifA(x,y) = ∅ at time t and A(x,y) = Ti at time t+1.• 

In this paper, the authors have shown one possible answer to this challenge: Instead of insisting on optimal strategies, an algorithm that trades off completeness and optimality for polynomial run-time was presented. 

Then the number of different placements of droplets on the array is )(mnd , which for modest numbers m=n=10 and d=10 yields more than 1.7×1013 possibilities. 

e.g., a control strategy for a complex DMFS can be generated in polynomial time that is guaranteed to be at most twice as long as an optimal solution then this might be sufficient for most practical purposes. 

More recently, there has been increased interest in microfluidic devices that handle discrete droplets, with volumes usually in the sub-microliter range. 

The discussion above has shown that droplet motion planning for DMFS has two main aspects: generating efficient droplet motion plans, and finding efficient algorithms to generate these plans. 

Droplet 3 (Figure 4d) has to evade droplets 1 and 2 and therefore turns left in steps 10 and 13, instead of choosing the shorter path towards the right.