JOURNAL OF ELECTRONIC TESTING: Theory and Applications 22, 199–210, 2006
c
2006 Springer Science + Business Media, LLC. Manufactured in The United States.
DOI: 10.1007/s10836-005-1256-3
Test Planning and Test Resource Optimization for Droplet-Based Microfluidic
Systems
∗
FEI SU, SULE OZEV AND KRISHNENDU CHAKRABARTY
Department of Electrical & Computer Engineering, Duke University, Durham, NC, 27708
fs@ee.duke.edu
sule@ee.duke.edu
krish@ee.duke.edu
Received September 30, 2004; Revised March 29, 2005
Editor: C. Landrault
Abstract. Recent years have seen the emergence of droplet-based microfluidic systems for safety-critical biomedical
applications. In order to ensure reliability, microsystems incorporating microfluidic components must be tested adequately. In
this paper, we investigate test planning and test resource optimization for droplet-based microfluidic arrays. We first formulate
the test planning problem and prove that it is NP-hard. We then describe an optimization method based on integer linear
programming (ILP) that yields optimal solutions. Due to the NP-hard nature of the problem, we develop heuristic approaches
for optimization. Experimental results indicate that for large array sizes, the heuristic methods yield solutions that are close
to provable lower bounds. These heuristics ensure scalability and low computation cost.
Keywords: droplet-based microfluidic systems, concurrent testing, microfluidic arrays, test planning, test resource opti-
mization
1. Introduction
Next-generation system-on-chip designs are expected to be
composite microsystems with microelectromechanical and
microfluidic components [15, 23]. These mixed-signal and
mixed-technology systems monolithically integrate micro-
electronics with microsensors and microactuators, thereby
leading to chips that can not only compute and commu-
nicate, but also sense and actuate. This high level of inte-
gration is enabling a new class of microsystems targeted
at health care, environmental monitoring, biomedical anal-
ysis, harmful agent detection for countering bio-terrorism,
and precision fluid dispensing [13].
In recent years, novel droplet-based microfluidic systems
have been developed to analyze nanoliter volumes of agents
[18]. These systems reduce the rate of reagent consump-
tion, thereby enabling continuous sampling and analysis for
on-line, real-time biological/chemical analysis. By scaling
down the concentration of the samples, simple sensing tech-
∗
This research was supported in part by the National Science Foundation
under grant number IIS-0312352. A preliminary version of this paper
appeared in Proc. European Test Symposium. pp. 72–77, 2004
niques can be utilized to replace conventional, costly, and
time-consuming practices involving batch analysis, sample
pre-treatment and frequent calibration. Droplet-based mi-
crofluidic systems therefore offer a promising platform for
massively parallel DNA analysis, and real-time molecular
detection and recognition.
As microfluidic systems become widespread in safety-
critical biomedical applications, system reliability emerges
as an essential performance parameter. In order to ensure
reliability, composite microsystems incorporating microflu-
idic components must be tested adequately. Therefore, there
is a pressing need for efficient test methodologies for these
microsystems. The ITRS 2003 document recognizes the
need for new test methods for disruptive device technologies
that underly microelectromechanical systems and sensors,
and highlights it as one of the five difficult test challenges
beyond 2009 [27].
Recently, a fault classification and a unified test method-
ology for droplet-based microfluidic systems has been de-
veloped [22]. Faults are classified as either catastrophic or
parametric, and they are detected by electrostatically con-
trolling and tracking droplet motion. This cost-effective test
methodology facilitates concurrent testing, which allows
200 Feietal.
fault testing and biomedical assays to run simultaneously on
a microfluidic system. Test planning and test resource opti-
mization are motivated by the need for concurrent testing.
In this paper, we investigate test planning and test
resource optimization problems for droplet-based microflu-
idic arrays. We first formulate the test planning problem
and prove that it is NP-hard. We then show how optimal
solutions can be obtained using integer linear programming
(ILP). Due to the NP-hard nature of the problem, the ILP
model is not applicable to large microfluidic arrays. We
therefore develop heuristics to solve this problem in a
computationally efficient manner. Experiments show that
for large array sizes, the results obtained from the heuristic
method are close to provable lower bounds.
The organization of the remainder of the paper is as fol-
lows. In Section 2, we present an overview of droplet-based
microfluidic systems. Related prior work is discussed in
Section 3. Section 4 describes the problem of test planning
and test resource optimization. This problem is shown to be
NP-hard in Section 5. An optimal solution based on inte-
ger linear programming is proposed in Section 6. Section
7 presents several heuristic algorithms, which are evalu-
ated through simulation experiments in Section 8. Finally,
conclusions are drawn in Section 9.
2. Background: Droplet-Based Microfluidic Systems
The operation of droplet-based microfluidic systems is
based on the principle of electrowetting actuation. By vary-
ing the electrical potential along a linear array of electrodes,
electrowetting can be used to move liquid droplets of nano-
liter volume along this line of electrodes [18]. Droplets can
also be transported, in user-defined patterns under clocked-
voltage control, over a two-dimensional array of electrodes
without the need for pumps and valves.
The basic component of a droplet-based microfluidic sys-
tem is shown in Fig. 1. The droplet, usually containing
biomedical samples, and the filler medium, such as sili-
cone oil, are sandwiched between two parallel glass plates.
The bottom plate contains a patterned array of individually
controllable electrodes, while the top plate is coated with
a ground electrode. The hydrophobic dielectric insulator is
added to the top and bottom plates to decrease the wettabil-
ity of the surface and to add capacitance between the droplet
and the control electrode.
Fig. 1. Basic component of a droplet-based microfluidic
system.
Fig. 2. Droplet transport in a two-dimensional array.
The basic principle underlying droplet transportation is
the electrostatic control of the interfacial tension at the
droplet/insulator interface. A control (actuating) voltage
is applied to an electrode adjacent to the droplet and, at
the same time, the electrode just under the droplet is de-
activated. This causes an accumulation of charge in the
droplet/insulator interface, resulting in an interfacial ten-
sion gradient across the gap between the adjacent electrodes,
which consequently causes the transportation of the droplet.
The velocity of the droplet can be controlled by adjusting
the actuation voltage (0–90V), and droplets can be moved
at speeds of up to 20 cm/s. Based on this principle, mi-
crofluidic droplets can be moved freely to any location of a
two-dimensional array; see Fig. 2. This design, which has
been fabricated on PCBs at Duke University [18], is ideally
suited for a large-scale integrated microfluidic system. Such
a system is expected to be common in the near future for
various biomedical applications, such as DNA sequencing
and bimolecular detection. A droplet can be easily detected
using the capacitive sensing circuit shown in Fig. 3.
Using a two-dimensional microfluidic array, many com-
mon operations for different biomedical assays can be
performed, such as sample introduction (dispense), sam-
ple movement (transport), temporarily sample preservation
(store), and mixing of different samples (mix). Note that
these operations can be performed anywhere on the array,
whereas in continuous-flow systems they must operate in a
specific micromixer or microchamber. The configurations
of the microfluidic array, i.e., the routes that sample droplet
travel and the rendezvous points of droplets, can be obtained
using software running on a PC or an ASIC [20, 21]; they
are then programmed into a microcontroller that controls
the voltages of the electrodes in the array. Test planning for
Fig. 3. Simple capacitive sensing circuit.
Test Planning and Test Resource Optimization for Droplet-Based Microfluidic Systems 201
a microfluidic array can also be implemented using a PC or
an ASIC.
3. Related Prior Work
Over the past decade, the focus in testing research has broad-
ened from logic and memory test to include the testing of
analog and mixed-signal circuits. MEMS is a relatively
young field compared to IC design, and MEMS testing
is still in its infancy. Recently, fault modeling and fault
simulation in surface micromachined MEMS have received
attention [4, 11, 14]. Researchers at Carnegie Mellon Uni-
versity are developing a comprehensive testing methodol-
ogy for a class of MEMS known as surface micromachined
sensors.
However, test techniques for MEMS cannot be directly
applied to microfluidic systems, since the techniques and
tools currently in use for MEMS testing do not handle
fluids. Hence they are of limited use for testing microflu-
idic devices. Most recent work in this area has been lim-
ited to the testing of continuous-flow microfluidic systems
[8, 9, 10]. Researchers at the MESA+Research Institute of
the University of Twente have applied mixed-signal testing
techniques to the problem of testing a microanalysis system.
Also, a design-for-testability (DFT) technique for Flow-
FET-based microfluidic systems has been proposed [9].
Similar to the MOSFET, a Flow-FET has source and drain
electrodes over which a relatively large voltage (∼100 V)
is applied. Due to the principle of electro-osmotic flow, the
electric field moves the charge accumulated between the
fluid and the surface of channel, dragging the bulk liquid
through the channel.
Optimal strategies for moving droplets in a microfluidic
system are proposed in [3]. The A
∗
algorithm from artificial
intelligence is used as the basis of a systematic search,
which is performed to generate a sequence of control signals
for moving one or multiple droplets from the start to the
goal positions in the shortest number of steps. This method
is closely related to the optimization problem of motion
planning with multiple moving robots [1, 12]. There are
two different groups of path planning problems for moving
robots. Navigation problem attempts to find a path from a
start position to a goal position through the shortest path,
whereas coverage problem focuses on finding the path of
coverage of an environment by mobile robots.
4. Problem Definition
In the test methodology proposed in [22], test stimuli
droplets are dispensed into the microfluidic system from the
droplet source and transported through the array (travers-
ing the cells) by following the designed testing scheme. As
described in [22], most catastrophic faults in droplet-based
microfluidic systems can lead to a complete cessation of
droplet transportation. Thus, for the faulty case, the test
stimuli droplet is stuck at an intermediate point during mo-
tion. On the other hand, the detection of all test stimuli
droplets at the droplet sinks indicates fault-free operation.
This methodology allows fault testing and biomedical as-
says to run concurrently on a microfluidic system. An ef-
ficient test plan not only ensures that the testing operation
does not conflict with the normal biomedical assay, but it
also guides test stimuli droplets to cover all the cells avail-
able for testing. This test plan can be optimized to minimize
the total testing time cost for a given test hardware over-
head, which refers here to the number of droplet sources
and droplet sinks. Note that some faults such as electrode
shorts affect two adjacent electrodes [19, 22]. To detect such
faults, defect-oriented test procedures are required, which
focus on pairs of cells and the traversal of droplets from
one cell to all its neighbors [19]. For simplicity, we do not
take into account such types of faults in this paper; only
catastrophic faults related to a single cell are targeted.
We can formulate the test planning problem in terms of
graph partitioning and the Hamiltonian path problem from
graph theory [5]. The key idea underlying this optimiza-
tion approach is to model the two-dimensional microfluidic
array as a directed graph, and then partition it into non-
overlapping subgraphs. Each part of the microfluidic array
is represented by a subgraph that is tested concurrently and
independent of the other parts. In this way, the total test
application time is reduced.
First we model the array of microfluidic cells using a
directed graph G = (V, E ) where the set of vertices V repre-
sents the set of available microfluidic cells, droplet sources
and droplet sinks, and e
ij
∈ E is a directed edge from ver-
tex i to vertex j if and only if these two vertices represent
two adjacent microfluidic cells and they satisfy the criterion
described below.
Note that unlike V, E is not determined apriori; rather
the set of edges is a variable, and the edges are determined
through the optimization procedure.
Definition 1. A Hamiltonian path from vertex s to vertex t
inagraphG is a path that starts at vertex s, ends at vertex t,
and visits every vertex of G exactly once.
We define e
ij
as follows:
e
ij
=
1 if a Hamiltonian path from a droplet source to a
droplet sink includes vertex i and vertex j
in consecutive order
0 otherwise
If a Hamiltonian path exists in an array with n cells, then
for any cell i in the array,
n
j=1
e
ij
=
n
j=1
e
ji
= 1.
The problem of finding a Hamiltonian path in graph G
from one source to one sink can be expressed as the follow-
ing problem: find a numerical instance of the set of binary
variables E = {e
ij
}, e.g., {e
12
= 1, e
21
= 0, ... , e
ij
= 1,
...}, that represents a Hamiltonian path from one source to
one sink.
202 Feietal.
Fig. 4. Graph model for a 4× 4 array.
If a Hamiltonian path exists, the cost C for this path
is defined as C =
n
i=1
n
j=1
e
ij
w
ij
, where i represents
any vertex in this path, j is the vertex adjacent to i in the
path, and w
ij
is the weight of e
ij
. Without loss of gener-
ality, we set w
ij
to be a constant value, assuming that the
transportation velocity between any two adjacent microflu-
idic cells is the same. For simplicity, let w
ij
=1. Therefore,
C =
n
i=1
n
j=1
e
ij
=
n
i=1
1 = n, i.e., the number of ver-
tices on the Hamiltonian path. If G has no Hamiltonian path,
the cost C is infinite.
Figure 4 gives an example of a graph model for single
source and single sink. In the graph model of this 4×4 array,
a black arrow between vertices i and j denotes that e
ij
= 1,
while the gray arrow between vertices i and j denotes that
e
ij
= 0. The cost C for this example is 11.
Based on the above definitions, we now develop the test
planning problem for multiple sources and multiple sinks.
We attempt to partition the directed graph representing the
microfluidic array into subgraphs, such that in each sub-
graph there exists a Hamiltonian path from one source to
one sink. In this way, the testing of the different partitions
can be performed independently and simultaneously in non-
overlapping parts of the microfluidic array. The total cost
for the array is the maximum of the cost for any of these sub-
graphs. This leads us to the following optimization problem
for minimizing the total cost:
• Optimal Partitioning Problem (OPP): Given N
source/sink pairs, determine an optimal partition that
divides the available cells in the array into N non-
overlapping partitions, such that in each partition there
exists a Hamiltonian Path from one source to one sink
and the maximum of the cost for these Hamiltonian paths
is minimized.
5. Analysis of Computational Complexity
In this section we prove that OPP is NP-hard. We first re-
view the following definition from computational complex-
ity theory:
Definition 2 [17]. Let L1 and L2 be two decision prob-
lems. L1 is polynomial-time reducible to L2 (L1 ≤ L2) if a
polynomial-time reduction f from L1 to L2 exists, subject
to
• f(x) is a yes-input for L2 if and only if x is a yes-input for
L1
• f is computable in polynomial-time.
We next note that if L1 is NP-complete, and L1 ≤ L2,
then L2 is NP-hard. This is a common technique to prove
that a given optimization problem is NP-hard.
We first consider the decision version D-PP of OPP, which
is expressed as follows.
• D-PP: Given N source/sink pairs and an upper limit D on
the cost, is it possible to partition array into N parts such
that there exists a Hamiltonian path of cost C
i
for each
partition and max
1≤i≤N
{C
i
} < D?
Theorem 1. OPP is NP-hard.
Proof: We first show that D−PP ∈ NP. We can non-
deterministically generate a N-partition and then verify in
polynomial time that max
1≤i≤N
{C
i
} < D. To show that
D-PP is NP-hard, we reduce the problem of determining a
Hamiltonian cycle in grid graph (HC-GG), which is known
to be NP-complete [7 ]. A grid graph G is a finite, induced
subgraph of the infinite two-dimensional grid. It has a finite
set of vertices V= {v
1
, v
2
, ..., v
n
}, where v
i
represents a
grid point (x, y). Note that x and y are positive integers,
denoting the x and y coordinates, respectively. An edge
exists in G between point (x, y) and (x
y
) if and only if
|x −x
|+|y − y
|=1.
We next define a polynomial-time reduction f from an
arbitrarily-chosen instance of HC-GG to an instance of D-
PP with N = 1 and D =∞. Given a grid graph G,any
vertex v
i
in G is mapped to a cell c
i
in array A, such that
cell c
i
= f(v
i
) and c
j
= f(v
j
) are adjacent in A if and only if
there exists an edge between v
i
and v
j
in G. We define the
vertices with the maximum (or minimum) value x of the x-
coordinate (or the y-coordinate y) in the corresponding grid
graph to be boundary vertices in G. Similarly, the cells in
the array obtained by mapping from the boundary vertices
in G are defined as boundary cells in A. Next we attempt
to add a droplet source s
1
and a droplet sink s
2
to this
array. There are two possible cases. In Case 1, there exist
two adjacent boundary vertices (noted as v
1
and v
n
)inG,
such that there also exist two adjacent cells (noted as c
1
and c
n
) on the boundary of array A. We then add s
1
next
Test Planning and Test Resource Optimization for Droplet-Based Microfluidic Systems 203
Fig. 5. (a) Illustration of Case1; (b) Illustration of Case2.
to c
1
and s
2
next to c
n
; see Fig. 5(a). In Case 2, if there
are no adjacent boundary vertices in G and neither are there
adjacent boundary cells in A, we select a single boundary
cell denoted by c
1
, and place s
1
and s
2
together adjacent
to c
1
; see Fig. 5(b). It is obvious that the transformation
described above can be carried out in polynomial time.
Next we prove that there exists a Hamiltonian path from
s
1
to s
2
of cost C < ∞ in A if and only if there exists a
Hamiltonian cycle in G of cost less than ∞.
1. Proof for Case 1: Assume there exists a Hamiltonian
cycle in G, denoted by v
1
v
2
...v
n
v
1
, where v
1
and v
n
are
two adjacent boundary vertices. Due to the mapping f: G→
A, c
1
= f (v
1
), c
n
= f (v
n
) and they are two adjacent cells
on the boundary of array A. In this way, there exists a path
f (v
1
) f (v
2
)...f (v
n
) from c
1
to c
n
that visits every cell exactly
once. In addition, s
1
is adjacent to c
1
and s
2
adjacent to c
n
.
Therefore, there is a Hamiltonian path from s
1
to s
2
in A
and cost C = n < ∞.
On the other hand, if there exits a Hamiltonian path s
1
c
1
... c
n
s
2
from s
1
to s
2
in array A, a Hamiltonian path from
c
1
to c
n
also exists. Now by the inverse transformation f
−1
:
A→G, it is seen that there exists a Hamiltonian path f
−1
(c
1
)
... f
−1
(c
n
) from f
−1
(c
1
)tof
−1
(c
n
). Moreover, f
−1
(c
1
) and
f
−1
(c
n
) are two adjacent vertices. Therefore, there exists a
Hamiltonian cycle f
−1
(c
1
)...f
−1
(c
n
) f
−1
(c
1
)inG.
2. Proof for Case 2: If there exist no adjacent cells on
the boundary of A, we place s
1
and s
2
together next to one
boundary cell c
1
. This implies that in any path from s
1
to
s
2
, c
1
is visited at least twice. Therefore, there exists no
Hamiltonian path in A for this case and C =∞. Similarly
in G, since there are no adjacent vertices on the boundary,
some boundary vertices have only degree one. This violates
the necessary condition for the existence of a Hamiltonian
cycle, i.e., every node should have a degree of at least two.
Hence there is also no Hamiltonian cycle in G.
Thus we have shown that any instance of HC-GG is
polynomial-time reducible to an instance of D-PP (N =
1 and D =∞). Since HC-GG is NP-complete, D-PP is at
least NP-hard. Moreover, since D-PP is in NP, it is also
NP-complete. The optimization version of D-PP, i.e. the
Optimal Partitioning Problem is therefore NP-hard.
6. Integer Linear Programming Model for OPP
Although OPP has been proven in Section 5 to be NP-hard,
we show in this section that it can be solved exactly using
integer linear programming (ILP) for a microfluidic array
of modest size. An ILP model can be described as follows:
Minimize: Ax (objective function)
Subject to: Bx ≤ C (constraint inequalities),
where x is a vector of variables, A is an objective function
vector, B is a constraint matrix and C is a column vector
of constraints. We used a popular public domain ILP solver
called lpsolve for our work [2 ].
We formulate the ILP model for OPP as follows. It is
obvious that when N = 1, OPP is equivalent to the Hamil-
tonian path problem for a single source and a single sink
described in the earlier section.
For N>1, we define a binary variable S
ik
as follows:
S
ik
=
1ifvertexi is in subgraph k, i.e., microfluidic
cell i belongs to partition k.
0 otherwise
where 1≤ k ≤ N. Since every vertex only belongs to one
subgraph,
N
k=1
S
ik
= 1 ∀i.
Definition 3. Vertex j is the connected neighbor of vertex i,
if there is an edge between i and j, and either e
ij
= 1ore
ji
= 1.
Next we impose the constraint that vertex i is in partition
k if and only if its connected neighbor is also in partition k.
This is expressed as follows:
S
ik
= 1 if and only if
n
j=1
e
ij
S
jk
= 1 ⇒ S
ik
=
n
j=1
e
ij
S
jk
.
The existence of Hamiltonian paths in non-overlapping
partitions ensures that, for every cell i in array,
n
j=1
e
ij
=
n
j=1
e
ji
= 1.
Finally, we incorporate the objective function into the
ILP model. The objective of this optimization problem is to
minimize the total cost C = max
k
{C
k
}=max
k
{n
k
}, k =
1,2...N, where n
k
is the number of vertices visited by
Hamiltonian path k. It is easily seen that n
k
=
n
i=1
S
ik
.
Therefore, C = max
1≤k≤N
n
i=1
S
ik
.
