An optimized knight traversal technique to detect multiple faults and Module Sequence Graph based reconfiguration of microfluidic biochip

Mukta Majumder, Department of Computer Science and Application, University of North Bengal, Siliguri – 734013, West Bengal, India. Email: mukta_jgec_it_4@yahoo.co.in Abstract Conventional biomedical analysers are replaced by digital microfluidic biochips and they are adequate to integrate different biomedical functions, essential for diverse bioassay operations. From the last decade, microfluidic biochips are getting plenty of acceptances in the field of miscellaneous healthcare sectors like DNA analysis, drug discovery and clinical diagnosis. These devices are also bearing a vital role in the area of safety critical applications such as food safety testing, air quality monitoring etc. As these devices are used in safety critical applications, clinical diagnosis and real‐time biomolecular assay operations, these must have properties like precision, reliability and robustness. To accept it for discriminating purposes, the microfluidic device must endorse its preciseness and strength by following sublime testing strategy. Here, an optimized droplet traversal technique is proposed to investigate the multiple defective electrodes of a digital microfluidic biochip by embedding boundary cum row traversal and KNIGHT traversal procedure (based on the famous Knight Tour Problem). The proposed approach also enumerates the traversal time for a fault‐free biochip. In addition to identifying the faulty electrodes, a Module Sequencing Graph based reconfiguration technique is proposed here to reinstate the device for normal bioassay operation.


| INTRODUCTION
A couple of decades ago, the digital microfluidic biochip (DMFB) was introduced as a new category of a lab-on-a-chip (LOC) device [1], and it has become very popular in the present day for biochemical analysis or bioassay operations. This biochip is also called biomedical microelectromechanical systems as it replaces highly repetitive laboratory tasks by substituting traditional large equipment with composite micro-system [2,3]. It offers the advantages of design flexibility, higher sensitivity, smaller size and lower cost [4]. In the last few years, these biochips grab the attraction of a huge number of researchers.
In earlier days, microfluidic biochip was manipulated by continuous fluid flow carried out by using micro-valves, micro-pumps and micro-channels [2,5,6]. An alternate approach is to manipulate liquid using discrete droplet [7]. The droplet-based chip is referred to as 'DMFB' [8][9][10]. The digital microfluidic system is advantageous over the continuous flow system [8,11]. The droplet size can be manipulated by varying the voltage on the electrodes [12][13][14]. A number of methods for manipulating microfluidic droplets have been proposed in the literature [15,16].
The fault of a DMFB can be categorized as either catastrophic or parametric [8,17]. Due to the catastrophic fault, the test droplet is stuck during its movement at the fault position, resulting in a malfunction. On the other hand, parametric or the soft fault degrades the system's performance. In a fault-free system, all the test droplets can be seen at the sink in stipulated time by using a capacitive detection circuit [11]. As this chip is used as a medical microsystem and other safety and clinical appliances, reliability, dependability and correctness are the major issues. A detailed discussion about the applications of DMFB in pharmacogenomics, drug development, tissue engineering and other areas were described in [18][19][20]. The proposed approach not only enumerates the traversal time for a fault-free biochip in an optimized duration, but also reconfigures the defective cells of a faulty biochip by using a novel Module Sequence Graph (MSG)-based reconfiguration technique to boost its application.
To find a fault within a newly manufactured biochip, it is essential to route a set of test droplets to every edges and cells of entire biochip by controlling the voltage of the electrodes [7,9,[21][22][23]. If the device under test contains any defect, it is prerequisite to identify the defective electrodes. As soon as the location of the fault is identified it is obligatory to reconfigure the faulty module to reinstate its normal bioassay operation [24,25]. As the biochip contains a number of unused cells, the reconfiguration of defective cell can be overhaul by swapping it with one of those spare cells. Since these biochips are targeting the extremely competitive and inexpensive market place and applied to numerous bioassay operations, assessment and diagnosis methods must be economic, prompt and effectual [9,26]. After reconfiguration of the defective cells of a biochip, the device can work as a fresh one.
In this article, we introduce a time-effective testing methodology for digital microfluidic device using multiple droplets. The proposed method allows offline testing framework using parallel droplets routing on two-dimensional electrode array. It optimizes the droplet traversal technique to investigate multiple faulty electrodes by embedding boundary cum row traversal and KNIGHT traversal procedure, based on the famous Knight Tour Problem of Chess Board. It also computes the traversal time for a fault-free biochip. In addition to identifying the defective location, an MSG-based reconfiguration technique is illustrated here to reinstate the device for usual bioassay operation. The proposed framework is simulated in OpenMP-Cþþ on Linux multicore environment.
The proposed article contributes to the literature in several ways. The salient features of the article are as follows: • This article introduces an efficient Knight traversal technique to assess the electrodes of a DMFB using multiple droplets. • A novel MSG-based reconfiguration technique is proposed to reinstate the defective cells.
• The suggested result indicates that the propound technique enumerates the traversal time for a fault-free biochip in an optimized duration to equip the biochip cost effective and market demandable.

| RELATED WORKS
In recent times, the organized movement of droplets in DMFB, used for different bioassay operations to achieve higher sensitivity and better throughput, has attracted the interest of various researchers.
A DMFB utilizes the phenomenon of electrowetting to manipulate and move nanolitre droplets containing biological samples on a two-dimensional electrode array [5]. Some wellpronounced biochip models were described in [6,27,28].
Since the bio-assay operation requires a safe and secure performance, the biochip must be fault free. A few methods on fault modelling and fault simulation for continuous-flow microfluidic biochips were investigated in [9,29]. To classify the defects and test application, procedures for digital biochip were discussed by Su et al. [29]; they categorized the faults in biochip as catastrophic and parametric. For detecting catastrophic faults in digital microfluidic arrays, some effective techniques were found in [6,17], where a test droplet routed through the suspected cells and edges using a predefined path towards the sink. Li et al. defined some defects such as damage to the hydrophobic layer, dielectric breakdown, short-circuited microelectrodes, broken wire, transistor failure and parasitic leakage [30]. They also investigated fault models like dispensing failure, transportation failure, mixing failure and splitting failure on modern DMFB like micro-electrode-dot-array.
Roy et al. proposed a cost-effective technique for mixing and splitting the droplet sample without using storage unit [31]. Bera et al. also derived a sample preparation technique by using mix-split operations based on weighted dilution process [32]. Nahar et al. [33] investigated another droplet splitting mechanism using the phase separation performance technique. Several research works demonstrated other movement strategies like uphill climbing or coalescence of droplets by electrowetting-on-dielectric actuation [34,35].
In [36], a technique was proposed to use Hamiltonian path to detect catastrophic faults in microfluidic arrays. One problem with this approach is that although finding Hamiltonian paths in grid structures is well known, but checking the existence of Hamiltonian path in a given graph is an nondeterministic polynomial time complete problem [37]. Therefore, it would be very expensive to determine such paths in the microfluidic array. An alternative method for testing DMFB based on Euler paths was found in [38]. This method mapped a DMFB into an undirected graph and a Euler path was determined for testing the biochip. A technique of multiple faults detection and identification of their locations was proposed by Davids et al. [8].
A concurrent testing methodology for detection of catastrophic faults in digital microfluidic systems was presented and the problems of test planning and resource optimization were discussed by Su and Chakraborty [11]. Xu et al. discussed a parallel scan-like testing using multiple droplets for detection of faults and investigated their diagnosis methods in a biochip [9]. Fault detections in microfluidic biochips with multiple droplets in parallel were also addressed in [21,39]. In [7,8,23], the authors proposed some techniques where they first selected some start electrodes (pseudo sources) or base nodes, then the traversal of the microarray was done by moving the droplets from these pseudo sources or base nodes to pseudo sink and then to sink reservoir.
Mukherjee et al. [40] proposed a fault identification technique using LED light, and Ghosh et al. [41] used CCD camera and image segmentation technique to identify faults in microarray. Hu et al. identified errors in droplet transportation in the biochip using capacitive sensors in real time, and software-based recovery was accomplished using dynamic reconfiguration [17]. Reconfiguration technique was involved in different approaches like local, partial and full, which were described in [42,43].

| PROPOSED TECHNIQUE
In this article, we have segmented our proposed technique into two sections. In the first section, we are traversing multiple test droplets from source to sink using different base nodes or pseudo sources and pseudo sinks to detect whether the microarray contains any defective electrode. And in the second section, we are identifying the location of defective electrode in the biochip.

| Fault detection
In this work, traversal of each electrode is prerequisite with minimum number of overlapping paths. Therefore, we have traversed multiple droplets using several pseudo sources and pseudo sinks. This step also involves routing each electrode using the test droplet at least once, by either boundary cum row traversal or KNIGHT traversal. As soon as the fault is detected, it is required to locate it for reconfiguration. We have used the backtracking procedure to uncover the fault position. As the forward movement of the droplet is ceased at fault location, identification of multiple faults is not feasible using single droplet, hence we are using multiple test droplets.

| Traversal procedure
In this article, we use multiple droplets to traverse the N � M microarray, where N is the number of rows and M is the number of columns. The concept of the KNIGHT traversal has originated from Knight Tour Problem, as to visit a rectangular chessboard [44]. To implement the technique for visiting each edge at least once, we have used boundary cum row traversal (Right-right movement as shown in Figure 1 To reduce the traversal time, we are applying parallel routing of multiple droplets for the microarray. We differentiate the rows as R 1 , R 2 , R 3 , …, R N , where N is the total number of rows and the columns as C 1 , C 2 , C 3 ,…, C M , where M is the total number of columns of the (N � M) microarray.
To implement the journey process across the microarray, initially, we start with boundary cum row traversal where we circulate the test droplets through the boundary and internal rows. In boundary traversal, we move two droplets in the opposite direction through the boundary from source reservoir to sink reservoir as shown in Figure 2. After the completion of boundary traversal, we place the internal row traversal droplets on the pseudo sources (R 2 , R 4 as shown in Figure 3). Then move these test droplets from pseudo sources to sink reservoir through pseudo sinks as shown in Figure 3. If any test droplet does not reach the sink, this implies there is a defect within the microarray. To recognize the defective location, backtracking procedure is applied as illustrated in Section 3.3. The summarized algorithm of boundary cum row traversal is described in Algorithm 1 and the time required to complete the traversal process is T 1 from Equation (1).
Algorithm 1 boundary cum row traversal 1: Dispense two droplets from source reservoir and start travelling opposite direction as clockwise and anti-clockwise along the boundary electrodes towards sink. 2: After completion of step 1, |N/2| droplets placed at row pseudo sources one by one. 3: The test droplets move towards right boundary in parallel, then pseudo sinks to sink reservoir. 4: If any droplet does not reach the sink reservoir after stipulated time, then backtrack the droplet through the same path to the source.
After successful completion of boundary cum row traversal, we initiate internal electrode traversal by circulating the test droplets according to KNIGHT traversal. The test droplet starts its journey from column pseudo source (C i ) or row pseudo source (R j ) using DDR movement and comes back as soon as encounter other boundary electrode to its initial positions using UUL movement. This movement is termed as KNIGHT traversal and the pseudo source and the pseudo sink are the same location in this Knight traversal for each droplet. In KNIGHT traversal, if there is any premature execution of DDR movement as the test droplets reach microarray boundary, UUL movement is applied immediately from that location via the same path, otherwise via the next possible path as shown in Figures 4  and 5. In case of odd-row microarray, we select C i as C 1 , C 4 , C 7 , …, up to C MÀ 1 and C 1 , C 3 , C 5 , …, up to C MÀ 1 for even-row microarray, where M is the number of columns of microfluidic array. After t time units of starting column-wise knight movement (⌈(NÀ 1)/6⌉ À 1) droplets are dispatched from reservoir for row pseudo sources (R j ) and execute the KNIGHT traversal process in parallel as shown in Figure 4a,b. Row pseudo sources (R j ) are defined as R 7 , R 13 , R 19 , …, up to R NÀ 1 for odd-row and R 5 , R 9 , R 13 , …, up to R NÀ 1 for even-row microarray, where N is the number of rows of microfluidic array. After reaching the pseudo sink, move the test droplets of C i and R j in clockwise and anticlockwise, respectively, to attain the sink reservoir through the microarray boundary. The KNIGHT traversal algorithm is presented in Algorithm 2, and the droplet traversal path is shown in Figures 4 and 5. The required time 'T 2 ' to complete the traversal is determined by Equation (2), where M is the number of column and N is the number of row as mentioned earlier.

7: Else
Apply UUL movement to fetch the test droplet at pseudo sink via the next possible path. 8: Move the test droplets from C i to sink in clockwise and from R j to sink in anti-clockwise direction. 9: If any droplet does not reach the sink after stipulated time, then backtrack the droplet through the same path to the source.

| Identify fault location
Our technique not only works efficiently to traverse the microarray, it also exposes and identifies the defective electrode by using the 'backtracking' procedure similar to [3,8,45]. Fault is detected if any of the test droplets is stopped and does not reach the sink after the stipulated time. In such an erroneous situation, the stalled droplet is rolled back to the source through the same path. Consider a 5 � 6 microarray and apply the boundary cum row traversal. While passing through the microarray, the motion of the droplet is stopped due to the existence of fault at boundary electrode as shown in Figure 6a. To get back the droplet and to identify the faulty location, we are applying a control voltage in the opposite direction. We have assumed earlier that the test droplet takes 1 unit time for one edge movement, hence if the droplet takes 7 units time for backtracking from defective location to source reservoir, this implies defect is located at 7 edges distance on boundary from the source reservoir. The backtracking procedure is illustrated in Figure 6b.
Similarly, we apply backtracking procedure in case of KNIGHT traversal as shown in Figure 7, where Figure 7a shows the KNIGHT traversal procedure with stuck droplet and Figure 7b shows the backtracking process from the defective location to reservoir. According to Figure 7b, the test droplet takes 6 units time for backtracking from defective location to source reservoir.

| RESULTS AND DISCUSSIONS
The proposed method is applied to detect the faults and identify their locations in a DMFB. Starting from the source to sink, the test droplets are travelling throughout the microarray, and seek out the existence of defective electrode. Due to any fault, test droplet stops and further movement of the particular droplet is ceased. The proposed technique not only detects the fault and identifies the faulty electrode, but also measures the overall traversal time for fault-free microarray. Since there is a chance of mixing two droplets at the time of parallel execution, the authors had to be very sincere to identify the pseudo sources. Taking each edge movement of the droplet as 1 unit of time, we have calculated the proposed time for traversal.
We have simulated the proposed algorithms in OpenMP-Cþþ on Linux multicore environment for a large number of microarrays and collected the results for the analysis. In this experiment, we use the source reservoir at (1,1) and sink at (N,M) of the microarray as shown in Figure 2.
The simulated result is segregated into two parts, boundary cum row traversal (T 1 ) and KNIGHT traversal (T 2 ) as given in Table 1, where the total time for traversing a fault-free biochip is calculated as T 1 þ T 2 .
Next, the authors have presented a comparative study of our proposed technique with the existing techniques [9,43] in Table 2. It clearly shows that the proposed technique outperformed the existing techniques in most of the cases. The authors have considered the droplet's placement time from source to pseudo sources, traversal time from pseudo sources to pseudo sinks and, finally, pseudo sinks to sink for all cases.
Our proposed technique uses lesser droplets to test the microarray compared to [8,39] where the researchers threw a large number of droplets to traverse the microarray. We incorporate a comparative study of number of droplets with its associated routing time in droplet traversal time (DTT), where DTT is defined by the ratio of total number of used droplet and total time of traversal. The DDT is calculated as follows: The result of the comparative study over DDT is given in Table 3, and it evidently shows that the proposed technique gives better enhancement with respect to the number of test droplets used in this work with the other existing techniques. The proposed technique reduces the installation cost effectively compared to the existing technique [40], where the authors used multiple photodiode sensors (combination of LED and photodiode) to simulate the experiment. Here the LED light and the sensor were placed at the opposite ends of each other of every row of the microarray. The LED light passed through the droplet and fell on the sensor. The proposed technique is also economic with respect to [41], where a CCD camera was used to capture three images in different time instances to identify the exact fault location of the microarray. These types of set-up are costly which will increase the testing and fabrication cost of the biochip. As the biochips are designed for a very completive and low-cost market segment, the testing and fault identification techniques must be cost effective.

| FEASIBLE RECONFIGURATION FOR THE PROPOSED FAULT DETECTION TECHNIQUE
Reconfiguration is used to bypass the defective cell of the microfluidic array for yield enhancement to tolerate faults by changing the control voltages of some electrodes. In this work, the authors relocate the faulty cell by fault-free spare cell and identify the reconfiguration possibility by Reconfiguration with Spare (RS), which defines the ratio of spare cell and active cell of the microfluidic array.

RS ¼ number of spare cells of n � n array number of active cells of n � n array ð4Þ
In Figure 8a, an example of microarray is shown with size 9 � 8, which has 26 spare cells and 46 active cells. By using Equation (4), the RS of given microarray is 0.5652, which is 56% of active cells, and the microarray contains five different modules of different dimensions.
Local reconfiguration technique involves only relocating provincially the defective module [42]. Since the microfluidic biochip is small in size, the dimension is one of the significant issues to reduce the space. However, the reconfiguration is difficult by replacing the faulty module if there is no accessible spare cell. In such situation, the partial reconfiguration is beneficial where the whole module is relocated to spare module [24]. In this article, we design the Module Spare Ratio (MSR) to define the ratio between the adjacent spare cell and the number of cell of the faulty module, and it is calculated by From Equations (4) and (5), if RS〈1 and MSR〈1 of a particular module, partial and local reconfiguration may suffer to relocate the faulty module. From Figure 8a, the MSR of module 3 is 1, and from Figure 8b, the MSR of module 4 is 0.67. In case of module 3, the local or partial reconfiguration is possible, but for the case of module 4, these reconfiguration techniques are not possible due to lack of spare module. The proposed reconfiguration technique is superior than that in [24,25,42] in terms of MSR. In this paper, we are proposing an MSG-based reallocation technique, which reduces the overhead, as it does not reconfigure the whole module rather; it will only relocate the required cells of the affected module.

| Reconfiguration technique
The proposed technique reconfigures the required defective cells to their adjacent spare cells according to the cell sequence  (Figure 9 shows MSG of module 4 of Figure 8b). MSG shows the sequence number of a particular module according to the droplet pathway like S 1 , S 2 , …, S 12 . Each spare cell marked as 'F' and defective cell or dead cell as 'd' in the MSG as shown in Figure 9.
The proposed MSG-based technique relocates a cell, S 1 to its reconfigured position as S 1 ' using any one of the spare cells (F) adjacent to the path as illustrated in Figure 9. According to cell sequence number, we repeat this reconfiguration process until each cell is relocated. We consider the cells as vertices in the MSG-based reconfiguration technique which is described in Algorithm 3 and the steps are shown in Figure 10, where Figure 10a-d illustrates the reconfiguration of S 2 , S 3 (dead cell), S 4 and S 5 to their designated locations S 2 ', S 3 ', S 4 ' and S 5 ', respectively. Similarly, the rest of the cells relocate their positions sequentially to spare cells (F), which are horizontally or vertically adjacent to recently relocated cells (S iÀ 1 ') and, finally, the whole module is reconfigured to its remodelled location according to the sequence number as shown in Figure 11a. The proposed MSG-based algorithm finds an alternate path for the defected module by avoiding the defective cells. Since the proposed MSG-based reconfiguration uses the minimum number of adjacent spare cells, the cost of the technique is negligible as compared to relocate the whole module to another spare module. After executing this MSG-based algorithm, the biochip can reinstate its normal bioassay operation. By applying the proposed technique, it is possible to get multiple potential pathways for the defective module which is shown in Figure 11.

| CONCLUSION
An effectual multi-droplets traversing technique for detecting several faults and to identify or uncover these faulty locations within DMFB has been presented here. It has also been shown that the proposed technique is susceptible to