Abstract
Increased use of digital microfluidic (DMF) biochips has fueled the replacement of expensive healthcare and biochemical laboratory procedures with lowcost, fullyautomated, miniaturized integrated systems. Dilution and mixing of fluid samples in a certain ratio are two fundamental primitives needed in sample preparation, which is an essential component of almost all protocols. Most of the existing dilution algorithms used in dropletbased microfluidic systems deploy a sequence of (1 : 1) mixsplit steps, where two unit volume droplets of different concentrations are mixed, followed by a balanced split operation to obtain two equalsized droplets. In this work, we introduce a simulationguided optimization procedure (SIMOP) for achieving the target concentrations while optimizing multiple factors according to userspecified priority levels. The SIMOP algorithm produces a given concentration while optimizing each criterion as desired. Experimental results favorably demonstrate the performance of the proposed procedure compared to many previous algorithms used for singletarget dilution. We also study the impact of errors caused by unbalanced droplet splitting on the accuracy of target concentration obtained by SIMOP sequences. Experimental study reveals that SIMOP outperforms previous algorithms from the perspective of volumetric error management that may occur in reaction paths. The proposed technique may find many potential applications to robust sample preparation needed in diverse areas of biomedical engineering and healthcare domains.
Introduction
In recent years, the introduction of digital microfluidic biochips (DMFB) has come as a favorable and effective evolution in clinical diagnosis and healthcare domain [1, 2, 10, 20, 24, 25, 30]. These biochips effectively enable us to realize the concept of LabonaChip (LoC). An LoC incorporates one or more laboratory protocols or assays on a compact biochip to provide immediate investigation at the location of the chip. Application areas include pointofcare diagnosis, protein, DNA, enzymatic assessment and many such important practices.
We have extended our research on the basis of DMFBs and (1 : 1) sample dilution algorithms [6, 7, 15, 26, 27, 31]. In a DMF biochip, discrete fluid droplets can move on a substrate of 2D array of electrodes by effectively inducing electrowetting (EWOD) technique [10, 20, 24, 25]. This procedure ensures minimal human intervention to precisely manipulate and move droplets on biochips, resulting in higher sensitivity, portability, lower cost, and lower power consumption [1, 2].
In digital microfluidics, droplets are maneuvered in a series of dilutions to achieve target concentration factors (CF) for any liquid. In a chemical assay, mixing and splitting of fluid droplets are basic steps. For an effective sample preparation [12, 13], we engage a series of (1 : 1) balanced mixsplit operations. Consider two oneunit volume droplets (say, \(V_1\) and \(V_2\)) with CFs (say, \(C_1\) and \(C_2\)) participating in first mix and then split to generate two droplets each possessing oneunit volume. The concentration factor of the newly created droplets are computed as the arithmetic mean value \(\left( \frac{C_1 \times V_1 + C_2 \times V_2}{V_1 + V_2}\right) \) of the parent droplets. This balanced dilution technique is used in the industry to generate a range of target concentrations using established schemes.
We have organized our work into two parts. PartI has extensive analysis and investigation of the SIMOP simulation and optimization procedure reported through the first four sections. PartII extends the SIMOP algorithm to consider the application of unbalanced mixsplit errors on droplet volumes in the reaction path. Addressing PartI, in Section 2, prior work in the domain of algorithmic microfluidics is reviewed and the motivation to realize the SIMOP model is discussed. The third section explains the problem formulation. We present the methodology utilized to address the challenge of optimizing (1 : 1) balanced dilution in the fourth section. The experimental results of SIMOP simulation are illustrated in the fifth section. PartII starts with the sixth section where we review if our SIMOP algorithm can sustain volumetric errors (VE) to reach target concentrations. The first subsection in the sixth section presents how errors on reaction path affect the performance of multiple SIMOP sequences. The second subsection lays out the experimental observations of SIMOP Error Optimization scheme that talks about application of volumetric errors on optimal SIMOP sequences at multiple steps or onestepatatime. In our final note, conclusion and future scope are narrated in the last section.
Prior Art and Motivation
A number of (1 : 1) sample dilution algorithms are available and each of these adheres to its own set of sample preparation rules. Nevertheless, to generate target concentrations, none of these algorithms has explored to find out if any alternative ordering of steps better than any of the available sequences exists for preparing a particular concentration. Several approaches exploring a number of optimization schemes handling singletarget preparation on DMF biochips have been proposed:

1.
Bit Scanning (BS) [31] is considered as one of the pioneer methods in algorithmic microfluidics.

2.
Dilution and mixing with reduced wastage (DMRW) [26].

3.
Improved dilution and mixing algorithm (IDMA) [27].

4.
Graphbased optimal reactant minimization (GORMA) [7].

5.
Reactant minimization algorithm (REMIA) [15].

6.
Weighted dilution (WD) [5].
BS, DMRW and IDMA
BS scheme [31] converts the target concentrations (\(C_t\)) into their respective binary representations and then scans the generated bitstream. The droplets with current concentration (termed as \(C_\mathrm{intermediate}\)) are mixed with either the buffer (\(C_\mathrm{Buffer}\) = \(0\%\)) or the sample (\(C_\mathrm{Sample}\) = \(100\%\)). As for example, if we aim to achieve a target concentration of \(78\%\) in 10 (1 : 1) dilution steps, then based on the precomputed denominator, d = \(2^{10}\) = 1024, we can generate \(C_t\) = \(\frac{799}{1024}\) = \(78.03\%\) = 0.1100011111. Based on the rightmost bit (i.e., 1), the dilution process is started with sample \(\left( \frac{1024}{1024}\right) \) and buffer \(\left( \frac{0}{1024}\right) \), scanning the bit string from right to left. If a 1bit is scanned, we fetch a sample droplet for a mix/split and if a 0bit is encountered, we use a buffer droplet to perform the mix/split operation.
Compared to bitscanning, the DMRW algorithm [26] engages a strategy analogous to binary search to generate the fractions. At each dilution step, the target concentration is compared with the current \(C_\mathrm{intermediate}\) and if it is more (less), we perform a mixsplit operation between one sample (buffer) droplet and one droplet with concentration \(C_\mathrm{intermediate}\).
Improved DMRW algorithm (IDMA) [27] optimizes the use of droplets with intermediate concentration factors generated in the reaction path, which reduces waste generation and thereby results in the reduction of sample/reagent requirement compared to DMRW.
GORMA and REMIA
GORMA [7] employs an exhaustive inspection of many viable dilution possibilities and an optimal dilution graph is generated with the best possible droplet sharing to minimize reactant usage.
The reaction minimization algorithm REMIA [15] focuses on optimal usage of sample/reactant in dilution process.
WD
A recent paper also exhibited the power of simulation to establish the potential for using weighted dilution (WD) [5] over (1 : 1) balanced mixsplit. Weighted dilution requires more advanced technology and thus is not commercially available yet. However, note that our SIMOP work here uses only (1 : 1) balanced mixsplits.
Other Work for Optimizing Sample Preparation on Digital Microfluidic Biochips
Dinh et al. [9] presented an optimal sample preparation algorithm using a minimumcost maximumflow model. This scheme is based on a minimumcost maximumflow model. The authors claim that using the proposed model the optimal cost will be achieved for (a) sample and buffer usage, and (b) waste amount for multipletarget concentrations. Fan et al. [11] presented a multitarget sample preparation algorithm for reactant minimization on digital microfluidic biochips (DMFBs). The proposed method first converts the reactant minimization problem into a network flow model with a set of target CFs and then solves the converted model by integer linear programming (ILP). Experimental results fairly illustrate that the new algorithm can lessen the reactant consumption by up to \(31\%\) as compared with the then stateoftheart. An innovative ILPbased synthesis for sample preparation on DMFBs [32] is proposed that focuses to derive an optimal scheduling for practical sample preparation applications within a reasonable time. In this procedure, a new ILP formulation is utilized for scheduling operations on digital microfluidic biochips.
A novel attempt was made to solve the problem of generating samples of multipletarget concentrations with high utilization of intermediate droplets on DMFBs [12]. Simulation results show that this approach can achieve up to \(71\%\) reduction in the number of waste droplets and up to \(50\%\) reduction in sample preparation time. Addressing sample preparation on DMFBs, Hsieh et. al. [13] presented an automated design methodology for sample preparation, including architectural and layout synthesis followed by dynamic error recovery. The authors aimed to evaluate the proposed algorithm on reallife biochemical applications to exhibit its efficacy to achieve up to \(48.39\%\) reduction in sample preparation time.
The Proposed Scheme
Our analysis reveals that BS method tends to minimize dilution steps but ends up with notable sample and buffer consumptions leading to produce significant waste when compared to other dilution methods. Unlike BS, IDMA and DMRW schemes use up fewer input droplets, (such as reagent/sample and buffer) which is promising, however, both these methods spend extra time in the dilution process. These algorithms were developed for waste optimization. GORMA and REMIA focus on reactant minimization. All these different methods do not focus on optimizing the entire combination, i.e., steps, sample, buffer, waste together, which led us to explore an approach that can produce a consolidated improvement on the existing schemes [9, 12, 13].
Hence, we started checking for the existence of a new order of mixsplit operations applied on a set of fractions that may become remarkably advantageous over the existing methods (BS, DMRW, IDMA, GORMA, REMIA and WD). A test simulation was carried out to generate the fraction \(C_\mathrm{t}\) = \(\frac{799}{1024}\) in a new way. Note that in Fig. 1, we start the SIMOP algorithm mixing oneunit volume of \(C_\mathrm{Sample}\) = \(\frac{1024}{1024}\) and oneunit volume of \(C_\mathrm{Buffer}\) = \(\frac{0}{1024}\) to generate first intermediate concentration, \(C_1\) = \(\frac{512}{1024}\). In the subsequent discussion in this paragraph, we omit mentioning the denominator, which is obviously \(2^{10}\) = 1024. We further mix concentrations 512 and 1024 to generate second intermediate, \(C_2\) = 768. As we advance in the corresponding SIMOP dilution graph left to right generating target CFs stepbystep, target numerator \(C_\mathrm{t}\) = 799 is attained at the 10th step. The origin of the two directed edges incident on a particular node designate the two concentrations from which the new concentration is being generated and the two outgoing edges from the same node (considering full utilization of the two droplets) exhibit which other concentrations will be generated further using this concentration. As for example, 808 has 2 outgoing edges, first one such droplet is used to generate 796, and then the second produces 802 combining 808 and 796. We have (1 : 1) ratio of mixing being followed in SIMOP to generate 2 droplets of 808, this particular intermediate concentration (\(C_\mathrm{intermediate}\)) is then fully utilized as compared to 672 and 802, where one waste droplet is generated for each.
Comparing the number of waste droplets, sample droplets and buffer droplets used, the mixing sequence shown in Fig. 1 shows a significant improvement over the existing sample dilution schemes such as BS, DMRW, IDMA, GORMA, REMIA and WD. This observation indicates that such alternative methods might exist for a large number of cases. Moreover, there may be many different sequences for optimizing different parameters for even a single concentration [11, 32]. With these objectives in mind, we aimed to develop a simulationbased algorithm to generate an optimal method for each target concentration. Although our method is based on exhaustive search, the fact that the number of mixing steps used in the present stateoftheart is as low as 10, further gave us the impetus to follow it up. However, it had to be implemented quite carefully to make it viable in real time.
In many computing domains involving query processing, some preprocessing of data (offline) is performed so that the queries can be answered faster whenever required (online). In SIMOP, we adopt a similar approach to search for an optimal mixsplit sequence required to generate a given target CF. Even a significant increase in the time complexity of the preprocessing phase is acceptable, if it can speed up the query step by a small amount. We, therefore, propose here to generate all the possible sequences of mixsplit steps to generate each target fraction, then identify the optimal sequence for each criterion from that exhaustive list with respect to each fraction and finally store them in a lookup table during the preprocessing phase. Shao et al. [28] proposed a reliabilityaware sample preparation (RASP) method for generating the optimized mixsplit sequence. RASP presents a lookup table construction algorithm along with the table query method to generate the droplets with target CFs within the error tolerance of \(0.1\%\). LUTOSAP [29] is a lookup table method enabling fast query response where the weighted sum of sample consumption, buffer consumption, and the number of mixsplit operations is optimized. Experimental results using LUTOSAP obtain optimal sample preparations in microseconds within the accuracy tolerance of \(0.2\%\). However, they did not consider the problem of volumetric split errors while producing target droplets.
Our Solution Methodology
We present below the SIMOP algorithm starting with two initial concentration factors (CFs) with a goal to generate a target fraction (\(C_\mathrm{f}\)). How the algorithm executes every mixsplit step is elaborated as follows:

1.
Any mixsplit operation involves two droplets with different CFs, known as the boundary CFs (\(C_\mathrm{l}\), \(C_\mathrm{h}\)) that first get merged and then get split and dispensed into two droplets with target concentration factor \(C_\mathrm{t}\) = \(\frac{(C_\mathrm{l} + C_\mathrm{h})}{2}\).

2.
A particular concentration (i.e., \(C_\mathrm{f}\)) can be obtained from the set of available CFs (numerators varying from 1 to \(2^n\)1), where n is the maximum number of allowable steps.

3.
SIMOP creates unique mixsplit sequences involving different CFs and the selected mixsplit sequences significantly affect the side effects of the algorithm.
We designed SIMOP algorithm to accept only one user input value denoting the maximum allowable limit of mixsplit operations in the dilution process. It then moves forward to pick up all the optimal possibilities to generate all possible fractions, applying different optimizing criteria. Figure 2 illustrates how our algorithm works for an input of n = 3, computing denominator, d = \(2^n\) = \(2^3=8\). The first step starts with a mixsplit combining buffer (\(\frac{0}{8}\)) and sample (\(\frac{8}{8}\)) droplets resulting in two droplets of concentration, \(C_1\) = \(\frac{4}{8}\). Whenever a new concentration (say, \(\frac{2}{8}\)) is generated, we combine this with all the parent concentrations generated in the system so far, i.e., \(\frac{2}{8}\) is mixed with \(\frac{0}{8}\), \(\frac{4}{8}\) and \(\frac{8}{8}\) separately. This means we have a new branch of SIMOP tree with three nodes. By attaining the maximum steplimit, the algorithm halts to produce leaf nodes with target concentrations. Figure 2 shows us the leaf node values \(\frac{1}{8}\), \(\frac{3}{8}\) and \(\frac{5}{8}\) in the left branch of the tree. Moving forward, we can maintain an unique path starting from root node (say, \(\frac{0+8}{8 \times 2}\) = \(\frac{4}{8}\)) to a target fraction (any value from \(\frac{1}{8}\) to \(\frac{7}{8}\)). Further, we can check for an optimal path among different unique paths generated for a particular fraction. As shown in Fig. 2, we generate all the fractions from \(\frac{1}{8}\) to \(\frac{7}{8}\), where each of the fractions \(\frac{3}{8}\) and \(\frac{5}{8}\) were created in two different unique ways. Therefore, even in this small example, the utility of running simulation for finding better alternatives is vindicated. While computing the best possible route to each concentration, our simulation aims at minimizing four different factors,

1.
Number of steps

2.
Number of waste droplets

3.
Number of sample droplets

4.
Number of buffer droplets
A costly or rare sample might need minimization of sample usage, whereas a cheaper sample along with a costly buffer would require buffer optimization. Reducing the number of mixsplit steps as well as number of waste droplets will also benefit the process both from time and cost perspective. The SIMOP algorithm has a preprocessing step where we run a simulation algorithm and store all results beforehand in a repository, from which they can be retrieved during use. If there exist different optimal solutions to each of the four factors mentioned above, we store each solution independently. Thus, we not only ascertain optimality over all other algorithmic solutions, but we do so over all the four factors.
SIMOP Implementation Details
This algorithm starts with an user input that is the accuracy limit (x) of concentration factors. Longest path length in the mixing graph is determined by the value of x. With this limit in hand, our simulation exercise is run generating all numerators lying between 0 and \(2^x\) while we seek to optimize one or more of the factors (number of steps, sample droplets, buffer droplets, waste droplets) selected by user. Given the denominator having a value of \(2^x\), we bifurcate from the root node (with sample and buffer concentrations) extending a tree where every branch represents a separate mixsplit sequence. In addition, we branch out each path of the SIMOP tree and compare the nodes at each mixsplit level against a final repository updated dynamically during runtime.
We initiate SIMOP model with buffer (\(0\%\)) and sample (\(100\%\)) and these concentrations are treated as fractions with powers of 2 as we have the precomputed denominator, d = \(2^x\) where x = accuracy limit. Thus, if x = 3, then utilizing all three dilution steps a minimum of \(\frac{1}{8}\) = \(12.5\%\) concentration can be achieved and at the same time we can consider a maximum of \(\frac{7}{8}\) = \(87.5\%\). We proceed by mixing the available CFs at each dilution step. We start with 0 and \(2^x\) and then add on those fractions created in intermediate steps to expand the list of generated concentrations. Progressing this way, an exhaustive search is carried out carefully avoiding repetitions to determine all feasible approaches and establish the best one.
Whenever a new concentration is generated, we compare it against the present optimal sequences stored in the output datastore for each of the different optimization factors. If there is no sequence stored yet, all the four locations for storing sequences for that concentration are immediately updated with the current method. If a previous optimal path is stored, we perform comparisons of that against the current process and replace if necessary. In this way, the final datastore is updated dynamically. The replacement algorithm does not simply replace the stored sequence on the basis of a single criterion, but evaluates the new sequence generated against the stored one along a chain of criteria as illustrated below. For example, if we were to be optimizing waste and we find another process with the same waste droplets as the one stored, we would then compare it against steps, then sample then buffer. The different orderings of comparison criteria we used for the four parameters are the following—

1.
steps \(\triangleright \) steps>waste>sample>buffer

2.
waste \(\triangleright \) waste>steps>sample>buffer

3.
sample \(\triangleright \) sample>waste>steps>buffer

4.
buffer \(\triangleright \) buffer>waste>steps>sample
This ordering can always be customized by the user as per the required circumstances. In the worst case, the datastore size = \(2^{x  1} \times m \times L\), where x = accuracy level, m = optimizing factors, L = maximum length of mixsplit sequence. However, after the initial datastore is created, a second pass is performed to remove the redundant copies of sequences, for example, if all four optimized sequences for a certain target concentration are the same, we need to retain only one copy. This greatly reduces the number of sequences to be stored.
We observed that the number of steps for SIMOP always remains same with BS as well as DMRW methods. Hence, we will not have a different process to optimize steps. That leaves us with \(m = 3\). In addition, after the second pass, when we get all the spurious copies removed, the space requirement comes down considerably. Table 1 shows us the space saved for different inputs of maximum steps.
Experimental Observation of SIMOP Simulation
To exhibit that our SIMOP simulation is indeed an improvement over existing sample dilution algorithms such as DMRW, BS, IDMA, REMIA, and GORMA, we have compared their results with ours and found significant savings concerning sample, buffer and waste droplets.
For example, when the number of steps = 10 (illustrated in column1 of Table 2), SIMOP generated only 19 cases out of \(2^{10}\) = 1024, where the number of waste droplets was the same as that of BS method, while in each of the rest there was an improvement. Similarly, for REMIA our method has shown improvement in waste generation for all fractions except in 27 cases (shown in column5 of the same Table 2). Our method has shown a decrement in the amount of waste generated in \(97\%\) cases as compared to REMIA, and in \(98\%\) cases when compared to the BS method (see Table 2).
Waste reduction is also found to be significant when compared to DMRW (\(85\%\) cases). Column2 of Table 3 illustrates that SIMOP and DMRW have 151 cases where waste droplet generation was the same and for the rest, SIMOP performed better. On a similar note, for \(79\%\) cases, SIMOP performed better than IDMA in waste generation (column5 of the same Table 3). Consider column8 of Table 3. For \(34\%\) cases, SIMOP performed better than GORMA in waste droplet generation. Figure 3 presents the percentage of fractions, where an improvement of at least one droplet was obtained using SIMOP over BS and DMRW, where the maximum number of steps used were 5, 9, 10 and 11, respectively. The number of steps, however, did not show any improvement, remaining same for all three methods (BS, DMRW, SIMOP).
Table 4 gives a glimpse of the improvement obtained by the proposed method considering average droplet reduction per fraction when compared to five balanced (1 : 1) dilution schemes (BS, DMRW, IDMA, REMIA and GORMA).
For the BS method, waste optimization can be as high as 8 drops for some fractions. As we can see, there are many cases where the waste, sample, or buffer is optimized greatly than that found by the BS or DMRW method. Moreover, there have been no cases where our algorithm was worse compared to the given algorithms (as expected).
Moreover, it is very easy to set the optimization goal using multiple criteria with the priority amongst them being strictly specified. For example, when \(C_\mathrm{t}\) = \(\frac{313}{1024}\), SIMOP produces two separate optimal paths, one aim to optimize buffer (as illustrated in Fig. 4a)) and the other focus on sample optimization (shown in Fig. 4b)) against those supplied by BS and DMRW. The reactant cost is much improved for SIMOP (as shown in Table 5). Still, we have \(\frac{799}{1024}\) that generated a unique mixing scheme optimizing all the factors simultaneously (already reflected in Fig. 1). This demonstrates the real power of using simulation.
To establish the efficacy of SIMOP, the algorithm is compared with the six aforementioned singletarget sample preparation methods—BS, DMRW, IDMA, REMIA, GORMA and WD. We adopt the same experimental environment setup used in a number of earlier works [15]−[31], where various target concentrations are in the range between \(\frac{1}{1024}\) and \(\frac{1023}{1024}\), i.e., the number of fractional bits is set to ten (for BS, DMRW, IDMA, REMIA, GORMA and SIMOP following (1 : 1) balanced mix model). Note that when comparing for sample usage, in case of SIMOP, we consider the 1023 sequences (out of a total of 1411) that optimize the usage of sample droplets and likewise for buffer usage and waste. Besides, we have included WD that uses composite dilution scheme using (1 : 1) and (1 : 2) mixsplits, and the target concentration factors are in the range of \(\frac{1}{1152}\)–\(\frac{1151}{1152}\). We analyze the experimental results reported in Table 6 as follows:

The results noted in Table 6 indicate that SIMOP as expected performs better than all the existing five balanced dilution techniques (BS, DMRW, IDMA, REMIA, and GORMA) in waste, sample, and buffer minimization. ^{Footnote 1}

Considering waste (W) generation, the average waste per fraction for BS is the maximum, \(W_\mathrm{BS}\) = 8.01 and the average waste generated per fraction using SIMOP is the minimum, \(W_\mathrm{SIMOP}\) = 2.59. The percentage savings in wastage can be computed as \(\frac{W_\mathrm{BS}W_\mathrm{SIMOP}}{W_\mathrm{BS}} \times 100\%\) = \(\frac{8.012.59}{8.01} \times 100\%\) = \(67.67\%\). This means SIMOP achieves \(67.67\%\) savings over BS, \(48.41\%\) savings over DMRW, \(44.66\%\) savings over IDMA, \(60.46\%\) and \(32.38\%\) savings over REMIA and GORMA, respectively.

When compared to the same set of algorithms (BS, DMRW, IDMA, REMIA, and GORMA), SIMOP performs better in terms of the buffer (B) usage. Average buffer usage per SIMOP fraction is the minimum 2.3. In terms of \(\%\) savings in buffer consumption, experimental results reveal that SIMOP is better than BS (\(\frac{B_\mathrm{BS}B_\mathrm{SIMOP}}{B_\mathrm{BS}} \times 100\%\) = \(\frac{5.012.3}{5.01} \times 100\%\) = \(54.09\%\)), DMRW (\(34.47\%\)), IDMA (\(31.34\%\)), REMIA (\(62.48\%\)), and GORMA (\(12.55\%\)).

SIMOP also consumes less amount of sample, i.e., average sample (S) usage per fraction, \(S_\mathrm{SIMOP}\) = 2.3, minimum among all the seven sample dilution schemes. We have SIMOP performing close to REMIA, where average sample usage per fraction, \(S_\mathrm{REMIA}\) = 2.42. The proposed scheme is achieving only \(4.99\%\) savings over REMIA in sample usage but performing much better in buffer consumption (\(62.48\%\) savings).

Both BS and SIMOP lead to a minimal number of dilution operations (\(\mathrm{DS}_\mathrm{BS}\) = \(\mathrm{DS}_\mathrm{SIMOP}\) = 9.01) where DS indicates average dilution steps per fraction.
The above analysis concludes that SIMOP not only performs well in waste, sample, and buffer minimization but also achieves optimality in dilution operation count.
We have compared the performance of SIMOP simulation with other balanced schemes (BS, DMRW, IDMA, REMIA, and GORMA) and one weighted scheme (WD). Consider Fig. 5. We present the corresponding frequency of fractions against the number of waste droplets generated for all seven dilution algorithms. Note that, for SIMOP, the frequency of target concentrations are more for low wastecounts (2, 3). Way forward in this paper, wastecount is referred to as Wcount. As for example, for 326 target fractions, SIMOP has generated Wcount = 2, and for 482 target fractions, the Wcount = 3. When the number of waste droplets is \(\le 3\), the frequency of target concentrations for those classes has been the highest for SIMOP, and the frequency values decrease monotonically as we proceed along the xaxis in Fig. 5. In addition, note that for any value of Wcount from 6 to 9, the number of target concentrations for SIMOP is the lowest among all the different methods. For example, for Wcount = 9, we get a maximum of 512 target fractions affected for BS and in contrast we get a minimum of 2 target fractions for SIMOP.
The bar diagrams shown in Fig. 6 illustrate the performance of seven sample dilution algorithms SIMOP, BS, DMRW, IDMA, REMIA, GORMA, and WD with respect to the quantity of usage of sample droplets. For 1023 targets, the frequency of target concentrations is reported against the count of sample droplets being used. We consider SIMOP, DMRW, and BS bars to report the target frequencies in Table 7 and Fig. 7. Note that a maximum of 344 SIMOP target concentration factors needs only one sample droplet compared to 215 DMRW fractions and only 10 BS target fractions that require the same volume of sample droplet as input. SIMOP is performing better than DMRW and BS schemes for sample droplet counts 2, 3, 4, 5, 6, and 7 also.
Similarly, Fig. 8 shows the performance of all seven algorithms generating target concentration factors captured against buffer droplet count. As a typical example, Table 8 compares SIMOP with IDMA and REMIA target fractions against buffer utilization. SIMOP generates a maximum of 344 targets that need only 1 buffer droplet compared to 214 for IDMA and only 10 for REMIA as shown in Fig. 7. From the Fig. 8, it is quite obvious that the SIMOP scheme is far better than even the next best, which can be either DMRW or IDMA.
Note that in all three figures, Figs. 5, 6, and 8, SIMOP bars are becoming much shorter towards the right in each of the plots, i.e., for each optimization criterion, the values of the SIMOP scheme are relatively smaller compared to the other balanced schemes on an overall basis.
One other interesting thing to note is that targets \(C_\mathrm{t}\) = 311 and complement \(C_{\mathrm{t}'}\) = 713 are symmetrical w.r.t sample and buffer consumption. Consider the dilution sequence 1 shown in Fig. 9. Target \(C_\mathrm{t}\) = 311 is generated with 1 sample and 4 buffer droplets, whereas, \(C_{\mathrm{t}'}\) = 713 needs just the opposite, 4 sample and 1 buffer droplets. Waste generation is the same, 3 for both cases.
Figure 10 illustrates dilution sequence 2 to generate the same (\(C_\mathrm{t}\), \(C_{\mathrm{t}'}\)) as (311, 713) requiring (sample, buffer) as (2, 3) and (3, 2) respectively. This analysis motivated us to check two sets of SIMOP targets, (1) Set1: 1511; (2) Set2: complements of Set1, 5131023. These two sets are symmetrical around waste droplet generation in the dilution process as displayed in Fig. 11. It is not difficult to theoretically show that the generation of the complementary fraction f’ is always possible using a mirror imagelike sequence generated from the original sequence of a given fraction f. In addition, our SIMOP experiment clearly demonstrates the empirical result that such mirrorimage sequences of optimal sequences are also optimal for generation of complementary sequences.
We ran the simulation on an Intel Corei5 2.5 GHz processor with 8 GB of RAM and the running times for different accuracy limits (i.e., inputs) are reported in Table 9. The concentrations that are currently considered in the presentday labs deal with 10 dilution steps, whereas we went up to 11 steps. Our 10step simulation was completed in about 11 min, which is very encouraging considering the fact that the simulation will be run only once as a preprocessing step, and after that, during real operation, all results will be available via a table lookup in negligible time [28, 29]. Even the process with 11 mixsplit steps took a little over 2 h, which is quite acceptable.
Error Optimization Scheme
Prior Art and Motivation for Error Optimization Scheme in SIMOP
To achieve a target CF as discussed in previous sections, we follow a (1 : 1) balanced mixsplit model to mix two oneunit volume droplets followed by a subsequent split to generate two daughter droplets [22, 23]. However, a major source of error in sample dilutions is due to an unbalanced or erroneous split, where two unequalvolume droplets are generated during a dispense operation. Considering the prior literature for error recovery on DMFBs, we have several avenues to look forward as reported below:

1.
Addressing error recovery in DMFBs, Alistar et al. [3] proposed an online synthesis that ascertains appropriate recovery actions when errors are detected in the droplet volumes.

2.
We note biochip synthesis and dynamic error recovery proposed by Hsieh et al. [14]. To ensure the correctness of intermediate mixsplits and efficient error recovery during sample preparation, an optimization algorithm and associated chip design strategy have been illustrated.

3.
To address volumetric errors that occur during droplet manipulation onchip, an extensive survey on recently proposed techniques has been made and an experimental case study was presented to demonstrate the application of the errorrecovery scheme to reallife biochips [16].

4.
Luo et al. [18] presented a physicalaware system reconfiguration technique that uses sensor data at intermediate checkpoints to dynamically reconfigure the biochip.

5.
Luo et al. [19] proposed a hardwareassisted method that can be implemented in real time on a fieldprogrammable gate array (FPGA) to address error recovery in cyberphysical digitalmicrofluidic biochips using a compact dictionary.

6.
An innovative dropletrouting method is proposed [34] that avoids crosscontamination in the optimization of droplet flow paths.

7.
Zhao et al. [35] presented a synthesis method incorporating control paths and an errorrecovery mechanism in the design of a digital microfluidic labonchip.

8.
By exploiting MEDAspecific advances in droplet sensing, a novel errorrecovery technique [17] was presented to dynamically reconfigure the biochip using realtime data provided by onchip sensors.

9.
The authors aimed to propose the first sample preparation method [33] that exploits the MEDAspecific advantages of finegrained control of droplet sizes and realtime droplet sensing to effectively handle latency for error detection during sample preparation.
Based on the abovecited cases, we extend BS scheme and our SIMOP procedure to analyze and investigate sample preparation in the presence of volumetric errors (VE) [4, 21,22,23]. VE is caused by: (1) droplet volume variation in mixsplit operations (2) inadequate mixing, and (3) the presence of insoluble chemicals or contaminants in the droplets.
We propose an error optimization model [14] where we consider volumetric errors applied to the generated droplets during the splitting phase. In (1 : 1) mixsplit operation, we assume that splitting of a larger droplet of 2X unit volume causes two resulting droplets of size (\(1X + \epsilon \)) and (\(1X  \epsilon \)) where the fraction (\(\epsilon \)) denotes a volumetric error caused by unbalanced split operation [4, 23].
Note that the desired target CF (\(C_\mathrm{t}\)) may get altered due to an unbalanced split on any step of the reaction path causing an error. We have performed exhaustive simulation exercise on all numerators generated for the following number of mixsplit steps, \(n~=~4,5,9\) and 10 and applied errors—\(1\%, 2\%, 3\%, 5\%, 7\%\) and \(10\%\). Note that there is an inherent limitation on the accuracy of the target achievable, which is determined by the value of n. For example, when \(n~=~10\), at most 10 mixsplit steps are needed by both BS and SIMOP algorithms and for a choice of n an accuracy of more than \(\frac{1}{2^{n+1}}\) cannot be achieved (e.g., when n = 10, the minimum accuracy achievable is \(\frac{1}{2\times (2^{10})}\) = \(\frac{1}{2048}\) = 0.0004883).
Consider Fig. 12. With a small example, we present the propagation of unbalanced split error (\(\in ~=~5\%\)) through the reaction path to generate \(C_\mathrm{t}\) = \(\frac{9}{32}\) with \(n = 5\) mixsplits. Note that the volume error (VE) is applied on step \(S_4\) due to an erroneous split of the waste droplet at step \(S_3\). With VE, we compute the altered \(C_\mathrm{intermediate}\) = \(C_4^{'}\) = \(\frac{C_3 \times (1+\in ) + C_1 \times 1}{(1+\in ) + 1}\) = \(\frac{4 \times 1.05 + 16 \times 1}{1.05 + 1}\) = 9.85366. The droplet volume is also changed to \(V_4^{'}\) = \(\frac{(1+\in ) + 1}{2}\) = \(\frac{2 + .05}{2}\) = 1.025. At the final step, we carry out the mixsplit operation using (\(C_4^{'}\), \(V_4^{'}\)) and (\(C_2\), \(V_2\)) to generate final CF = \(C_{\rm f}\) = \(\frac{C_4^{'} \times V_4^{'} + C_2 \times V_2}{V_4^{'} + V_2}\) = \(\frac{9.85366 \times 1.025 + 8 \times 1}{1.025 + 1}\) = 8.93827. Thus, a \(5\%\) volume error that originates at step \(S_3\) successively affects steps \(S_4\) and \(S_5\). This ultimately alters the final \(C_f\) with target error = \(\frac{C_\mathrm{f}  C_\mathrm{t}}{2^5}\) \(< \frac{1}{2^6}\).
Definition 1
(Critical and noncritical errors) If the error of a final concentration induced by an unbalanced split is less than the inherent accuracy level that is used in the dilution algorithm then the error is called noncritical (nonCE), and may be ignored. An error whose magnitude is more than the inherent accuracy level is then considered as critical (CE) [21,22,23]. It is clear that only critical errors visibly affect the \(C_t\) and when they are detected by sensors, corrective measures are needed to restore the desired concentration factor of the diluted sample. In other way, when applying volume errors (say, \(\epsilon \) = \(\pm 0.01\), \(\pm 0.02\), \(\pm 0.03\), \(\pm 0.05, \pm 0.07\) or \(\pm 0.1\)) at any dilution step on the reaction path, if the final CF (\(C_\mathrm{f}\)) generated at the final dilution step gives the deviation, say x = \(C_\mathrm{f}  C_\mathrm{t} \ge 0.5\), then the generated CF is marked as CE. Otherwise, it is denoted as nonCE or Good.
Our simulation experiments reveal that there are many CE and nonCE split errors on the reaction path for a given target CF. Figure 13 shows the criticality of operations at different mixsplit steps for \(C_\mathrm{t}\) = \(\frac{341}{1024}\) with \(n~=~10\) mixsplit operations running both BS and SIMOP applying VE = \(+7\%\). Consumption of input sample and buffer for BS is computed as 5 droplets each. SIMOP has a much lower sample and buffer intake as 1 and 2 drops, respectively. We have only 1 waste droplet generated by SIMOP as compared to a large number of waste droplets (9) out of BS.
In Fig. 13, the final CFs generated out of BS and SIMOP schemes are plotted along yaxis against steps S1 to S9 shown along xaxis. Note that BS generated 9 waste droplets while preparing this target concentration and in 3 cases out of 9, where the volume error was introduced in the dilution steps S5, S7, and S9 because of erratic splits in steps S4, S6, and S8, respectively, the final concentration was critically affected with deviation \(x > 0.5\) (BS final CFs are 342.0312 (S5), 345.1852 (S7), and 357.2381 (S9)). In contrast, SIMOP generated the same \(C_\mathrm{t}\) = \(\frac{341}{1024}\) with a nonCE final CF having deviation = 0.0476, i.e., error = \(\frac{0.0476}{1024} < \frac{1}{2048}\).
Experimental Results
The performance of both BS and SIMOP methods as discussed above led us to inquire about the CE and nonCE deviations of all the CFs. Consider Table 10 showing the lists of SIMOP and BS targets critically deviated by volume errors. The value of the deviation x (i.e., \(C_\mathrm{f}  C_\mathrm{t}\)) is first compared against the threshold value of 0.5, crossing which the target concentration becomes nearer to the previous or the next possible fraction that can be generated. Clearly, this should be the natural choice for noise margin and using this threshold out of 1023 targets, we found that a large number of targets crossed CE say 993 for SIMOP and 1013 for BS with \(7\%\) VE (shown in 2nd and 3rd columns of Table 10). Therefore, we have also experimented by relaxing the deviation check by \(x \ge 1.0\), followed by \(x \ge 1.5\), and finally \(x \ge 2.0\).
Relaxing the threshold value from 0.5 to 1.0 improved the result as reported in 4th and 5th columns of Table 10.
We get 190 CEaffected SIMOP targets compared to 770 BS targets given the deviation, \(x \ge 1.5\) (reported in \(1\mathrm{st}\) row, \(6\mathrm{th}\) and \(7\mathrm{th}\) columns). As expected, the result is further improved when the final CFs are tested with \(x \ge 2.0\), in SIMOP 56 targets while in BS 613 targets were marked as critical (\(1\mathrm{st}\) row, \(8\mathrm{th}\) and \(9\mathrm{th}\) columns). Figure 14 pictorially represents the data reported in \(2\mathrm{nd}\) to \(9\mathrm{th}\) columns of Table 10, where the performance of SIMOP and BS targets are illustrated with the application of VE and critical deviation limits 0.5, 1.0, 1.5 and 2.0.
Apart from tracking waste onestepatatime, we thought if there exists any other order of introducing volume error during mixsplit operations involving waste droplets. We aim to present a rigorous analysis of SIMOP sequences (for 3 different cases—max number of mixsplit steps = 8, 9 or 10) to perform the following operations:

1.
Group the sequences based on generated waste droplets (W), say, 1W, 2W, 3W etc.

2.
Prepare combinations of \(\pm 3\%\) and \(\pm 5\%\) VE depending on 2W and 3W. Consider Table 11 showing \(2^2 = 4\) sets for 2W and \(2^3 = 8\) sets for 3W when \(\pm 3\%\) error is considered.

3.
Table 11 shows combinations of errors injected due to an unbalanced split applied on waste droplets located in the reaction path of SIMOP sequences. For each of the sequences with 2W or 3W, the SIMOP algorithm will pass through all of the multiple combinations stated in the table. Thus, we have two different ways to simulate the application of volume errors,

Scheme 1: Errors introduced onestepatatime or onesteperror

Scheme 2: Errors introduced at multiplesteps or multisteperror

OneStepErrors on Reaction Path: Performance of Multiple SIMOP Sequences
We are considering two targets, C(t) = \(\frac{343}{1024}\) and complement \(C'\)(t) = \(\frac{681}{1024}\) for \(n~=~10\) steps. On application of \(+10\%\) unbalanced split error onestepatatime, we try to find out how multiple sequences (shown in Table 12) of these two target CFs perform.
For both \(\frac{343}{1024}\) and \(\frac{681}{1024}\), sequence1 is generating 3 waste droplets where sequence2 is giving a higher waste value of 5 (reported in 4th and 7th columns of Table 12). Considering sample and buffer consumption for both the target numerators 343 and 681, note that sequence1 of 343 consumes 2 drops of sample and 3 drops of buffer droplets that is just the reverse in the case of sequence1 of 681 (shown in 2nd and 3rd columns of Table 12).
It is observed that for sequence2, both the target numerators 343 and 681 are generating final CFs that are affected by critical errors and the critical deviations are just the opposite as reflected in Fig. 15. Thus, on the reaction path, 5 waste droplets are located at 5 different dilution steps and we have applied \(+10\%\) VE onestepatatime to produce 5 different final CFs. Note that Fig. 15 displays a broad deviation range (\( 20\) to 20) on yaxis. Deviations for both 343 and 681 are plotted along yaxis, whereas the xaxis shows steps S5–S9. For step S5, if we select positive deviation 2.02181 (for 343) and corresponding negative deviation \( 2.02181\) (for 681), then the error in target CF will be given by, \(E_{0.1}\) = \(\frac{2.02181}{1024}\) = 0.00197 \(> \frac{2}{1024}\), i.e., critical error.
We extended our analysis on target CFs \(\frac{311}{1024}\) and \(\frac{713}{1024}\) each with 2 dilution sequences as shown in Figs. 9 and 10. Consider Fig. 16 showing the performance of 2 sequences of \(\frac{311}{1024}\) with the application of \(10\%\) volume errors at mixsplit steps S7, S8 and S9. Two sequences are generating final CFs that are critical. The critical deviations for sequence1 are just the opposite of the deviations computed for sequence2.
We further applied the same \(10\%\) VE on \(\frac{713}{1024}\) with 2 sequences. Note that in Table 13, 3rd column lists final CFs out of which 3 are critical shown in steps S2, S8, and S9. Whereas sequence2 of target numerator 713 is generating another set of final CFs and 3 of this set are critical, all tracked in the higher steps, S7, S8, and S9. All these 6 critical final CFs of \(\frac{713}{1024}\) are located in different places in the reaction path and the performance is illustrated in Fig. 17. It is observed that multiple dilution sequences of both C(t) = \(\frac{311}{1024}\) and complement \(C'\)(t) = \(\frac{713}{1024}\) (each with 3 wastes) are generating the same number of critical final CFs with \(10\%\) VE. But Figs. 16 and 17 are displaying different performance plots involving critical deviations and final CFs because of varied positions of waste droplets in dilution sequences of these target fractions.
MultistepErrors on SIMOP Sequences
We have considered a group of 326 SIMOP sequences each of which has 2 waste droplets (2W) in the dilution sequence. Our goal is to apply combinations of \(\pm 3\%\) volume errors at multiple steps for 2W as stated in Table 11 (under “Experimental Results” section) and check the performance of target CFs against the said volume error.
We have selected two target CFs \(\frac{235}{1024}\) and \(\frac{777}{1024}\) each with 2W. The generation sequence of C(t) = \(\frac{235}{1024}\) has wastes on 6th and 8th steps affecting successive steps 7th and 8th, respectively with VE. For C(t) = \(\frac{777}{1024}\), steps 7 and 8 emit waste droplets and with the application of unequal split errors, higher steps 8 and 9 are getting affected. For C(t) = \(\frac{235}{1024}\), it is observed that, combinations (\(0.03,  0.03\)) and (\( 0.03, 0.03\)) are generating final CFs with noncritical deviations. But, final CFs produced by multistep application of all positive (0.03, 0.03) and all negative errors (\( 0.03,  0.03\)) can not pass the critical error (CE) check, i.e., deviation from the desired target is outside the noncritical range of (\( 0.5\) to \(+ 0.5\)).
Unlike \(\frac{235}{1024}\), multisteperror scheme is applied on \(\frac{777}{1024}\) for all four combinations ((0.03, 0.03), (\(0.03,  0.03\)), (\( 0.03, 0.03\)) and (\( 0.03,  0.03\))) resulting in 4 final CFs, all passing CEcheck. Thus, the SIMOP sequence for C(t) = \(\frac{777}{1024}\) is ‘clear’ all through the error tolerance limit and can be considered as erroroptimized.
The above observation motivated us to extend our analysis on three SIMOP targets, say, C(t) = \(\frac{247}{1024}\) (2W), C(t) = \(\frac{694}{1024}\) (3W) and C(t) = \(\frac{939}{1024}\) (3W). We have applied combinations of \(\pm 3\%\) volume error at multiple steps to reach each of these target CFs. It is noted that target \(\frac{247}{1024}\) and its complement \(\frac{777}{1024}\) both pass critical deviation tests (dev. \(\le 0.5\)) for all four combinations applying multisteperror procedure as shown in Figs. 18 and 19.
The above is not the case for fractions \(\frac{694}{1024}\) and \(\frac{939}{1024}\), both having 3 waste droplets located at 3 different dilution steps on the reaction path. Therefore, multisteperror scheme is tried for 8 combinations of \(\pm 3\%\) volume errors (Table 11) and passing CEtest for all these 8 different cases is much more challenging. Figures 20 and 21 display the performance of final CFs generated for both targets with application of \(\pm 3\%\) VE. In Fig. 20, we have 4 noncritical final CFs, 694.218 (\(0.03, 0.03,  0.03\)), 693.647 (\(0.03,  0.03, 0.03\)), 694.466 (\( 0.03, 0.03,  0.03\)), 693.895 (\( 0.03,  0.03, 0.03\)) and rest 4 are critical. Thus, out of 8 combinations, we have 4 ‘Good’ and 4 ‘Bad’ targets. Similarly, Fig. 21 shows the performance plot for \(\frac{939}{1024}\), where, we have the final CFs arranged in a staircase like manner, first 2 are critical, followed by 4 noncritical, last 2 are critical.
At the end of this simulation exercise, we were able to determine if the generated final CFs are within the acceptable range of errors, i.e., error tolerance limit of the SIMOP sequences. Critical errors occurred in one scheme, say, onestep can be nullified by the other, i.e., multistep and vice versa.
The above example is remarkable in a sense that it clearly demonstrates that the error quantum highly depends on the sequence used to generate a particular target. Hence, for a particular application, if the error tolerance is the most important aspect, which is to be focused, then the SIMOP algorithm may also be tuned keeping it as the topmost optimization criterion. We, however, plan to explore it elsewhere.
Experimental Observations
OneStepataTime Application of Volume Errors on SIMOP Sequences
Final CFs with unbalanced split errors become more critical on the higher dilution steps of SIMOP reaction path when the error is applied onestepatatime. Consider the bar diagrams in Fig. 22. For 8, 9 and 10 steps, we are obtaining similar results for SIMOP sequences, e.g., dilution step3 is the minimum for three cases, 30 (for 8steps), 18 (for 9steps) and 44 (for 10steps). Thus, minimum number of sequences have critical deviation at dilution step3 irrespective of lower denominator \(2^{8} = 256\) (step8) to higher denominators \(2^{9} = 512\), \(2^{10} = 1024\) (steps 9 and 10). On the other hand, the last but one step is having the maximum number of sequences with critical deviations, like, 101 (for step6), 109 (for step7) and 310 (for step8), irrespective of the number of steps.
Multistep Application of Volume Errors on SIMOP Sequences
The final CFs becomes almost symmetrically distributed with critical deviations by introducing combinations of unbalanced split error (e.g., \(\pm3\%\)) on multiple dilution steps carrying waste droplets. Note that Fig. 23 illustrates the 8 combinations to test the targets generated under 8, 9, and 10 steps. It is observed that for (8 Steps, 3W), we have 99 critical final CFs for all positive errors (0.03, 0.03, 0.03) and 100 critical final CFs for all negative errors (\( 0.03,  0.03,  0.03\)). Similarly, (9 Steps, 3W) displays 248 critical targets for (0.03, 0.03, 0.03) and 251 critical targets for (\( 0.03,  0.03,  0.03\)). Finally, (10 Steps, 3W) has 581 critical fractions for (0.03, 0.03, 0.03) and 588 critical fractions are captured for (\( 0.03,  0.03,  0.03\)). Each of the above cases illustrates the almost symmetrical distribution.
Conclusion
In this paper, we have presented a simulationbased algorithm to obtain a target concentration using (1 : 1) mixsplit steps, minimizing the number of buffer, sample or waste droplets as prioritized by the user. By efficiently storing the optimal mixsplit graphs needed to obtain all target concentrations, one can easily retrieve and use them whenever required using table lookup. One also has the option of retrieving a result as per the specific optimization needs, as multiple solutions optimizing different parameters are made available through SIMOP. Our simulation results can be used as benchmarks to evaluate the efficacy of a new algorithm. Besides, the proposed error optimization model aims to find out the error tolerance of SIMOP sequences concerning volumetric errors encountered in the reaction path of sample dilution.
Future work in this direction may include extending the simulationbased method for generating multiple targets. One may also try to decrease the simulation time further so that dilutions with higher resolutions can be handled.
Notes
 1.
Note that comparison of the above parameters (waste, sample and buffer) directly with WD dilution scheme will not be fair, since in the last step, the WD scheme may generate either 2 droplets or 3 droplets of output, whereas SIMOP being a balanced scheme always generates 2 droplets of output. Therefore, to make an apple to apple comparison between SIMOP and WD schemes, we should ideally normalize the average values of the waste, sample and buffer by dividing each of them with 2 for SIMOP and a factor x for WD, where x is the number of output droplets averaged over all the fractions (\(2~<~x~<~3\)). Hence, though we have also presented the values for WD in the tables, we have not directly compared them with SIMOP values.
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A preliminary version of this paper appeared in Proc. DISCOVER, Page No. \(16\) (2016) [8].
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Bera, N., Majumder, S., Das, S. et al. SIMOP: A SIMulationGuided OPtimization Mechanism for Sample Preparation with Digital Microfluidic Biochip. SN COMPUT. SCI. 2, 84 (2021). https://doi.org/10.1007/s4297902100506x
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Keywords
 Algorithmic microfluidics
 Waste optimization
 Simulation
 Error correction