- Email: [email protected]
An adaptive neural-fuzzy approach for microfluidic droplet size prediction
Microelectronics Journal 78 (2018) 73–80
Contents lists available at ScienceDirect
Microelectronics Journal journal homepage: www.elsevier.com/locate/mejo
An adaptive neural-fuzzy approach for microfluidic droplet size prediction Ali Lashkaripour a , Masoud Goharimanesh b , Ali Abouei Mehrizi c , Douglas Densmore d,* a
Boston University, Biomedical Engineering Department, 44 Cummington Mall, Boston, USA Mechanical Engineering Department, Ferdowsi University of Mashhad, Mashhad, Iran c Department of Life Sciences Engineering, Faculty of New Sciences and Technologies, University of Tehran, Tehran, Iran d Boston University, Department of Electrical and Computer Engineering, 8 Saint Mary’s Street, Boston, USA b
A R T I C L E
I N F O
Keywords: Microfluidics Multiphase flow Droplet size Neural networks Fuzzy systems ANFIS
A B S T R A C T
Droplet-based microfluidics is capable of being a superior platform for biological and biomedical applications due to their higher accuracy, throughput and sensitivity, low reagent consumption and fast reaction time. However, the complex and nonlinear behavior of multiphase microfluidic devices and several parameters affecting droplet size simultaneously necessitates costly design iterations to generate droplets with the radius of interest for a certain application. To address this, we exploit soft computing methods to bridge fuzzy systems and neural networks in order to build an adaptive neural-fuzzy inference system (ANFIS) that predicts the droplet size generated in a flow-focusing microfluidic device based on six major parameters, that includes geometry, flow, and fluid properties. This model shows a significant accuracy with a coefficient of determination of 0.96 when compared to the observed data points. Once the ANFIS model is built and verified, we use it to study the effect of each input parameter on droplet size which is challenging and/or expensive to determine experimentally.
1. Introduction Microfluidics has provided new opportunities and accelerated the progress in several fields including chemistry, medicine, pharmaceutical and biomedical engineering [1–4]. Micrometer-scale feature size, high surface to volume ratio and predictable laminar flow inherent to microfluidic devices provide a unique environment for researchers in the field of life sciences to explore the novel physics that governs the field [5,6]. Droplet-based microfluidic devices offer numerous advantages over continuous-flow microfluidics, such as accurate volume and concentration control, high throughput, high sensitivity, low sample and reagent consumption and low thermal mass, which lead to faster, cheaper and more accurate results [7,8]. However, microfluidic devices are still not widely used in life science laboratories, as the community would have expected a decade ago [5,9]. High barrier of entry to the fabrication process, the complexity of the governing physics and costly and time-consuming device iterations to reach a desired performance, have kept the field of microfluidics from being extensively deployed by most of the research groups in the field of life sciences. Some studies have addressed the fabrication cost issues by proposing cost-efficient substitutes to photo-lithography [10,11]. Nonetheless, designing a droplet generator is heavily reliant on sev-
eral design iterations and experience of the designer to perform as required for a certain application. This is due to the fact that there are several parameters acting simultaneously which determine the droplet size [12]. To clarify the complex governing physics of droplet generation, some studies explained droplet generation with dimensionless numbers through a number of simplifying assumptions [13]. Scaling laws were also suggested for T-junction droplet generation [14], however, weak control over droplet size and undefined droplet break-up point led researchers to adopt flow-focusing droplet generators as the standard method of microfluidic droplet production [15]. However, the complexity of the governing physics in flow-focusing devices has prevented accurate prediction of droplet size through scaling laws [16]. Therefore, predictive models which explain the role of each parameter on the performance of the device and approximate the droplet size based on given inputs can be extremely useful. These models are capable of clarifying the effects of the parameters that are hard to determine experimentally. Moreover, they can play a significant role in reducing the development time of droplet-based microfluidic devices. Soft computing methods such as fuzzy logic and neural networks can be used to make predictions of systems behavior and performance [17]. Fuzzy systems are capable of converting logical statements to math-
* Corresponding author. E-mail address: [email protected] (D. Densmore). https://doi.org/10.1016/j.mejo.2018.05.018 Received 15 December 2017; Received in revised form 9 May 2018; Accepted 29 May 2018 Available online XXX 0026-2692/© 2018 Elsevier Ltd. All rights reserved.
A. Lashkaripour et al.
Microelectronics Journal 78 (2018) 73–80
Fig. 1. The overall flow of building an ANFIS predictive model. In this study, we identified six effective parameters in determining droplet size in microfluidic flow-focusing droplet generation. We built a dataset of input-output relations using an experimentally verified numerical model. Using this dataset we trained and verified our proposed ANFIS model. Through the ANFIS model we studied the effects of each parameter on droplet size which is difficult and/or expensive to carry out experimentally.
ematical equations, thus, serving as a tool for simplifying the representation and modeling of complex systems [18]. More importantly, fuzzy logic is capable of grouping and clustering data to several categories. However, fuzzy systems are unable to learn from data and evolve to become more accurate overtime [19]. On the contrary, artificial neural networks are able to learn from data to accurately capture input-output relations. The ability of neural networks in discovering the nonlinear relations between inputs and output and establishing a complex dynamic model of these data is significant [20,21]. However, when dealing with a system that behaves differently depending on the state of the system, neural networks will lose their accuracy, unless, multiple neural networks are proposed for each state of the system. To avoid this, and exploit the clustering power of fuzzy systems and learning capabilities of neural networks, adaptive neural-fuzzy inference systems (ANFIS) were introduced [22]. ANFIS models enable researchers to match any given input-output relation regardless of its complexity or nonlinearity, even if the system has multiple modes. In this paper, we present an accurate predictive ANFIS model that converts six major inputs of a microfluidic droplet generator to an output droplet radius. Once we verify the accuracy of the ANFIS model with an observed dataset, we investigate how changes in input parameters affect the droplet size to further clarify the governing physics of microfluidic flow-focusing droplet generation, as shown in Fig. 1. Also, this model can be further utilized to reduce the number of costly design iterations required for developing microfluidic droplet-based devices. The remainder of this study is as follows. In Section 2 and 3, we introduce microfluidic flow-focusing droplet generation and its design space that we used to build an input-output dataset. In Section 4, we build and train the proposed ANFIS model, also, we introduce the underlying fuzzy membership functions for each input parameter. The accuracy of the ANFIS model is verified in Section 5, by comparing the data predicted by ANFIS to the dataset derived from the numerical model. Finally, we use this verified model to study the effects of each parameter on droplet radius.
Fig. 2. Microfluidic droplet generation can be achieved by flowing an aqueous and a non-aqueous phase through a narrow channel called orifice. (a) The schematic and the flow direction of water and oil in a flow-focusing geometry. (b) A numerical model of microfluidic droplet generation in COMSOL Multiphysics environment (c) An experimental snapshot of microfluidic droplet generation, using Mineral oil as the non-aqueous phase and DI water as the aqueous phase.
channel called orifice as shown in Fig. 2. The geometry of the microfluidic flow-focusing device is adopted from Ref. [15]. Due to the presence of a nozzle, which creates a unique velocity field, this geometry produces monodispersed droplets with a superior control over droplet size and its breaking point. Normally, to characterize how variations in droplet generation parameters affect the droplet size, one would have to build numerous different devices, testing each device at different flow conditions and fluid properties. This process can be very expensive and time-consuming. Moreover, although some parameters such as geometry, oil, and water flow rate can be varied and studied experimentally, the effects of other parameters such as surface tension, oil viscosity, and density are hard to study experimentally due to the limited number of available water and oil combinations. Therefore, to clarify the impact of these parameters on droplet size, a numerical model of microfluidic droplet generation was developed and veri-
2. Microfluidic droplet generation Microfluidic flow-focusing droplet generation can be achieved by flowing an aqueous phase and a non-aqueous fluid through a narrow 74
A. Lashkaripour et al.
Microelectronics Journal 78 (2018) 73–80
fied experimentally [23]. Numerical studies of multiphase flows such as droplet generation are computationally intensive and time-consuming. As a result, studying a large design space in this field would require a considerable amount of computation power, time and resources not available to most. Therefore, predictive models built based on a small number of numerical simulations that can predict the system’s performance for a wide range of inputs are of major importance and can replace large-scale numerical simulations. As demonstrated in Ref. [24], flow-focusing droplet generation behaves differently for droplets larger than 5 μm and droplets smaller than 5 μm in radius. As a result, neural networks alone cannot capture systems behavior accurately, without proposing two separate neural network models. This necessitates a need for an ANFIS model to predict droplet radius for all ranges of size.
Table 1 The range of six major parameters considered in this study that affects droplet size. For each parameter four equally-spaced values (levels) are considered between the upper- and lower-bound to build the input-output dataset, using the verified numerical model. Parameter
3. Design space
Qc Qw
Lower-bound
Upper-bound
Orifice width Oil viscosity Surface tension Oil velocity Water velocity Oil density
(μm) (mPa.s) (N/m) (m/s) (m/s) (kg/m3 )
3 2 0.003 0.02604 0.001736 700
48 41 0.02175 0.10416 0.009548 1900
4. ANFIS predictive model As shown in Fig. 1, to build an ANFIS model that predicts droplet size based on the inputs of a microfluidic droplet generator, the first step is to build a dataset of observed data points. As explained in the previous sections, the built input-output dataset includes parameters including water velocity, surface tension, oil velocity, oil viscosity, oil density and orifice size as inputs and droplet radius as the output. Fig. 3 shows the structure of our proposed ANFIS model, and its input-output relation through a black box, which converts the aforementioned inputs of the microfluidic device to droplet radius. The first layer of the ANFIS model is the inputs of a microfluidic droplet generator and the middle layers or the black box section is where the rules are generated
(1)
where 𝜇c is oil viscosity, uc is oil velocity and 𝜎 is surface tension.
Λ=
Unit
To explore and characterize the design space with the minimum number of experiments, we used an orthogonal array of inputs (i.e., response surface method [28]) using Taguchi method [29]. Through Taguchi method, to build an orthogonal array of six parameters each having four levels, only thirty-two experiments are needed, whereas, a full-factorial design of experiments requires 4096 (i.e., 46 ) experiments. Having the input-output values of these thirty-two experiments through an experimentally validated numerical model, alongside with nineteen experimentally obtained input-output data-points used to verify our numerical model a total number of 51 data-points that relates the values of six inputs to a droplet radius output were acquired.
As shown previously, out of eight parameters defining the geometry, flow properties, and fluid properties of a microfluidic flow-focusing droplet generator, only six of them affect droplet size [24]. We varied the range of these major parameters to represent real-world applications. For each parameter, four different values (i.e., levels) were considered to be equally spaced within a certain range, as given in Table 1. The orifice width was varied to include both fabrication limits and previous studies [15,25]. Oil viscosity and density were varied to cover different oils commonly used in microfluidic droplet generation (i.e., FC-40, Silicone oil, and Oleic acid oil) [15,26,27]. Surface tension was varied to include the surface tension between water and the aforementioned oils. The two major dimensionless numbers that affect droplet size are Capillary number (Ca.) as given in Eq. (1), and flow rate ratio (Λ), as given in Eq. (2).
𝜇u Ca = c c 𝜎
Range
Name
(2)
where Qc is the flow rate of oil and Qw is the flow rate of water. Oil and water velocities were chosen to cover a wide range of Capillary numbers (i.e., 0.01–0.1) and flow-rate ratios (i.e., 2–60), which represent common conditions in microfluidic flow-focusing droplet generation.
Fig. 3. The proposed ANFIS model for microfluidic droplet radius prediction. This model has six inputs and five layers where the input will be fuzzified, weighted, interconnected, defuzzified and finally aggregated to generate a single output. In this model, the user can enter the input values and through a black box the output will be predicted. 75
A. Lashkaripour et al.
Microelectronics Journal 78 (2018) 73–80
Fig. 5. Accuracy verification of the ANFIS model with the existing dataset. The predicted droplet radius using ANFIS model shows an R2 of 0.96 with the observed data points which clarifies its significant power in predicting microfluidic droplet size.
using neural networks. In the first layer, each input (node) is associated with an adaptive node through a unique function, 𝜇(i), called a membership function (MF). In this study, a Sugeno model [30] alongside Takagi–Sugeno type fuzzy “If-Then” rules [31] are exploited as given in Eq. (3). g1 = p1 a + q1 b + r1 c + s1 d + t1 e + u1 f + v
(3)
where, membership function for input 1 is a, for input 2 is b, for input 3 is c, for input 4 is d, for input 5 is e and for input 6 is f . This description of the membership functions and Eq. (3) shows the Takagi-Sugeno type If-Then rules. The second layer of the proposed ANFIS model includes the fuzzy membership functions of the six inputs. Bell-shaped membership functions with a range of zero to one are utilized as given in Eq. (4). g (x; n, p) =
( 1+
1 x −p 1
)2n
(4)
Each membership function was considered to be a third-degree function, as shown in Fig. 4. These Gaussian bell-shaped membership functions will fuzzify and categorize the inputs to three zones (i.e., the degree of the membership function). These functions cluster the input values into different zones where Y-axis demonstrates the corresponding value of each input to the designated zone. Each input can have a zero to one corresponding value for each zone, where zero demonstrates no membership and one shows a full membership. The second and third layer are connected through allocated weights for each of the membership functions defined in the second layer. As a result, the inputs that were fuzzified will be multiplied by their allocated weight and directed to the third layer as shown in Eq. (5). wi = 𝜇(i)i × 𝜇(i)i+1
(5)
where 𝜇(i) is the membership function and w is the corresponding weight of the membership function. In the third layer, each node generated in the previous layers will be interconnected through a neural
Table 2 Input parameters considered for deriving the color-maps of Figs. 6–10, except the two variables considered in each figure.
Fig. 4. Fuzzy third-degree membership functions that cluster the inputs for each parameter to three overlapping zones. Therefore, capturing possible multiple modes of the system. Fuzzy membership functions are given for six input parameters including (a) orifice size (b) water velocity (c) oil velocity (d) oil density (e) oil viscosity and (f) surface tension.
76
Parameter
Value
Unit
Orifice size Water velocity Oil velocity Oil density Oil viscosity Surface tension
25.5 0.011284 0.0651 1300 0.0215 0.012375
[μm] [m/s] [m/s] [kg/m3 ] [Pa.s] [N/m]
A. Lashkaripour et al.
Microelectronics Journal 78 (2018) 73–80
Fig. 6. Alterations of droplet radius (𝜇m) with respect to changes in water velocity and oil viscosity. While keeping other parameters according to Table 2.
Fig. 7. Variations of droplet radius (𝜇m) with respect to changes in water velocity and oil density. While keeping other parameters according to Table 2.
function will be normalized against the summation of other membership functions weights, as given in Eq. (6). These normalized weights of each input membership function represent the relative firing strength of each neuron in an actual neural network.
network. These connections are made through rules generated by the neural networks. Considering the third-degree membership functions and six input parameters, there will be a total of 36 = 729 rules generated by neural networks. In this layer, the weight of each membership
Fig. 8. Variations of droplet radius (𝜇m) with to changes in surface tension and oil velocity. While keeping other parameters according to Table 2. 77
A. Lashkaripour et al.
Microelectronics Journal 78 (2018) 73–80
Fig. 9. Variations of droplet radius (𝜇m) with respect to changes in oil viscosity and oil velocity. While keeping other parameters according to Table 2.
Fig. 10. Variation of droplet radius (𝜇m) with to changes in orifice size and oil viscosity. While keeping other parameters according to Table 2.
wi ∗ =
wi w1 + w2 + w3 + w4 + w5 + w6
(6)
model can explore a large design space to clarify how parameter variations affect droplet size while requiring negligible computational power and resources. This was previously almost impossible (i.e., significantly expensive or computationally intensive) with either experimental or numerical methods. In the next section, we will verify the accuracy of this model and implement this ANFIS model to study the effects of all the six inputs on droplet size generated in a microfluidic flow-focusing device.
In the fourth layer of the ANFIS model, the signals received from the previous layers will be defuzzified through inferring the 729 rules generated by the neural network as given in (7). Qi,4 = wi ∗ × g
(7)
where, Qi,4 is the signal received by the fourth layer of the ANFIS model. In the fifth and the final layer of the ANFIS model, all the defuzzified signals will be gathered and the aggregated signals will be the output of the ANFIS model. Through this, the ANFIS model will take six input signals of orifice size, oil velocity, oil viscosity, oil density, water velocity, and surface tension and convert these to an output signal which is the droplet radius, as given in Eq. (8). ∑ wi × g ∑ ∗ Qi,5 = wi × g = i ∑ (8) wi i
5. Results and discussion Once the ANFIS predictive model is built, its accuracy is evaluated using the dataset of the observed data points generated by the verified numerical model. To ensure that a predictive model is accurate, the coefficient of determination (i.e., R2 ) is a commonly used measure. Assuming a dataset that consists of n observed data points, noted as y1 , …, yn and n predicted data points that noted as f1 , …, fn , we can calculate mean of the observed data points using Eq. (9).
i
where Qi,5 is the signal received by the fifth layer of the ANFIS model. Through this, we build an ANFIS model that predicts the droplet radius in a microfluidic flow-focusing droplet generation device. This ANFIS
y= 78
n 1∑ y n i=1 i
(9)
A. Lashkaripour et al.
Microelectronics Journal 78 (2018) 73–80
An inherent measure of a dataset is its variability. Two variabilities are sufficient to compare two datasets. One considering the variability of the observed data points relative to the mean of the observed data points, as shown in Eq. (10). The second variability can be defined by considering the variations of the observed data points in comparison to the predicted data points, which is calculated by Eq. (11). Using these two variabilities, through Eq. (12) the coefficient of determination is calculated. ∑ (yi − yi )2 (10) SStot =
complex multiphase flows such as droplet generation. This model offers new insights into droplet generation physics and provides accurate predictions that we hope reduces the development time of droplet-based microfluidic devices. Additionally, this model can be integrated as an input into other models that predict the behavior and the path of a droplet in a network of channels based on the droplet size, such as the discrete model presented in Ref. [33]. 6. Conclusion
i
SSres =
∑
(yi − fi )2
In this study, we developed a predictive model which predicts the droplet radius in a microfluidic flow-focusing device. By exploiting soft computing techniques, we designed an adaptive neural-fuzzy inference system (i.e., ANFIS), trained by a dataset which was generated through an experimentally verified numerical model. The ANFIS model takes six inputs including orifice size, oil velocity, water velocity, oil density, oil viscosity and surface tension and predicts the radius of the produced droplet. We demonstrate that the designed ANFIS model has a significant accuracy with a coefficient of determination of 0.96 in comparison to the observed data points. This model was further utilized to consider a large and continuous design space and study the effect of variations in the input parameters on the droplet radius. This was previously considerably expensive or computationally heavy. ANFIS models, however, are able to explore a large and continuous design space in an efficient manner and make accurate predictions on how input parameters affect the output of a microfluidic droplet generator.
(11)
i 2
R =1−
SSres SStot
(12)
It can be concluded from Eq. (12) that the closer R2 value gets to 1, the model is predicting the behavior of the system more accurately. As demonstrated in Fig. 5, this model shows an R2 = 0.96 when comparing the predicted droplet radius which is significant considering the fact that multiphase flows such as droplet generators are highly complex, nonlinear and with multiple modes of behavior. Now that we have established the accuracy of our proposed ANFIS model, we can study a large continuous design space of the effective parameters on droplet radius which is challenging or expensive to study experimentally. Fig. 6 demonstrates the variations of droplet radius with the variations of water velocity (i.e., water flow rate considering a constant geometry) and oil viscosity. It can be seen that increasing the water flow rate, in general, will increase the droplet size. However, at very high flow rates, it can be observed that there is a sudden decrease in droplet size. This may be due to a change in generation regime to the thread formation regime which has been observed experimentally in previous studies [13]. The variations of droplet size in regards to water velocity and oil density is shown in Fig. 7. It shows increasing water velocity will increase the droplet size in general. Interestingly, it can be observed that changes in oil density affect droplet size, however, density is usually not considered in microfluidic droplet generation. Density differences can be considered through Atwood number as given in Eq. (13). We observed that at Atwood numbers around 0.3 and higher, the effects of density should be taken into account. A=
𝜌1 − 𝜌2 𝜌1 + 𝜌2
Acknowledgements This work was supported by the NSF Living Computing Project Award #1522074 and NSF CAREER Award #1253856. Appendix A. Supplementary data Supplementary data related to this article can be found at https:// doi.org/10.1016/j.mejo.2018.05.018. References [1] [2] [3] [4]
(13)
In Fig. 8, we show the variations of droplet size for a wide range of oil velocities and surface tensions. This figure demonstrates that increasing the oil velocity will result in smaller droplets. This can be verified through the definition of the Capillary number as given in Eq. (1). It is known that increasing Capillary number will reduce the droplet size [32]. Therefore, increasing oil velocity will decrease the droplet size, as expected. Moreover, this model predicts that increasing surface tension will reduce the droplet size. This cannot be explained using Capillary number but it can be due to the variations in droplet generation regimes from dripping to jetting where a sudden reduction in droplet size is observed. In Fig. 9 we demonstrate the variations of droplet size by altering oil viscosity and oil velocity. It can be observed that increasing oil velocity will decrease the droplet size as expected. In addition, it can be seen that increasing oil viscosity increases droplet size. Also, two extremes are obvious in this figure, first, when the oil is very viscous and second when the oil velocity is very high in which we can see new behaviors in microfluidic droplet generation. Finally, we illustrate how droplet radius changes with variations in the orifice size and oil viscosity. As expected, larger orifice sizes will result in larger droplets. Also, as it can be concluded from Eq. (1), increasing the oil viscosity will decrease the droplet size. The proposed ANFIS model behaved accurately for a large and continuous design space of input parameters. Therefore, soft computing methods, ANFIS specifically, are capable of modeling and simulating
[5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21]
[22] [23] [24]
79
T.M. Squires, S.R. Quake, Rev. Mod. Phys. 77 (2005) 977. E.K. Sackmann, A.L. Fulton, D.J. Beebe, Nature 507 (2014) 181. D.J. Beebe, G.A. Mensing, G.M. Walker, Annu. Rev. Biomed. Eng. 4 (2002) 261. M. Rasouli, A. Abouei Mehrizi, A. Lashkaripour, Transport Phenom. Nano Micro Scales 3 (2015) 68. G.M. Whitesides, Nature 442 (2006) 368. T.A. Duncombe, A.M. Tentori, A.E. Herr, Nat. Rev. Mol. Cell Biol. 16 (2015) 554. X.C. i Solvas, et al., Chem. Commun. 47 (2011) 1936. S.Y. Teh, R. Lin, L.H. Hung, A.P. Lee, Lab a Chip 8 (2008) 198. C.M.B. Ho, S.H. Ng, K.H.H. Li, Y.J. Yoon, Lab a Chip 15 (2015) 3627. A. Lashkaripour, R. Silva, D. Densmore, Microfluid. Nanofluidics 22 (2018) 31. D.J. Guckenberger, T.E. de Groot, A.M. Wan, D.J. Beebe, E.W. Young, Lab a Chip 15 (2015) 2364. V. Cristini, Y.C. Tan, Lab on a Chip 4 (2004) 257. W. Lee, L.M. Walker, S.L. Anna, Phys. Fluids 21 (2009) 032103. P. Garstecki, M.J. Fuerstman, H.A. Stone, G.M. Whitesides, Lab a Chip 6 (2006) 437. Y.C. Tan, V. Cristini, A.P. Lee, Sensor. Actuator. B Chem. 114 (2006) 350. G.F. Christopher, S.L. Anna, J. Phys. D Appl. Phys. 40 (2007) R319. M. Goharimanesh, A. Lashkaripour, S. Shariatnia, A. Akbari, Int. J. Fuzzy Syst. 16 (2014) 133. L.A. Zadeh, Fuzzy Set Syst. 90 (1997) 111. M. Goharimanesh, A. Lashkaripour, A.A. Akbari, J. World’s Electr. Eng. Technol. (JWEET) 4 (2015). H.B. Demuth, M.H. Beale, O. De Jess, M.T. Hagan, Neural Network Design, Martin Hagan, 2014. A. Krogh, J. Vedelsby, Neural network ensembles, cross validation, and active learning, in: Advances in Neural Information Processing Systems, 1995, pp. 231–238. J.S. Jang, IEEE Transactions on Systems, Man, and Cybernetics, vol. 23, 1993, p. 665. A. Lashkaripour, A. Abouei Mehrizi, M. Rasouli, M. Goharimanesh, J. Comput. Appl. Mech. 46 (2015) 167. A. Lashkaripour, A.A. Mehrizi, M. Goharimanesh, M. Rasouli, S.R. Bazaz, J. Mech. Med. Biol. 18 (2018) 1850002.
A. Lashkaripour et al.
Microelectronics Journal 78 (2018) 73–80 [30] C. Fantuzzi, R. Rovatti, On the approximation capabilities of the homogeneous Takagi-Sugeno model, in: Fuzzy Systems, 1996., Proceedings of the Fifth IEEE International Conference on, vol. 2, IEEE, 1996, pp. 1067–1072. [31] Y. Chen, B. Yang, A. Abraham, L. Peng, IEEE Trans. Fuzzy Syst. 15 (2007) 385. [32] J.D. Tice, H. Song, A.D. Lyon, R.F. Ismagilov, Langmuir 19 (2003) 9127. [33] A. Grimmer, W. Haselmayr, A. Springer, R. Wille, A discrete model for Networked Labs-on-Chips: linking the physical world to design automation, in: Design Automation Conference (DAC), 2017 54th ACM/EDAC/IEEE, IEEE, 2017, pp. 1–6.
[25] P. Kim, K.W. Kwon, M.C. Park, S.H. Lee, S.M. Kim, K.Y. Suh, Soft Lithography for Microfluidics: A Review, The Korean BioChip Society, 2008. [26] L. Mazutis, A.D. Griffiths, Lab a Chip 12 (2012) 1800. [27] L.H. Hung, K.M. Choi, W.Y. Tseng, Y.C. Tan, K.J. Shea, A.P. Lee, Lab a Chip 6 (2006) 174. [28] S.H. Kim, S.W. Na, Struct. Saf. 19 (1997) 3. [29] R.K. Roy, A Primer on the Taguchi Method, Society of Manufacturing Engineers, 2010.
80
Contents lists available at ScienceDirect
Microelectronics Journal journal homepage: www.elsevier.com/locate/mejo
An adaptive neural-fuzzy approach for microfluidic droplet size prediction Ali Lashkaripour a , Masoud Goharimanesh b , Ali Abouei Mehrizi c , Douglas Densmore d,* a
Boston University, Biomedical Engineering Department, 44 Cummington Mall, Boston, USA Mechanical Engineering Department, Ferdowsi University of Mashhad, Mashhad, Iran c Department of Life Sciences Engineering, Faculty of New Sciences and Technologies, University of Tehran, Tehran, Iran d Boston University, Department of Electrical and Computer Engineering, 8 Saint Mary’s Street, Boston, USA b
A R T I C L E
I N F O
Keywords: Microfluidics Multiphase flow Droplet size Neural networks Fuzzy systems ANFIS
A B S T R A C T
Droplet-based microfluidics is capable of being a superior platform for biological and biomedical applications due to their higher accuracy, throughput and sensitivity, low reagent consumption and fast reaction time. However, the complex and nonlinear behavior of multiphase microfluidic devices and several parameters affecting droplet size simultaneously necessitates costly design iterations to generate droplets with the radius of interest for a certain application. To address this, we exploit soft computing methods to bridge fuzzy systems and neural networks in order to build an adaptive neural-fuzzy inference system (ANFIS) that predicts the droplet size generated in a flow-focusing microfluidic device based on six major parameters, that includes geometry, flow, and fluid properties. This model shows a significant accuracy with a coefficient of determination of 0.96 when compared to the observed data points. Once the ANFIS model is built and verified, we use it to study the effect of each input parameter on droplet size which is challenging and/or expensive to determine experimentally.
1. Introduction Microfluidics has provided new opportunities and accelerated the progress in several fields including chemistry, medicine, pharmaceutical and biomedical engineering [1–4]. Micrometer-scale feature size, high surface to volume ratio and predictable laminar flow inherent to microfluidic devices provide a unique environment for researchers in the field of life sciences to explore the novel physics that governs the field [5,6]. Droplet-based microfluidic devices offer numerous advantages over continuous-flow microfluidics, such as accurate volume and concentration control, high throughput, high sensitivity, low sample and reagent consumption and low thermal mass, which lead to faster, cheaper and more accurate results [7,8]. However, microfluidic devices are still not widely used in life science laboratories, as the community would have expected a decade ago [5,9]. High barrier of entry to the fabrication process, the complexity of the governing physics and costly and time-consuming device iterations to reach a desired performance, have kept the field of microfluidics from being extensively deployed by most of the research groups in the field of life sciences. Some studies have addressed the fabrication cost issues by proposing cost-efficient substitutes to photo-lithography [10,11]. Nonetheless, designing a droplet generator is heavily reliant on sev-
eral design iterations and experience of the designer to perform as required for a certain application. This is due to the fact that there are several parameters acting simultaneously which determine the droplet size [12]. To clarify the complex governing physics of droplet generation, some studies explained droplet generation with dimensionless numbers through a number of simplifying assumptions [13]. Scaling laws were also suggested for T-junction droplet generation [14], however, weak control over droplet size and undefined droplet break-up point led researchers to adopt flow-focusing droplet generators as the standard method of microfluidic droplet production [15]. However, the complexity of the governing physics in flow-focusing devices has prevented accurate prediction of droplet size through scaling laws [16]. Therefore, predictive models which explain the role of each parameter on the performance of the device and approximate the droplet size based on given inputs can be extremely useful. These models are capable of clarifying the effects of the parameters that are hard to determine experimentally. Moreover, they can play a significant role in reducing the development time of droplet-based microfluidic devices. Soft computing methods such as fuzzy logic and neural networks can be used to make predictions of systems behavior and performance [17]. Fuzzy systems are capable of converting logical statements to math-
* Corresponding author. E-mail address: [email protected] (D. Densmore). https://doi.org/10.1016/j.mejo.2018.05.018 Received 15 December 2017; Received in revised form 9 May 2018; Accepted 29 May 2018 Available online XXX 0026-2692/© 2018 Elsevier Ltd. All rights reserved.
A. Lashkaripour et al.
Microelectronics Journal 78 (2018) 73–80
Fig. 1. The overall flow of building an ANFIS predictive model. In this study, we identified six effective parameters in determining droplet size in microfluidic flow-focusing droplet generation. We built a dataset of input-output relations using an experimentally verified numerical model. Using this dataset we trained and verified our proposed ANFIS model. Through the ANFIS model we studied the effects of each parameter on droplet size which is difficult and/or expensive to carry out experimentally.
ematical equations, thus, serving as a tool for simplifying the representation and modeling of complex systems [18]. More importantly, fuzzy logic is capable of grouping and clustering data to several categories. However, fuzzy systems are unable to learn from data and evolve to become more accurate overtime [19]. On the contrary, artificial neural networks are able to learn from data to accurately capture input-output relations. The ability of neural networks in discovering the nonlinear relations between inputs and output and establishing a complex dynamic model of these data is significant [20,21]. However, when dealing with a system that behaves differently depending on the state of the system, neural networks will lose their accuracy, unless, multiple neural networks are proposed for each state of the system. To avoid this, and exploit the clustering power of fuzzy systems and learning capabilities of neural networks, adaptive neural-fuzzy inference systems (ANFIS) were introduced [22]. ANFIS models enable researchers to match any given input-output relation regardless of its complexity or nonlinearity, even if the system has multiple modes. In this paper, we present an accurate predictive ANFIS model that converts six major inputs of a microfluidic droplet generator to an output droplet radius. Once we verify the accuracy of the ANFIS model with an observed dataset, we investigate how changes in input parameters affect the droplet size to further clarify the governing physics of microfluidic flow-focusing droplet generation, as shown in Fig. 1. Also, this model can be further utilized to reduce the number of costly design iterations required for developing microfluidic droplet-based devices. The remainder of this study is as follows. In Section 2 and 3, we introduce microfluidic flow-focusing droplet generation and its design space that we used to build an input-output dataset. In Section 4, we build and train the proposed ANFIS model, also, we introduce the underlying fuzzy membership functions for each input parameter. The accuracy of the ANFIS model is verified in Section 5, by comparing the data predicted by ANFIS to the dataset derived from the numerical model. Finally, we use this verified model to study the effects of each parameter on droplet radius.
Fig. 2. Microfluidic droplet generation can be achieved by flowing an aqueous and a non-aqueous phase through a narrow channel called orifice. (a) The schematic and the flow direction of water and oil in a flow-focusing geometry. (b) A numerical model of microfluidic droplet generation in COMSOL Multiphysics environment (c) An experimental snapshot of microfluidic droplet generation, using Mineral oil as the non-aqueous phase and DI water as the aqueous phase.
channel called orifice as shown in Fig. 2. The geometry of the microfluidic flow-focusing device is adopted from Ref. [15]. Due to the presence of a nozzle, which creates a unique velocity field, this geometry produces monodispersed droplets with a superior control over droplet size and its breaking point. Normally, to characterize how variations in droplet generation parameters affect the droplet size, one would have to build numerous different devices, testing each device at different flow conditions and fluid properties. This process can be very expensive and time-consuming. Moreover, although some parameters such as geometry, oil, and water flow rate can be varied and studied experimentally, the effects of other parameters such as surface tension, oil viscosity, and density are hard to study experimentally due to the limited number of available water and oil combinations. Therefore, to clarify the impact of these parameters on droplet size, a numerical model of microfluidic droplet generation was developed and veri-
2. Microfluidic droplet generation Microfluidic flow-focusing droplet generation can be achieved by flowing an aqueous phase and a non-aqueous fluid through a narrow 74
A. Lashkaripour et al.
Microelectronics Journal 78 (2018) 73–80
fied experimentally [23]. Numerical studies of multiphase flows such as droplet generation are computationally intensive and time-consuming. As a result, studying a large design space in this field would require a considerable amount of computation power, time and resources not available to most. Therefore, predictive models built based on a small number of numerical simulations that can predict the system’s performance for a wide range of inputs are of major importance and can replace large-scale numerical simulations. As demonstrated in Ref. [24], flow-focusing droplet generation behaves differently for droplets larger than 5 μm and droplets smaller than 5 μm in radius. As a result, neural networks alone cannot capture systems behavior accurately, without proposing two separate neural network models. This necessitates a need for an ANFIS model to predict droplet radius for all ranges of size.
Table 1 The range of six major parameters considered in this study that affects droplet size. For each parameter four equally-spaced values (levels) are considered between the upper- and lower-bound to build the input-output dataset, using the verified numerical model. Parameter
3. Design space
Qc Qw
Lower-bound
Upper-bound
Orifice width Oil viscosity Surface tension Oil velocity Water velocity Oil density
(μm) (mPa.s) (N/m) (m/s) (m/s) (kg/m3 )
3 2 0.003 0.02604 0.001736 700
48 41 0.02175 0.10416 0.009548 1900
4. ANFIS predictive model As shown in Fig. 1, to build an ANFIS model that predicts droplet size based on the inputs of a microfluidic droplet generator, the first step is to build a dataset of observed data points. As explained in the previous sections, the built input-output dataset includes parameters including water velocity, surface tension, oil velocity, oil viscosity, oil density and orifice size as inputs and droplet radius as the output. Fig. 3 shows the structure of our proposed ANFIS model, and its input-output relation through a black box, which converts the aforementioned inputs of the microfluidic device to droplet radius. The first layer of the ANFIS model is the inputs of a microfluidic droplet generator and the middle layers or the black box section is where the rules are generated
(1)
where 𝜇c is oil viscosity, uc is oil velocity and 𝜎 is surface tension.
Λ=
Unit
To explore and characterize the design space with the minimum number of experiments, we used an orthogonal array of inputs (i.e., response surface method [28]) using Taguchi method [29]. Through Taguchi method, to build an orthogonal array of six parameters each having four levels, only thirty-two experiments are needed, whereas, a full-factorial design of experiments requires 4096 (i.e., 46 ) experiments. Having the input-output values of these thirty-two experiments through an experimentally validated numerical model, alongside with nineteen experimentally obtained input-output data-points used to verify our numerical model a total number of 51 data-points that relates the values of six inputs to a droplet radius output were acquired.
As shown previously, out of eight parameters defining the geometry, flow properties, and fluid properties of a microfluidic flow-focusing droplet generator, only six of them affect droplet size [24]. We varied the range of these major parameters to represent real-world applications. For each parameter, four different values (i.e., levels) were considered to be equally spaced within a certain range, as given in Table 1. The orifice width was varied to include both fabrication limits and previous studies [15,25]. Oil viscosity and density were varied to cover different oils commonly used in microfluidic droplet generation (i.e., FC-40, Silicone oil, and Oleic acid oil) [15,26,27]. Surface tension was varied to include the surface tension between water and the aforementioned oils. The two major dimensionless numbers that affect droplet size are Capillary number (Ca.) as given in Eq. (1), and flow rate ratio (Λ), as given in Eq. (2).
𝜇u Ca = c c 𝜎
Range
Name
(2)
where Qc is the flow rate of oil and Qw is the flow rate of water. Oil and water velocities were chosen to cover a wide range of Capillary numbers (i.e., 0.01–0.1) and flow-rate ratios (i.e., 2–60), which represent common conditions in microfluidic flow-focusing droplet generation.
Fig. 3. The proposed ANFIS model for microfluidic droplet radius prediction. This model has six inputs and five layers where the input will be fuzzified, weighted, interconnected, defuzzified and finally aggregated to generate a single output. In this model, the user can enter the input values and through a black box the output will be predicted. 75
A. Lashkaripour et al.
Microelectronics Journal 78 (2018) 73–80
Fig. 5. Accuracy verification of the ANFIS model with the existing dataset. The predicted droplet radius using ANFIS model shows an R2 of 0.96 with the observed data points which clarifies its significant power in predicting microfluidic droplet size.
using neural networks. In the first layer, each input (node) is associated with an adaptive node through a unique function, 𝜇(i), called a membership function (MF). In this study, a Sugeno model [30] alongside Takagi–Sugeno type fuzzy “If-Then” rules [31] are exploited as given in Eq. (3). g1 = p1 a + q1 b + r1 c + s1 d + t1 e + u1 f + v
(3)
where, membership function for input 1 is a, for input 2 is b, for input 3 is c, for input 4 is d, for input 5 is e and for input 6 is f . This description of the membership functions and Eq. (3) shows the Takagi-Sugeno type If-Then rules. The second layer of the proposed ANFIS model includes the fuzzy membership functions of the six inputs. Bell-shaped membership functions with a range of zero to one are utilized as given in Eq. (4). g (x; n, p) =
( 1+
1 x −p 1
)2n
(4)
Each membership function was considered to be a third-degree function, as shown in Fig. 4. These Gaussian bell-shaped membership functions will fuzzify and categorize the inputs to three zones (i.e., the degree of the membership function). These functions cluster the input values into different zones where Y-axis demonstrates the corresponding value of each input to the designated zone. Each input can have a zero to one corresponding value for each zone, where zero demonstrates no membership and one shows a full membership. The second and third layer are connected through allocated weights for each of the membership functions defined in the second layer. As a result, the inputs that were fuzzified will be multiplied by their allocated weight and directed to the third layer as shown in Eq. (5). wi = 𝜇(i)i × 𝜇(i)i+1
(5)
where 𝜇(i) is the membership function and w is the corresponding weight of the membership function. In the third layer, each node generated in the previous layers will be interconnected through a neural
Table 2 Input parameters considered for deriving the color-maps of Figs. 6–10, except the two variables considered in each figure.
Fig. 4. Fuzzy third-degree membership functions that cluster the inputs for each parameter to three overlapping zones. Therefore, capturing possible multiple modes of the system. Fuzzy membership functions are given for six input parameters including (a) orifice size (b) water velocity (c) oil velocity (d) oil density (e) oil viscosity and (f) surface tension.
76
Parameter
Value
Unit
Orifice size Water velocity Oil velocity Oil density Oil viscosity Surface tension
25.5 0.011284 0.0651 1300 0.0215 0.012375
[μm] [m/s] [m/s] [kg/m3 ] [Pa.s] [N/m]
A. Lashkaripour et al.
Microelectronics Journal 78 (2018) 73–80
Fig. 6. Alterations of droplet radius (𝜇m) with respect to changes in water velocity and oil viscosity. While keeping other parameters according to Table 2.
Fig. 7. Variations of droplet radius (𝜇m) with respect to changes in water velocity and oil density. While keeping other parameters according to Table 2.
function will be normalized against the summation of other membership functions weights, as given in Eq. (6). These normalized weights of each input membership function represent the relative firing strength of each neuron in an actual neural network.
network. These connections are made through rules generated by the neural networks. Considering the third-degree membership functions and six input parameters, there will be a total of 36 = 729 rules generated by neural networks. In this layer, the weight of each membership
Fig. 8. Variations of droplet radius (𝜇m) with to changes in surface tension and oil velocity. While keeping other parameters according to Table 2. 77
A. Lashkaripour et al.
Microelectronics Journal 78 (2018) 73–80
Fig. 9. Variations of droplet radius (𝜇m) with respect to changes in oil viscosity and oil velocity. While keeping other parameters according to Table 2.
Fig. 10. Variation of droplet radius (𝜇m) with to changes in orifice size and oil viscosity. While keeping other parameters according to Table 2.
wi ∗ =
wi w1 + w2 + w3 + w4 + w5 + w6
(6)
model can explore a large design space to clarify how parameter variations affect droplet size while requiring negligible computational power and resources. This was previously almost impossible (i.e., significantly expensive or computationally intensive) with either experimental or numerical methods. In the next section, we will verify the accuracy of this model and implement this ANFIS model to study the effects of all the six inputs on droplet size generated in a microfluidic flow-focusing device.
In the fourth layer of the ANFIS model, the signals received from the previous layers will be defuzzified through inferring the 729 rules generated by the neural network as given in (7). Qi,4 = wi ∗ × g
(7)
where, Qi,4 is the signal received by the fourth layer of the ANFIS model. In the fifth and the final layer of the ANFIS model, all the defuzzified signals will be gathered and the aggregated signals will be the output of the ANFIS model. Through this, the ANFIS model will take six input signals of orifice size, oil velocity, oil viscosity, oil density, water velocity, and surface tension and convert these to an output signal which is the droplet radius, as given in Eq. (8). ∑ wi × g ∑ ∗ Qi,5 = wi × g = i ∑ (8) wi i
5. Results and discussion Once the ANFIS predictive model is built, its accuracy is evaluated using the dataset of the observed data points generated by the verified numerical model. To ensure that a predictive model is accurate, the coefficient of determination (i.e., R2 ) is a commonly used measure. Assuming a dataset that consists of n observed data points, noted as y1 , …, yn and n predicted data points that noted as f1 , …, fn , we can calculate mean of the observed data points using Eq. (9).
i
where Qi,5 is the signal received by the fifth layer of the ANFIS model. Through this, we build an ANFIS model that predicts the droplet radius in a microfluidic flow-focusing droplet generation device. This ANFIS
y= 78
n 1∑ y n i=1 i
(9)
A. Lashkaripour et al.
Microelectronics Journal 78 (2018) 73–80
An inherent measure of a dataset is its variability. Two variabilities are sufficient to compare two datasets. One considering the variability of the observed data points relative to the mean of the observed data points, as shown in Eq. (10). The second variability can be defined by considering the variations of the observed data points in comparison to the predicted data points, which is calculated by Eq. (11). Using these two variabilities, through Eq. (12) the coefficient of determination is calculated. ∑ (yi − yi )2 (10) SStot =
complex multiphase flows such as droplet generation. This model offers new insights into droplet generation physics and provides accurate predictions that we hope reduces the development time of droplet-based microfluidic devices. Additionally, this model can be integrated as an input into other models that predict the behavior and the path of a droplet in a network of channels based on the droplet size, such as the discrete model presented in Ref. [33]. 6. Conclusion
i
SSres =
∑
(yi − fi )2
In this study, we developed a predictive model which predicts the droplet radius in a microfluidic flow-focusing device. By exploiting soft computing techniques, we designed an adaptive neural-fuzzy inference system (i.e., ANFIS), trained by a dataset which was generated through an experimentally verified numerical model. The ANFIS model takes six inputs including orifice size, oil velocity, water velocity, oil density, oil viscosity and surface tension and predicts the radius of the produced droplet. We demonstrate that the designed ANFIS model has a significant accuracy with a coefficient of determination of 0.96 in comparison to the observed data points. This model was further utilized to consider a large and continuous design space and study the effect of variations in the input parameters on the droplet radius. This was previously considerably expensive or computationally heavy. ANFIS models, however, are able to explore a large and continuous design space in an efficient manner and make accurate predictions on how input parameters affect the output of a microfluidic droplet generator.
(11)
i 2
R =1−
SSres SStot
(12)
It can be concluded from Eq. (12) that the closer R2 value gets to 1, the model is predicting the behavior of the system more accurately. As demonstrated in Fig. 5, this model shows an R2 = 0.96 when comparing the predicted droplet radius which is significant considering the fact that multiphase flows such as droplet generators are highly complex, nonlinear and with multiple modes of behavior. Now that we have established the accuracy of our proposed ANFIS model, we can study a large continuous design space of the effective parameters on droplet radius which is challenging or expensive to study experimentally. Fig. 6 demonstrates the variations of droplet radius with the variations of water velocity (i.e., water flow rate considering a constant geometry) and oil viscosity. It can be seen that increasing the water flow rate, in general, will increase the droplet size. However, at very high flow rates, it can be observed that there is a sudden decrease in droplet size. This may be due to a change in generation regime to the thread formation regime which has been observed experimentally in previous studies [13]. The variations of droplet size in regards to water velocity and oil density is shown in Fig. 7. It shows increasing water velocity will increase the droplet size in general. Interestingly, it can be observed that changes in oil density affect droplet size, however, density is usually not considered in microfluidic droplet generation. Density differences can be considered through Atwood number as given in Eq. (13). We observed that at Atwood numbers around 0.3 and higher, the effects of density should be taken into account. A=
𝜌1 − 𝜌2 𝜌1 + 𝜌2
Acknowledgements This work was supported by the NSF Living Computing Project Award #1522074 and NSF CAREER Award #1253856. Appendix A. Supplementary data Supplementary data related to this article can be found at https:// doi.org/10.1016/j.mejo.2018.05.018. References [1] [2] [3] [4]
(13)
In Fig. 8, we show the variations of droplet size for a wide range of oil velocities and surface tensions. This figure demonstrates that increasing the oil velocity will result in smaller droplets. This can be verified through the definition of the Capillary number as given in Eq. (1). It is known that increasing Capillary number will reduce the droplet size [32]. Therefore, increasing oil velocity will decrease the droplet size, as expected. Moreover, this model predicts that increasing surface tension will reduce the droplet size. This cannot be explained using Capillary number but it can be due to the variations in droplet generation regimes from dripping to jetting where a sudden reduction in droplet size is observed. In Fig. 9 we demonstrate the variations of droplet size by altering oil viscosity and oil velocity. It can be observed that increasing oil velocity will decrease the droplet size as expected. In addition, it can be seen that increasing oil viscosity increases droplet size. Also, two extremes are obvious in this figure, first, when the oil is very viscous and second when the oil velocity is very high in which we can see new behaviors in microfluidic droplet generation. Finally, we illustrate how droplet radius changes with variations in the orifice size and oil viscosity. As expected, larger orifice sizes will result in larger droplets. Also, as it can be concluded from Eq. (1), increasing the oil viscosity will decrease the droplet size. The proposed ANFIS model behaved accurately for a large and continuous design space of input parameters. Therefore, soft computing methods, ANFIS specifically, are capable of modeling and simulating
[5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21]
[22] [23] [24]
79
T.M. Squires, S.R. Quake, Rev. Mod. Phys. 77 (2005) 977. E.K. Sackmann, A.L. Fulton, D.J. Beebe, Nature 507 (2014) 181. D.J. Beebe, G.A. Mensing, G.M. Walker, Annu. Rev. Biomed. Eng. 4 (2002) 261. M. Rasouli, A. Abouei Mehrizi, A. Lashkaripour, Transport Phenom. Nano Micro Scales 3 (2015) 68. G.M. Whitesides, Nature 442 (2006) 368. T.A. Duncombe, A.M. Tentori, A.E. Herr, Nat. Rev. Mol. Cell Biol. 16 (2015) 554. X.C. i Solvas, et al., Chem. Commun. 47 (2011) 1936. S.Y. Teh, R. Lin, L.H. Hung, A.P. Lee, Lab a Chip 8 (2008) 198. C.M.B. Ho, S.H. Ng, K.H.H. Li, Y.J. Yoon, Lab a Chip 15 (2015) 3627. A. Lashkaripour, R. Silva, D. Densmore, Microfluid. Nanofluidics 22 (2018) 31. D.J. Guckenberger, T.E. de Groot, A.M. Wan, D.J. Beebe, E.W. Young, Lab a Chip 15 (2015) 2364. V. Cristini, Y.C. Tan, Lab on a Chip 4 (2004) 257. W. Lee, L.M. Walker, S.L. Anna, Phys. Fluids 21 (2009) 032103. P. Garstecki, M.J. Fuerstman, H.A. Stone, G.M. Whitesides, Lab a Chip 6 (2006) 437. Y.C. Tan, V. Cristini, A.P. Lee, Sensor. Actuator. B Chem. 114 (2006) 350. G.F. Christopher, S.L. Anna, J. Phys. D Appl. Phys. 40 (2007) R319. M. Goharimanesh, A. Lashkaripour, S. Shariatnia, A. Akbari, Int. J. Fuzzy Syst. 16 (2014) 133. L.A. Zadeh, Fuzzy Set Syst. 90 (1997) 111. M. Goharimanesh, A. Lashkaripour, A.A. Akbari, J. World’s Electr. Eng. Technol. (JWEET) 4 (2015). H.B. Demuth, M.H. Beale, O. De Jess, M.T. Hagan, Neural Network Design, Martin Hagan, 2014. A. Krogh, J. Vedelsby, Neural network ensembles, cross validation, and active learning, in: Advances in Neural Information Processing Systems, 1995, pp. 231–238. J.S. Jang, IEEE Transactions on Systems, Man, and Cybernetics, vol. 23, 1993, p. 665. A. Lashkaripour, A. Abouei Mehrizi, M. Rasouli, M. Goharimanesh, J. Comput. Appl. Mech. 46 (2015) 167. A. Lashkaripour, A.A. Mehrizi, M. Goharimanesh, M. Rasouli, S.R. Bazaz, J. Mech. Med. Biol. 18 (2018) 1850002.
A. Lashkaripour et al.
Microelectronics Journal 78 (2018) 73–80 [30] C. Fantuzzi, R. Rovatti, On the approximation capabilities of the homogeneous Takagi-Sugeno model, in: Fuzzy Systems, 1996., Proceedings of the Fifth IEEE International Conference on, vol. 2, IEEE, 1996, pp. 1067–1072. [31] Y. Chen, B. Yang, A. Abraham, L. Peng, IEEE Trans. Fuzzy Syst. 15 (2007) 385. [32] J.D. Tice, H. Song, A.D. Lyon, R.F. Ismagilov, Langmuir 19 (2003) 9127. [33] A. Grimmer, W. Haselmayr, A. Springer, R. Wille, A discrete model for Networked Labs-on-Chips: linking the physical world to design automation, in: Design Automation Conference (DAC), 2017 54th ACM/EDAC/IEEE, IEEE, 2017, pp. 1–6.
[25] P. Kim, K.W. Kwon, M.C. Park, S.H. Lee, S.M. Kim, K.Y. Suh, Soft Lithography for Microfluidics: A Review, The Korean BioChip Society, 2008. [26] L. Mazutis, A.D. Griffiths, Lab a Chip 12 (2012) 1800. [27] L.H. Hung, K.M. Choi, W.Y. Tseng, Y.C. Tan, K.J. Shea, A.P. Lee, Lab a Chip 6 (2006) 174. [28] S.H. Kim, S.W. Na, Struct. Saf. 19 (1997) 3. [29] R.K. Roy, A Primer on the Taguchi Method, Society of Manufacturing Engineers, 2010.
80