A Grid search-based response surface methodology for the optimum biosynthesis of silver nanoparticles using Clerodendrum splendens

. In this paper, we propose a grid search algorithm using response surface methodology (RSM) for optimizing the biosynthesis of silver nanoparticles (AgNPs). A biological method using hot water (HW) Clerodendrum splendens ( C. splendens ) leaf extract is adopted for the discussion. The RSM integrated central composite design (CCD) is used to derive a quadratic objective function to model the yield of AgNPs by accounting the effects of the processing parameters, namely, C. splendens concentration, pH, and the reaction time. The optimization of the resulting objective function, which is a challenging multi-variate non-convex problem, is solved using a grid search approach. The respective optimum values of the parameters are found to be 54 mL, pH 12, and 23 min and the corresponding AgNPs yield is experimentally verified.


Introduction
AgNPs are widely used in many fields of science and technology due to its unique physical and chemical properties including antibacterial, antiviral, and antifungal activities 1 .Hence, the synthesis and the characterization of AgNPs is an active research topic and there are many recent attempts discussing the optimality of AgNPs synthesis [2][3][4][5] .These papers address the optimization of the yield, 2,3 particle size, 4 or the absorbance 5 of AgNPs with respect to various process parameters such as silver nitrate concentration, pH, reaction time, etc.The common strategy followed in such works is to adopt a response surface methodology (RSM), which is a statistical approach to optimize the reaction settings by accounting the interaction between the process parameters.This is often achieved by performing central composite design (CCD) experiments and then modelling the relationship between the parameters and the response of the reaction using a second order polynomial via regression analysis.The optimization of this regression models is a quadratic programming (QP), which, in the context of AgNPs synthesis, is a very challenging problem due the non-convex nature of the model.
To the best of our knowledge, most of the published works [3][4][5] do not put forward a strategy to obtain a global solution for the underlying QP; instead they discuss the optimum interaction between any two process parameters using three-dimensional (3D) response surface graphs, in which the other parameters are fixed to specific values.Such a strategy fails to address the multi-variate optimization problem in its general form, in which the simultaneous interactions between all the parameters are important.On the other hand, in the paper 2 , the optimum value is selected from the CCD parameter settings; which is a suboptimal method since the search space of QP solver is highly reduced.We remark that since the underlying QP is non-convex in nature any straight forward optimization routine will end up in local optimum points.As a remedy to all the roadblocks, we propose a simple brute force approach based on grid search algorithm.The proposed optimization strategy, with the help of MATLAB programming, maximizes the AgNPs yield over all the possible combinations of the process parameters sampled from a 3D grid.
Though the AgNPs can be synthesized using physical and chemical methods, we adopt a biological approach using plant leaf extract, which has received great attention in the recent years, as it is cost-effective, simple, environmentally friendly, and non-toxic 6 .Also, since the plant-based nanoparticles synthesis does not require the complicated process of maintaining the fungi or the bacterial cell cultures, it is simple and beneficial over other biological agents.
In this study, we make use of the leaves of Clerodendrum splendens, which is an evergreen climber or running shrub belonging to the family verbenaceae 7 .Its common name is Flaming Glorybower 8 .It is native to Western Africa and is widely distributed in the tropical regions throughout the world.Their leaves contain glycosides, reducing sugars, triterpenoids, unsaturated sterols, and flavonoids 9 .The plant is used in traditional medicine to treat wounds and burns 10 , haemorrhoids, diarrhea and dysentery 11 .Previous studies have reported that the C. splendens leaf extract has antioxidant, antimicrobial and anti-inflammatory properties 12 .The phytochemicals of the C. splendens leaf extract are vital for the synthesis of non-toxic silver nanoparticles.Our previous study discussed the synthesis of silver nanoparticles using HW C. splendens leaf extract without a grid search-based RSM 13 .
The aim of the study is to optimize the experimental conditions for the production of silver nanoparticles using C. splendens leaf extract.A grid search-based RSM algorithm is proposed to compute the optimum value of C. splendens concentration, pH, and reaction time.This study may present a better understanding of the role of C. splendens leaf extract for the production of AgNPs.

Materials
Silver nitrate (AgNO3) was purchased from Central drug house (CDH), India, and was of analytical grade.C. splendens leaves were collected from the surroundings of Chennai, Tamil Nadu, India.Double distilled water was used throughout the study.

Preparation of leaf extract
The collected C. splendens leaves were thoroughly washed and dried.The clean leaves were cut into fine particles.10 g of these finely cut clean leaves were boiled in 100 mL of distilled water for 2 min at 60 °C, and then the extract was filtered twice through Whatman filter paper grade No.1 (Pore size: 11 µm).The obtained leaf extract was stored at ambient temperature (30 °C) for further use.

Synthesis of silver nanoparticles
To estimate the formation of AgNPs, freshly prepared leaf extract was added to 100 mL of 1 mM AgNO3.The reaction was carried out in a dark room to decrease the photoactivation of AgNO3 at an ambient temperature of 30 °C.Within a short period, the colour of the resulting solution started changing from pale yellow to colloidal brown indicating the formation of AgNPs.The solution containing AgNPs was centrifuged at 15000 rpm for 15 min to obtain a pellet of AgNPs.The pellet was washed with distilled water to get rid of biomass residue and then dried in the oven at 60 °C for 24 h, which was used for further studies.

Optimization of silver nanoparticles production
The influence of three process parameters, i.e., the plant extract (C.splendens) concentration (A), pH (B) and the reaction time (C), on the yield of AgNPs (Y), was studied using a statistical method called RSM.A CCD experimental matrix consisting of three factors (A, B and C) at three levels was produced.We choose a Face-centred CCD having 6 centre points, 6 axial (star) points, and 8 factorial points resulting in 20-experimental runs.
The yields of AgNPs for all the 20 entries of the CCD matrix were experimentally measured.A second order regression equation model of the following form was generated to facilitate the proceeding optimization procedures: Where,   ,  = , , …  are the regression coefficients and (, , )is the predicted response variable as a function of A, B, and C. The development of CCD data and the regression model were done using MATLAB 8.6.
We formulate the following optimization problem to calculate the best values of A, B, and C: The optimization problem (2), which is commonly known as QP, is often non-convex for the AgNPs regression model.In order to solve this, a sufficiently large 3D grid representing all the possible discrete values of A, B and C, in a preferred range, is created and an extensive brute force search over the grid is performed.This grid search algorithm is implemented using MATLAB 8.6.

Statistical analysis
The significance of the model coefficients were statistically analyzed by ANOVA (Analysis of variance).The fit statistics of the quadratic model and ANNOVA table were generated using MATLAB 8.6.

Synthesis of AgNPs
The present study discusses the synthesis of AgNPs using leaves of C. splendens (Fig. 1a).
The AgNPs was synthesized by adding C. splendens extract in 100 mL of 1 mM AgNO3 solution.The successful production of AgNPs was visually confirmed by the colour change of AgNPs solution from pale yellow to reddish brown (Fig. 1b).The reddish-brown colour is due to the excitation of the surface plasmon vibrations in the AgNPs 14 .

Optimization of AgNPs formation by RSM
The proposed optimization strategy involves two major steps: i) an RSM step in which a second-order polynomial model, as per equation ( 1), corresponding to a CCD experimental matrix, is computed; ii) an optimization step in which the QP, described by equation ( 2) is solved using a grid search algorithm.
To design the CCD matrix, we considered three levels of the process parameters, coded as +1, 0, -1, representing high, intermediate, and low values, respectively as shown in Table 1.
The predicted values of the response variables (Y) generated according to equation (3), the observed response   , and CCD matrix are shown in Table 2.
Using the regression model (3), the optimization in equation ( 2) can be rewritten as  and the reaction time (C) for the synthesis of AgNPs are, respectively, 54 mL, 12, and 23 min, and the corresponding AgNPs yield is 1.0926.We repeated the experimental synthesis of AgNPs using the theoretically optimum parameter values of {54 mL, 12, 23 min} and obtained an AgNPs yield of 0.9715 (Fig. 3).This value is very close to the theoretical yield 1.0926, with a minor error of 0.1211.

Interactions among the process parameters
The relative interactions among the two parameters by keeping the third one to their optimum value was studied using 3D surface plot and 2D contour plot in Fig. 4. Figure 4a

ANOVA for RSM quadratic equation
The ANOVA for RSM quadratic model of AgNPs yield was highly significant with an F value of 3575.44 (Table 3).P-value less than 0.0001 indicates that the model terms are highly significant.In this case, A, B, C, AB, BC, A 2 , B 2 , and C 2 are highly significant model terms.
P-value greater than 0.1 indicate the model term is not significant.The fit statistics for the quadratic model is shown in Table 4. Coefficient of variation (CV) is the ratio of the standard deviation to mean.The lower the CV value of 0.0125 indicates a smaller deviation between the experimental and the predicted value.The coefficient of determination (R 2 ) is a statistical tool to measure the correlation between the experimental and the predicted responses.A high correlation is indicated by an R 2 value close to 1.In the present study, the R 2 value is 0.9997, which indicates a very good correlation.For a good model, the difference between the "predicted" and the "adjusted" R 2 should be less than 0.2 and in our case, the difference is 0.001.The Adeq Precision measures the signal to noise ratio.
An Adeq Precision greater than 4 is desirable and in the present study, the value of the ratio is 155.6777, which indicates an adequate signal level.

Conclusions
In this paper, we propose a new optimization strategy, in which the biosynthesis of AgNPs is optimized for the best yield.We discuss the synthesis of AgNPs using leaf extract of C. splendens and the optimum combination of the process parameters, i.e., the concentration of the leaf extract, pH, and the reaction time, is calculated.The proposed optimization strategy involves the computation of a second order regression model corresponding to a CCD experimental matrix and the optimization of the regression model using a grid search algorithm.The optimum combination of the process parameters is computed as 54 mL, pH 12, 30 min through a grid search algorithm and this reaction condition is experimentally verified.The proposed strategy is easily extendable to optimize the reaction conditions in any similar biosynthesis of nanoparticles.

Fig. 2
Fig.2 The 3D grid for the optimization search space

Fig. 3
Fig.3 UV-Visible spectrum of experimental synthesis of AgNPs under optimum condition shows the interactive effects of C. splendens concentration (A) and pH (B) in the production of AgNPs (Y) at the optimum reaction time (C) of 23 min.The graph shows that the yield of AgNPs is increasing until the C. splendens concentration (A) is 54 mL, and beyond 54 mL, the yield of AgNPs starts to decrease.With respect to pH (B), the yield is gradually increasing until pH12, and then drops slightly.

Figure 4b shows the
Figure 4b shows the interactive effects of C. splendens concentration (A) and reaction time

Fig. 4
Fig.4 The relative interactions between the parameters shown using 3D surface and 2D contour plots

Table 1 ,
is generated and the corresponding yield (  ) of AgNPs is experimentally calculated (see