All the analyses were conducted at a concentration of 40 µg/ml. STATISTICA software was used to analyze the results. The polynomial model of the second-order coefficients was determined by regression equation. For each response, the model was found to be significant when adjusted R value is near to predicted R value. Based on Eq. 1, Y1 is the suitable response for retention time and Y2 is for peak area, Y3 for tailing factor, and Y4 for theoretical plate. X1, X2 are the flow rate of mobile phase represented in ml/min, X2 is the internal temperature of column, and X3 is the degree of methanol in the design. In terms of coded factors, the model equation for responses (Y1-Y4) are as follows:

Variables effect on retention time (Y1)

Figures 2 and 3 (A1-A3 and B1-B3) graph shows the effect of independent factors (X1, X2 and X3) over dependent response retention time (Y1).

Fig. 2
figure 2

Plot a showing true versus predicted values, with residual plot b for response Y1

Fig. 3
figure 3

Relation plots (A1-A3; B1-B3) between true and measure values for Y1

Final equation in terms of coded factors

$$Y1left( {RT, , min } right) , = , 4.44 , – , 0.4194X1 , – , 0.5772X3 + , 0.1675X1X3 , + , 0.0972X1^{2}$$


Equation 2 shows the inverse relationship of RT and the functional parameters, i.e., flow rate of mobile phase (X1) and concentration of MeOH (X3). The solid positive relapse coefficient demonstrates a sharp increment in retention time, with equal a higher rate of flow and MeOH conc. To minimize the retention time, the flow rate can be increased. We can likewise accelerate the partition and lessen mobile phase thickness by expanding the temperature, in this way limiting backpressure. To expand the mobile phase variable i.e., methanol, the retention time will diminish, and every single other response will increase. This might be because of the mobile phase’s expanded extremity, making a quicker analyte balance among stationary phase and the mobile phase.

Variables effect on peak area (Y2)

Figures 4 and  5 (C1–C3 and D1-D3) show linear correlation plot for the response Y2 among true as well as measured values and the corresponding remaining graphs. Increase in flow rate makes the pinnacle zone (Area under curve) rise. Likewise, the peak area is additionally increased by increment in column temperature and amount of methanol in the mobile phase. Equation 3 shows impact of flow rate, column temperature and methanol concentration on peak area.

Fig. 4
figure 4

Plot a showing values true verses predicted with the residual plot b for the response Y2

Fig. 5
figure 5

Relation plot (C1-C3; D1-D3) between true with predicted values and residual plot for Y2

Final equation in terms of coded factors

$$begin{aligned} Y2(Peak;area) , = & , 3930.01 – 360.42X1 – , 0.2283X2 – , 2.83C – , 2.48X1X2 \ & + , 0.2233X1X3 + , 0.5558 , X2X3 + , 27.98X1^{2} – , 0.3528X2^{2} + , 1.32X3^{2} \ end{aligned}$$


Variables effect on tailing factor

Figures 6 and 7 (E1–E3 and F1-F3) show linear correlation plot for the response Y3 among true as well as measured values and the corresponding graphs.

Fig. 6
figure 6

Plot a showing true verses predicted values and residual plot b for the response Y3

Fig. 7
figure 7

Relation plot (E1-E3; F1-F3) between actual with predicted values and residual plot for Y3

Final Equation in Terms of Coded Factors

$$Y3 , left( {TF} right) , = , 0.7315 + 0.0106X1 – 0.0033X2$$


This equation shows the positive impact of flow rate (X1) so whenever increase in flow rate there will be elevation of TF value and negative impact of column temperature (X2). The increase in temperature value decreases the tailing factor.

Variables effect on theoretical plate

This equation expressed to predict values of response for coded variables. The response on theoretical plate is affected by flow rate (X1), temperature of column (X2), and methanol concentration in mobile phase (X3).

Figures 8 and  9 (G1-G3 and H1-H3) indicate linear correlation plot for the response Y4 among true as well as measured values and the corresponding graphs.

Fig. 8
figure 8

Plot a between true and measured values with residual plot b for the response Y4

Fig. 9
figure 9

Relation plot (G1-G3; H1-H3) between actual with predicted values and residual plot for Y4

Final equation in terms of coded factors

$$Y4 , left( {TP} right) , = 9546.59 – 526.72X1 + 136.06X2 – 190.78X3 – 143.08X1X2$$


Statistics of response

All the dependent critical analytical responses are analyzed statistically, i.e., RT, PA, TP, and TF. (Table 4).

Table 4 Statistics for the model of response

Analysis of variance for the responses (Y1-Y4)

Tables 5 , 6 , 7 and 8 show the ANOVA tables for various responses (RT, PA, TF, and TP).

Table 5 ANOVA results for Response 1: RT
Table 6 ANOVA results for Response 2: PA
Table 7 ANOVA results for Response 3: TF
Table 8 ANOVA results Response 4: TP

Graphical and numerical optimization

Figure 10 shows the overlay plots showing relationship of factor and responses for graphical optimization, whereas Figs. 11 and 12 show numerical optimized method having highest desirability of 0.918.

Fig. 10
figure 10

Overlay plots showing relationship of factors and responses

Fig. 11
figure 11

Graphs showing numerical optimization with highest desirability

Fig. 12
figure 12

Numerical optimization graphs showing most suitable method

Calibration curves

The span of the linearity can be analyzed by the standard solution of 10–50 µg/ml (r2 = 0.9997, slope = 70.72) (Fig. 13).

Fig. 13
figure 13

Calibration curve for flurbiprofen


During validation, all the graphs were clear, sharp, and very well without any impurities. Results for precision, the RSD percentage were less than 2. A recovery study is well utilized to determine the accuracy and the response of the peak area. The ICH limit decides different parameters of linearity with the system variables. In FLP, the linearity test was executed at 5 separate levels. The suggested approach shows a great linearity span of 10, 20, 30, 40, 50 μg/ml (r2 = 0.9997).

Intra-day and inter-day precision (n = 6).

This method is used to determine the precision values of inter-day and intra-day. % RSD was found 0.48 for inter-day and 0.081 for intra-day. The outcome (Table 9) has almost no effect on the parameters due to any little variation.

Table 9 Intra- and inter-day precision (n = 6)


Repeatability study was conducted and % RSD was found 0.05.


The process parameters were checked for robustness study; it is found acceptable % RSD value less than 2 percent within the limits. The fact that there were no obvious alterations in the chromatograms suggested that the HPLC procedures that have been developed are robust (Table 10).

Table 10 Robustness study

Limit of quantification (LOQ), limit of detection (LOD)

LOD and LOQ values determine the sensitivity of method. The lowest concentration can be detected by system is LOD, whereas LOQ is lowest concentration in analytes in stated sample determined under acceptable precision values. To obtain LOQ & LOD, actual drug concentration in linear range and calibration curve were used for 6 repetition assessments. LOD and LOQ values were 0.14 μg / ml and 0.42 μg / ml. (Table 11).

Table 11 Regression data for the calibration curve. (n = 6)

Analysis of the nanoparticles formulation and recovery study

The chromatogram of drug content from nanoparticles was obtained. A peak at RT 4.42 min was obtained with (% RSD 0.01). The formulation’s average recovery was discovered to be 101.28% with % RSD 0.08 (Tables 11 and 12).

Table 12 Formulation assay and recovery study results (n = 6)

Forced degradation analysis (stability-indicating methods)

Studies on force degradation were conducted out and the following results obtained (Table 13, Fig. 14).

Table 13 Results of forced degradation studies (n = 6)
Fig. 14
figure 14

Chromatograms of FBP and its degradation products

Oxidation degradation

FBP found to be extremely susceptible to degradation in the oxidation analysis. At RT values of 2.24, 2.34, 2.60, and 4.77, the FBP showed four additional degradation peaks.

Acid degradation

FBP very easily undergoes acid degradation; degradation graph showed 3 peaks more having values for RT of 2.27, 2.68, and 7.06 when solution was reacted with HCL for 15 min.

Basic degradation

FBP is found to undergo degradation very quickly in the base-induced degradation analysis. An additional 6 peak values are obtained in the drug content of 2.06, 2.42, 2.68, 2.77, 2.99, and 4.07.

Neutral degradation

No degradation peak observed in neutral degradation.

Photo degradation

At RT 3.16 min, the drug showed additional peaks.

Dry heat degradation

Just 6.32 percent of the medication was depleted under this condition. At RT values of 2.23 min and 3.13 min, the drug showed two additional peaks.

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