This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution license (
The aim of the present study was to optimize a chromatographic method for the analysis of atorvastatin (acid and lactone forms),
Atorvastatin, (3
Structures of atorvastatin and its metabolites and the metabolic pathways of atorvastatin [
To date, several methods have been published for quantification of atorvastatin and its metabolites in a biological matrix (plasma or serum), bulk drug, pharmaceuticals products, and aqueous samples. Various techniques employed include a derivative spectrophotometric method [
Some published papers have dealt with HPLC determination of atorvastatin alone [
Rapid, sensitive and effective methods for determination of drugs and metabolites in biological fluids are desirable [
Highperformance liquid chromatography with ultraviolet or electrochemical detection methods typically has a higher limit of quantification and is usually timeconsuming. Gas chromatography meets the required of limit of quantification but it needs complex derivatization steps. Enzyme inhibition assays are sensitive and easy to implement but are nonspecific and do not provide any information on the metabolite concentrations.
To the best of our knowledge, currently there is no HPLC method employing optimization techniques for determination of atorvastatin and its metabolites in biological sample. This work is an attempt of establishing an HPLC method which would be simple and sensitive and could be an alternative method to LC/MS/MS.
The goal was conducted by using the optimization step accomplished by Derringer’s desirability function. The main advantage of such approach is simultaneous optimization of influencing factors and response variables which enables prediction of chromatographic retention and postulation of optimum conditions for separation.
The presence of aromatic functional groups in the molecular structure of atorvastatin, such as phenyl and pyrol, makes a RPHPLC method with PDA detection suitable for the determination. Since the RPHPLC method is based on using a polar mobile phase, a complete description of the ionization profile of atorvastatin and its metabolites have been used for the evaluation of retention behavior and solubility. Considering the chemical structures and using MarvinSketch software, it is possible to establish a number of proton acceptor and donor groups (
Proton acceptor (A) and proton donor groups (D) of atorvastatin (
According to the obtained results, pKa values are: 4.33 (
In accordance with pKa values, acetate buffer pH 4.6 was used for dilution of these compounds. At this pH, compounds are presented in ionisable and unionisable forms (approximately 50:50). By applying MarvinSketch software logP values were calculated: 5.39 for ATO; 5.08 for
Developing and optimizing isocratic HPLC methods is a complex procedure that requires simultaneous determination of several factors. An HPLC method could be optimized by trial and error methodology, but this approach is time consuming and concerns the influence of one factor at time on response, while the information about other factors as well as interaction between factors is not available. In order to optimize more than one response at a time the chemometric methods which includes factorial design [
This paper deals with multiple response simultaneous optimizations using the Derringer’s desirability function for the development of a reversedphase HPLC method for the simultaneous determination atorvastatin (acid and lactone forms),
The selection of key factors examined for optimization was based on preliminary experiments and prior knowledge from literature. The factors selected for optimization process were: content of acetonitrile in the mobile phase (x_{1}); temperature of column (x_{2}) and flow rate (x_{3}). The variations of these parameters affect chromatographic behavior of substances: changing in a composition of the mobile phase induces a variation of the degree of ionization, and column temperature as well as flow rate affects retention behavior. Retention factor of first peak (
Experimental design of the facecentered central composite design and experimentally obtained responses; x_{1}content of acetonitrile in the mobile phase,
Factor levels  Responses  

Rt 
Rt ATOl  Sym 
Sym 
Sym ATO  Sym ATOl  RRt 

65  35  1  2.851  6.982  1.183  1.122  1.132  1.102  1.102 
65  35  1  2.856  7.089  1.146  1.098  1.107  1.121  1.126 
65  30  1  3.158  7.445  0.753  0.779  0.872  0.979  1.213 
65  35  1  2.874  7.251  1.128  1.168  1.232  1.246  1.207 
65  35  1  3.214  7.861  1.047  1.119  1.206  1.123  1.143 
70  30  1.2  2.905  7.652  0.441  0.503  0.427  0.913  0.556 
60  35  1  2.134  7.128  1.148  1.204  0.986  0.998  1.162 
65  35  0.8  2.302  7.636  0.897  0.962  1.165  0.946  1.237 
65  35  1  2.826  7.241  1.147  1.201  1.136  1.121  1.145 
70  30  0.8  2.342  8.359  0.595  0.436  0.661  0.626  1.362 
60  40  0.8  1.764  6.352  0.432  0.553  0.614  0.661  1.245 
60  30  0.8  2.089  6.452  0.559  0.651  0.801  0.867  1.268 
60  30  1.2  2.453  7.753  0.982  0.812  0.806  0.995  0.585 
70  40  1.2  1.472  6.152  1.284  0.993  0.898  0.962  0.565 
65  40  1.0  1.976  6.573  1.112  1.085  0.903  1.011  1.082 
60  40  1.2  2.543  8.459  1.052  1.021  0.984  0.924  0.581 
70  35  1.0  1.752  6.561  1.064  1.011  1.006  0.922  1.151 
65  35  1.0  2.847  7.139  1.152  1.117  1.127  1.098  1.142 
70  40  0.8  0.472  6.083  1.109  0.595  0.685  0.538  1.365 
65  35  1.2  3.367  8.124  1.203  1.206  1.167  1.121  0.536 
In the preliminary study, resolutions were satisfactory (Rs > 2) and were not considered as critical factors, but the problem was elution of the first peak (
The preliminary screening step was carried out according to a 2^{3} full factorial design with center point in order to identify the significant factors affecting the response. Three repetitions are generally carried out in order to know the experimental error variance and to test the predictive validity of the model. ANOVA generated for 2k Factorial design shows that curvature is significant for all the responses since the pvalue is less than 0.05.
Optimum chromatographic conditions were estimated by a facecentered central composite design using Derringer’s desirability function global optimization approach. Facecentered CCD is chosen due to its flexibility and high efficiency. The design required 2k + 2k + n = 20 runs, where k is the number of parameters studied (k = 3) and n the number of central points (n = 6). Replicates of the central points were performed to estimate the experimental error.
All experiments were conducted in randomized order to minimize the effects of uncontrolled variables that may introduce a bias on the measurements.
In order to get more realistic model, insignificant terms with corresponding pvalue > 0.05 at 95% confidence level, were eliminated from the model by a “backward elimination” process.
Response models with pvalue and statistical parameter (R^{2}) obtained from ANOVA.
b_{0}  b_{1}  b_{2}  b_{3}  b_{12}  b_{13}  b_{23}  b_{11}  b_{22}  b_{33}  R^{2}  R^{2} Adjusted  

Rt 
2.84  −0.20  −0.47  0.38  −0.38  −0.053  0.11  −0.79  −0.17  −0.099  0.9665  0.9363 
0.0001 ^{a}  0.0043  0.0001  0.0001  0.0001  0.4182  0.1175  0.0001  0.1429  0.3732  
Rt  7.27  −0.13  −0.40  0.33  −0.55  −0.51  0.20  −0.44  −0.27  0.6  0.9409  0.8878 
ATOl  0.0001  0.1042  0.0003  0.0014  0.0001  0.0001  0.0397  0.0121  0.0862  0.0018  
Sym

1.14  0.032  0.17  0.14  0.18  −0.13  0.066  −0.033  −0.21  −0.089  0.9794  0.9608 
0.0001  0.0834  0.0001  0.0001  0.0001  0.0001  0.0054  0.3236  0.0001  0.0187  
Sym

1.16  −0.070  0.11  0.13  0.067  −0.020  0.080  −0.086  −0.26  −0.11  0.9799  0.9618 
0.0001  0.0013  0.0001  0.0001  0.0036  0.2755  0.0012  0.0180  0.0001  0.0048  
Sym  1.16  −0.051  0.052  0.036  0.063  −0.049  0.1  −0.16  −0.27    0.9698  0.9426 
ATO  0.0001  0.0130  0.0126  0.0632  0.0079  0.0266  0.0003  0.0006  0.0001  (0.7937)  
Sym  1.12  −0.048  −0.028  0.13  0.030  0.040  0.034  −0.14  −0.11  −0.068  0.9546  0.9138 
ATOl  0.0001  0.0165  0.1222  0.0001  0.1448  0.0593  0.1007  0.0013  0.0077  0.0600  
RRt

1.14  0.016  −0.015  −0.37  0.005  −0.032  0.003  0.026  0.017  −0.24  0.9907  0.9823 
0.0001  0.2247  0.2593  0.0001  0.7284  0.0391  0.8235  0.2831  0.4718  0.0001 
^{a} pvalue.
Since the residuals of model describing RRt
Reduced response models and statistical data obtained from ANOVA (after backward elimination).
Response  Reduced response models *  R^{2}  Adj. R^{2}  Pred. R^{2}  Adequate precision  RSD (%) 

Rt 
2.83 − 0.02

0.9454  0.9258  0.8786  26.388  7.88 
Rt ATOl  7.23 − 0.40

0.9007  0.8428  0.7602  12.984  3.88 
Sym 
1.13 + 0.17

0.9695  0.9517  0.8050  22.114  6.01 
Sym 
1.16 − 0.07

0.9772  0.9606  0.9108  22.759  5.47 
Sym ATO  1.16 − 0.051

0.9564  0.9310  0.7508  19.841  6.25 
Sym ATOl  1.11 + 0.13

0.8370  0.8064  0.7443  15.555  8.27 
RRt 
0.94 − 1.5 × 10^{−3}

0.9960  0.9925  0.9806  45.543  1.59 
* Coefficients with
The qualities of the fitted mathematical models were examined by the coefficient of correlation R2. But, this coefficient always decreases when a variable is eliminated from a regression model. Because of that, the adjusted R2 (which takes the number of selected variables) was taken into account. The obtained adjusted R2 were within acceptable limits of (R2 ≥ 0.80), indicating that the experimental data were a good fit to the equations.
All the reduced models have
It is obvious from
In order to get the best chromatographic performance, the multicriteria methodology [
In both cases, d_{i} will vary nonlinearly while approaching the desired value. But with a weight of 1, d_{i} varies linearly. In this work we chose weights equal to 1 for all responses. The partial desirability functions are then combined into a single composite response, the socalled global desirability function D, defined as the geometric mean of the different d_{i}values:
A value of D close to 1 means that the combination of different criteria is globally optimal. If any of the responses or factors falls outside their desirability range, the overall function becomes zero. The goals of multicriteria optimization for seven responses are shown in
Criteria for multivariate optimization of the individual responses.
Response  Goal  Weight  Lower limit  Upper limit  Importance 

Acetonitrile  range  1  60  70  3 
Temperature  range  1  30  40  3 
Flow rate  range  1  0.8  1.2  3 
Rt 
range  1  0.472  3.367  3 
Rt ATOl  target = 7  1  6.083  8.459  4 
Sym 
target = 1  1  0.432  1.284  3 
Sym 
target = 1  1  0.436  1.206  3 
Sym ATO  target = 1  1  0.427  1.232  3 
Sym ATOl  target = 1  1  0.538  1.246  3 
RRt 
range  1  0.856  1.366  3 
The optimization procedure was conducted under listed conditions and restrictions. The partial desirability functions (d_{i}) of each of the responses, and the calculated geometric mean as the maximum global desirability function (D = 0.919) are presented in
Bar graph showing individual desirability values (
Desirability function calculations were performed using DesignExpert^{®} 7.0. Obtained results are graphically presented (
Graphical representation of the constraints accepted fot the determination of global desirabilty and obtained optimal conditions.
For better visualization of the results, the global desirability function D is presented in a form of a 3D plots (
Threedimensional graph: (
The coordinates related to the functions maximum are selected as the best operating conditions. The best chromatographic conditions are achieved with 61.93% of ACN, temperature of column 31.44 °C and flow rate 1.03 mL min^{−1}. A representative chromatogram is shown in
LCPDA chromatogram of
LCPDA chromatogram of
Atorvastatin,
Stock solutions of atorvastatin,
Biological samples were obtained from Wistar rats (
The HPLC analyses were done by using an Agilent Technologies 1200 Series (Santa Clara, CA, USA) chromatographic system equipped with a PDA detector, binary pumps G1312A, diode array detector G1315D, degasser G1379B and manual injector G1328B. The sample loop was 20 μL. Separations were performed on a ZORBAX Eclipse Plus C18 Analytical (4.6 mm × 250 mm, 5 μm particle size) column (Agilent) with detection at 254 nm. Experiments were prepared according to the plan given in
Experimental design, statistical analysis and desirability function calculation were performed by using MarvinSketch 5.8.2 (Chem Axon Ltd., Somerville, MA, USA, and Budapest, Hungary) and DesignExpert^{®} 7.0 (StatEase Inc., Minneapolis, MN, USA).
The chemometric approach for optimization of chromatographic separation of atorvastatin (acid and lactone) and its metabolite has been demonstrated. The chemometric methodology chosen for the particular objectives was very successful in the retention behavior exploration.
Full factorial design was used to screen the chromatographic factors that had significant effects on the analysis time response. The significant factors were optimized by applying central composite design and surface response methodology. Since there was a mix of responses with different targets, a multicriteria decisionmaking tool (Derringer’s desirability function) was applied. After defining a global desirability according to the accepted constraints, optimal chromatographic conditions were established.
The goal of this optimization of RPHPLC was not to compete with the more potent LC/MS/MS assay with very low LOQ values. The proposed optimized RPHPLC, as a method available worldwide, was applied to the quantitative analysis of real plasma samples, with satisfying analytical parameters. This investigation also showed that chromatographic techniques coupled with chemometric tools can provide a complete profile of a separation process, making this combined technique a powerful analytical tool.
We gratefully acknowledge the financial support from the Ministry of Science and Environment of the Republic of Serbia, grants OI 172041, OI 172 043, and TR 34031.