Analytical Quality by Design: Achieving Robustness of an LC-CAD Method for the Analysis of Non-Volatile Fatty Acids

An alternative to the time-consuming and error-prone pharmacopoeial gas chromatography method for the analysis of fatty acids (FAs) is urgently needed. The objective was therefore to propose a robust liquid chromatography method with charged aerosol detection for the analysis of polysorbate 80 (PS80) and magnesium stearate. FAs with different numbers of carbon atoms in the chain necessitated the use of a gradient method with a Hypersil Gold C18 column and acetonitrile as organic modifier. The risk-based Analytical Quality by Design approach was applied to define the Method Operable Design Region (MODR). Formic acid concentration, initial and final percentages of acetonitrile, gradient elution time, column temperature, and mobile phase flow rate were identified as critical method parameters (CMPs). The initial and final percentages of acetonitrile were fixed while the remaining CMPs were fine-tuned using response surface methodology. Critical method attributes included the baseline separation of adjacent peaks (α-linolenic and myristic acid, and oleic and petroselinic acid) and the retention factor of the last compound eluted, stearic acid. The MODR was calculated by Monte Carlo simulations with a probability equal or greater than 90%. Finally, the column temperature was set at 33 °C, the flow rate was 0.575 mL/min, and acetonitrile linearly increased from 70 to 80% (v/v) within 14.2 min.


Introduction
For decades, pharmaceutical analysis was essentially a discipline in which indispensable chromatographic methods were developed on a trial-and-error basis. In recent years, however, advances in computational tools have increasingly enabled analysts to efficiently select the most appropriate separation conditions, taking into account the entire experimental space. Due to its resource-conscious nature, chemical intelligence has been gaining momentum and is now a key dimension of both the established Pharma 4.0 as well as emerging Pharma 5.0 industry perspectives.
As the heart of the pharmaceutical industry, new digital technologies are bringing prosperity, improved process understanding, and better time management across all segments, with a special focus on already overburdened quality control labs [1][2][3][4]. A particular benefit is the supply of the more consistent quality method and ultimately patient safety, as failure of the analytical method can have dramatic implications for human health [5]. Consequently, a variety of in silico solutions for selecting fit-for-purpose analytical methods have been developed to date.

Analytical Target Profile: Defining the Scope of the Method
AQbD workflow begins by summarizing the characteristics of the analytical that ideally will be achieved to guarantee the desired quality. The first step of namic procedure thus includes defining the analytical target profile (ATP). The of an ATP is in line with the concept of the quality target product profile define ICH Q8 guideline and was recently introduced through the ICH Q14 guidelin The ATP involves a description of the method purpose, the selection of an app analytical technique that is able to serve this purpose, and finally, the selection quality attributes that are going to be measured along with method performan acteristics. Therefore, the declaration of the ATP represents the base step enabling definition of desirable analytical method attributes with associated acceptanc ria [3]. In the specific case presented in this study, achieving the baseline separ eight non-volatile FAs (see Figure 1) in a sufficiently short analysis run-time po purpose of the intended LC−CAD method.

Design of Experiments in Modeling of Critical Method Attributes
The essence of the ATP was more precisely presented by defining critical quality attributes, called critical method attributes (CMA), associated with t ceptance criteria. The selection of CMAs refers to a set of chromatographic sep descriptors that are going to be used as a proper indicator of the capability of a me reach the predefined ATP. In LC, these method attributes are derived from the sep of critical pair of analytes, specific requirements for the peak shape, and/or the nu

Analytical Target Profile: Defining the Scope of the Method
AQbD workflow begins by summarizing the characteristics of the analytical method that ideally will be achieved to guarantee the desired quality. The first step of this dynamic procedure thus includes defining the analytical target profile (ATP). The concept of an ATP is in line with the concept of the quality target product profile defined in the ICH Q8 guideline and was recently introduced through the ICH Q14 guideline [3,23]. The ATP involves a description of the method purpose, the selection of an appropriate analytical technique that is able to serve this purpose, and finally, the selection of the quality attributes that are going to be measured along with method performance characteristics. Therefore, the declaration of the ATP represents the base step enabling further definition of desirable analytical method attributes with associated acceptance criteria [3]. In the specific case presented in this study, achieving the baseline separation of eight non-volatile FAs (see Figure 1) in a sufficiently short analysis run-time posed the purpose of the intended LC-CAD method.

Design of Experiments in Modeling of Critical Method Attributes
The essence of the ATP was more precisely presented by defining critical method quality attributes, called critical method attributes (CMA), associated with their acceptance criteria. The selection of CMAs refers to a set of chromatographic separation descriptors that are going to be used as a proper indicator of the capability of a method to reach the predefined ATP. In LC, these method attributes are derived from the separation of critical pair of analytes, specific requirements for the peak shape, and/or the number of theoretical plates, etc., and may vary depending on the method purpose [24]. Accordingly, CMAs are influenced by the wide range of LC system operating parameters, which are therefore accordingly denoted as critical method parameters (CMP) [3,6,7,10]. In general, LC method parameters may be classified in a way that all aspects of intended LC analysis, such as sample, mobile phase, detection, or column-related parameters, are properly taken under consideration ( Figure 2). The analysis of plotted Ishikawa or fish bone diagram followed with an appropriate risk-based approach or based on previous knowledge was used to provide comprehensive method understanding, as well as a scientifically based definition of CMPs. Using an approach known as CNX, important decisions were made about which method parameters should be kept under Control, which could be considered Noise factors, and finally, which method parameters required eXperimental evaluation of the associated acceptable ranges. The parameters that needed to be kept under control were set to constant values. Apart from providing proper insight into the intended method properties, this approach provides inputs for prospective method control strategy [25,26]. rmaceuticals 2023, 16, x FOR PEER REVIEW 4 o ingly, CMAs are influenced by the wide range of LC system operating parameters, wh are therefore accordingly denoted as critical method parameters (CMP) [3,6,7,10] general, LC method parameters may be classified in a way that all aspects of intended analysis, such as sample, mobile phase, detection, or column-related parameters, properly taken under consideration ( Figure 2). The analysis of plotted Ishikawa or bone diagram followed with an appropriate risk-based approach or based on previ knowledge was used to provide comprehensive method understanding, as well as a sc tifically based definition of CMPs. Using an approach known as CNX, important decisi were made about which method parameters should be kept under Control, which could considered Noise factors, and finally, which method parameters required eXperimen evaluation of the associated acceptable ranges. The parameters that needed to be kept der control were set to constant values. Apart from providing proper insight into the tended method properties, this approach provides inputs for prospective method con strategy [25,26]. Design of Experiments (DoE), as one of the constitutive concepts of the AQbD p adigm, represents an excellent in silico tool for the resource-efficient developmen various chromatographic methods. Primarily, DoE provides high-quality information a consequence to the simultaneous variation of experimental LC−CAD variables. Ho ever, when the number of factors to be considered raises, the increase of the cost and time needed for the analyses follows as well. This is especially evident when categor variables (e.g., type of column) are involved. Adjusting all the potentially influen factors, thus, almost certainly ends up with the significant quantity of runs that is practical to be carried out [27]. Alternatively, since profound background knowledg available, the parameter optimization can be accomplished through a multistage strat that includes the following: (1) preliminary scouting of the categorical factors, nam column chemistry and solvent type; (2) fixing some of the high-risk factors at levels t ensure the fulfillment of practical requirements; (3) fine-tuning of secondary high-r factors via Response Surface Methodology (RSM) design; (4) construction of the MO for robust method performance [28]. This sequential strategy provides also independ assessment of potential interactions between the significant factors for a better und standing of the method.

Scouting Stage: Selection of Organic Solvent and Column Type
It is known that the chemistry of the stationary phase and the type of organic mo fier are primary factors in determining the retention behavior of compounds in rever phase (RP) chromatography [24]. In this study, the type of organic modifier (ACN) w Design of Experiments (DoE), as one of the constitutive concepts of the AQbD paradigm, represents an excellent in silico tool for the resource-efficient development of various chromatographic methods. Primarily, DoE provides high-quality information as a consequence to the simultaneous variation of experimental LC-CAD variables. However, when the number of factors to be considered raises, the increase of the cost and the time needed for the analyses follows as well. This is especially evident when categorical variables (e.g., type of column) are involved. Adjusting all the potentially influential factors, thus, almost certainly ends up with the significant quantity of runs that is impractical to be carried out [27]. Alternatively, since profound background knowledge is available, the parameter optimization can be accomplished through a multistage strategy that includes the following: (1) preliminary scouting of the categorical factors, namely, column chemistry and solvent type; (2) fixing some of the high-risk factors at levels that ensure the fulfillment of practical requirements; (3) fine-tuning of secondary high-risk factors via Response Surface Methodology (RSM) design; (4) construction of the MODR for robust method performance [28]. This sequential strategy provides also independent assessment of potential interactions between the significant factors for a better understanding of the method.

Scouting Stage: Selection of Organic Solvent and Column Type
It is known that the chemistry of the stationary phase and the type of organic modifier are primary factors in determining the retention behavior of compounds in reversed phase (RP) chromatography [24]. In this study, the type of organic modifier (ACN) was selected a priori. This decision was supported by the fact that ACN-based mobile phases generally have high elution strength, which is beneficial when working with lipophilic analytes such as FAs. Nevertheless, a better evaporation profile over frequently used MeOH was meaningful since volatility is a very desirable characteristic in terms of the adopted detection technique [29].
On the other hand, regarding the selection of the stationary phase as the core of the chromatographic separation, a screening with four RP columns (different chemistry and different dimensions) was carried out. Using a mix of the homologous series composed of myristic acid, palmitic acid, and stearic acid and the gradient program of Ilko et al. [21] (see Table S1), a C 8 , a C 12 , and two C 18 columns were compared. Since the Hypersil Gold C 18 columns showed the best results in terms of retention and peak shapes (see Figure 3 and Table S3), and at the same time had the lowest level of background current of the CAD, this column was selected for further experiments. Retention times obtained using the other C 18 core-shell column were notably longer (no elution of stearic acid within the gradient program) and, interestingly, greatest with the C 12 column (see Figure 3). Compared to the other columns, this column has the highest carbon load and a considerably higher specific surface area, both of which are factors that influence retention [24]. With the C 8 column, the three FAs were separated but showed inferior peak shapes and worse separation performance (see Table S3). MeOH was meaningful since volatility is a very desirable characteristic in terms of the adopted detection technique [29].
On the other hand, regarding the selection of the stationary phase as the core of the chromatographic separation, a screening with four RP columns (different chemistry and different dimensions) was carried out. Using a mix of the homologous series composed of myristic acid, palmitic acid, and stearic acid and the gradient program of Ilko et al. [21] (see Table S1), a C8, a C12, and two C18 columns were compared. Since the Hypersil Gold C18 columns showed the best results in terms of retention and peak shapes (see Figure 3 and Table S3), and at the same time had the lowest level of background current of the CAD, this column was selected for further experiments. Retention times obtained using the other C18 core-shell column were notably longer (no elution of stearic acid within the gradient program) and, interestingly, greatest with the C12 column (see Figure 3). Compared to the other columns, this column has the highest carbon load and a considerably higher specific surface area, both of which are factors that influence retention [24]. With the C8 column, the three FAs were separated but showed inferior peak shapes and worse separation performance (see Table S3).  Table S1). Injection of the test solution containing myristic acid (C14:0), palmitic acid (C16:0), and stearic acid (C18:0) using the Symmetry Shield RP8 column (100 × 3.0 mm, 3.5 µm), Synergi Max-RP C12 column (100 × 4.6 mm, 4 µm), Hypersil Gold C18 column (150 × 2.1 mm, 3.0 µm), and the Kinetex Evo C18 column (150 × 4.6 mm, 2.6 µm).

Fixing Some of the High-Risk Factors at Reasonable Levels
After fixing the stationary phase and the type of organic solvent, it was necessary to optimize the factors of subsequent importance/factors also posing high-risk toward baseline separation.
The addition of formic acid to the mobile phase is on the one hand necessary for the peak shape, on the other hand it affects the CAD's level of background current [29]. The concentration of 0.05% (v/v) was an acceptable compromise that ensures robust protonation of the FAs and a low level of background current (<1 pA). A decrease to 0.02% (v/v) was associated with peak tailing and an increase in the symmetry factor to values above 1.20.  [21] (see Table S1). Injection of the test solution containing myristic acid (C14:0), palmitic acid (C16:0), and stearic acid (C18:0) using the Symmetry Shield RP8 column (100 × 3.0 mm, 3.5 µm), Synergi Max-RP C 12 column (100 × 4.6 mm, 4 µm), Hypersil Gold C 18 column (150 × 2.1 mm, 3.0 µm), and the Kinetex Evo C 18 column (150 × 4.6 mm, 2.6 µm).

Fixing Some of the High-Risk Factors at Reasonable Levels
After fixing the stationary phase and the type of organic solvent, it was necessary to optimize the factors of subsequent importance/factors also posing high-risk toward baseline separation.
The addition of formic acid to the mobile phase is on the one hand necessary for the peak shape, on the other hand it affects the CAD's level of background current [29]. The concentration of 0.05% (v/v) was an acceptable compromise that ensures robust protonation of the FAs and a low level of background current (<1 pA). A decrease to 0.02% (v/v) was associated with peak tailing and an increase in the symmetry factor to values above 1.20.
In the next step before optimization by RSM, the initial and final ACN percentage of the gradient program should be defined. Since data from the scouting stage were available, these high-risk factors could be efficiently fixed to meet practical requirements. Thus, the FA test mixture was subjected to a gradient from 65% (v/v) to 85% (v/v) ACN in 15 min at Pharmaceuticals 2023, 16, 478 6 of 17 20 • C and 0.7 mL/min. Under these conditions, a baseline separation between all analytes was achieved, but the elution window was relatively wide and a large expenditure of time that preceded the first-eluting peak, i.e., that followed the last-eluting compound, was evident (see Figure S1a).
To remove empty space before the elution of the least retained FA, the initial and final percentages of ACN in the mobile phase required slight adaptation. It was calculated that at the retention time of the linoleic acid (4.94 min) and the stearic acid (13.23 min), the mobile phase contained approximately 69.8% (v/v) and 80.8% (v/v) of ACN, respectively. This calculation was performed using the Equation (1): where ϕ e , ϕ i , and ϕ f are the content of organic solvent at the elution, the beginning of gradient, and the end of gradient, respectively. In the same equation, t grad , t r , and t D refer to the gradient time, the retention time of the least (or most) retained analyte, and the dwell time (≈ 1.37 min at 700 µL/min), respectively. For simplicity, the initial ACN percentage was rounded to 70% (v/v) while the final ACN percentage was rounded to 80% (v/v). Consequently, the gradient range on Hypersil Gold C 18 stationary phase was modified from 65-85% (v/v) ACN to 70-80% (v/v) ACN. In the following test run with the modified gradient conditions, acceptable chromatographic behavior was achieved for all the analytes (see Figure S1b).

Fine-Tuning of CMPs via RSM
Once the combination of initial and final percentage of ACN in the mobile phase was set, secondary parameters were optimized to improve the separation. After subjecting model mixture to adopted settings, it was noted that the critical peak pairs were formed by α-linolenic acid and myristic acid (peaks 1 and 2), i.e., oleic acid and petroselinic acid (peaks 6 and 7). Hence, separation criteria, S 1-2 and S 6-7 between the adjacent peaks were proclaimed as the first two CMAs. Separation criterion S is recognized a convenient way of measuring the baseline separation in gradient RP−LC. This is achieved by avoiding some of the disadvantages that standard parameters, such as resolution and peak capacity, suffer, according to [30]. S criterion is calculated using Equation (2): In Equation (2) t 2,start represents the beginning of the later peak, while t 1,end is the end of the former peak of two consecutive peaks (t 2,start > t 1,end ).
To further fulfill the definition of the ATP, a third CMA was taken into consideration. It was related to the retention of the last-eluting peak, and quantified via the retention factor k.
When observed all together, the satisfactory separation of all critical pairs and the reasonable retention of the last-eluting peak were identified as the goals of utmost importance and the supreme ATP of the proposed method. To fulfill these pre-defined CMAs, we varied the flow rate (x 1 ), the gradient time (x 2 ), and the column temperature (x 3 ), as variables that, besides the above-considered (categorical and numerical) factors, highly impact the RP−LC behavior [24,31,32]. These factors were thus declared as relevant CMPs and were subjected to RSM experiments. In order to adequately describe the experimental space, registered CMPs were simultaneously varied according to a face-centered Central Composite Design (CCD) requiring 18 runs in total (see Table 1). Broader ranges of the listed factors were used to examine as wide a space as possible. Clearly, these conditions constituted a trade-off between baseline separation and reduced analysis run-time. Using Equation (2) the first two CMAs, S 1-2 and S 6-7 , were calculated and direct modeling of S criteria were attempted. However, this resulted in models with poor predictive performance. One possible reason for the inadequacy of direct modeling could lie in the fact that baseline separation depends on multiple factors (e.g., the size and shape of neighboring peaks). Different assessment of S criterion, in this regard, likely comes to the fore in the case of small time differences. The impact of CMPs on the mentioned CMAs was therefore modeled indirectly, which has the advantage that it does not suffer from faulty estimation of baseline separation [33]. Indirect modeling means that mathematical models were developed for the retention times on the chromatogram that correspond to the end of the first peak and the beginning of the second peak. After developing the mathematical models for the corresponding retention times, they were used to calculate separation criteria S. The baseline separation was achieved when S criterion was greater than zero. On the other hand, the third CMA, k of the last-eluting peak, was modeled directly (Table 1). Since it carried information about the analysis runtime, the goal was to minimize its value.
In order to develop desired RSM models, empirical mathematical functions y = f (x 1 , x 2 , x 3 ) were fitted to experimentally acquired data [7,34]. Mathematical models are obtained by applying multiple regression and the least squares method, resulting in polynomial equations. The general form of the polynomial expression is given by Equation (3): In the present function, y is the modeled response value (CMA), x i indicates independent variable values (x 1 , x 2 , x 3 represent CMPs as in Table 1), and b-s are model coefficients which indicate the magnitude and the trend of respective equation term's influence, with the exception of b 0 , which is the intercept. Thus, b i , b ij , and b ii indicate single-factor, two-factor interaction (where i = j), and second-order value of factor effects [34].
The model development process was governed by the analysis of variance, which compares level-dependent and random error variances and quantifies model quality by the adjusted coefficient of determination (adj. R 2 ) [34,35]. A high value of adj. R 2 implies concurrence of predicted and measured response values in form of explained variance ratio taking into account degrees of freedom [36] and can be tuned by excluding uninfor-Pharmaceuticals 2023, 16,478 8 of 17 mative model terms. Model validation is often performed with the statistical assessment of lack of fit (p > 0.05), a numerical estimation similarity of residual and experimental variance [31,35]. Obtained mathematical models with quality assessment parameters are presented in Table 2. The mathematical transformations of results were applied in line with the Box-Cox plot results.

Computation of the MODR via Monte Carlo Simulations
The combination of CMP values providing an optimal chromatogram could be estimated simply by overlapping the collected response surfaces from generated RSM models, but this method does not follow the risk management approach [35]. Given that the ICH Q8 guideline stated that the design space (DS) is a multidimensional combination of input variable values and parameters that ensure method quality, it is essential to apply probability-based tools for desired response estimation [23]. As postulated in the ICH Q9 and Q14 guidelines, the definition of DS within analytical method development should be accompanied with quality risk analysis as well as setting up of an appropriate risk control strategy [3,26]. In that respect, the zone of theoretical robustness or an MODR needs to be further defined using statistical tools based on the understanding of the CMAs' measurement uncertainty and the quantification of the risk of reaching the predefined ATP [3,24,33,35]. The initial step is the discretization of experimental space and creation of uniformly distributed grid points for gradient time [8:0.35 :15], flow rate [0.5:0.01:0.7], and column temperature [30:1:40]. Thus, a total number of combinations of CMPs was 21 × 21 × 11 = 4851. To achieve the assurance of quality in terms of meeting predefined acceptance criteria set by the ATP with desired probability, a Monte Carlo (MC) simulation was performed. It included 5000 iterations to propagate the error in model coefficients' calculation when the error distribution equal to the calculated standard error was added to the estimated model coefficients. In this way response distribution was obtained for each operating condition corresponding to the created 4851 grid points. The criteria for satisfactory CMAs' values were set as follows: S 1-2 > 0, S 6-7 > 0, and k 8 < 18.8, and MODR was computed for the probability of 90% to meet defined criteria.
From the acquired MODR graphical presentation (Figure 4), the margins of the safe zone of theoretical method robustness may be extracted, pointing out to the limits to which the reaching of the ATP would not be compromised. The whole MODR central figure is considered as a "safe zone" from which the working point can be selected randomly. However, it is recommendable to select the working point from the central part (illustrated in Figure 4 in yellow) since the blue colored part represents the edges at risk of falling out of the robust region. Setting up these boundaries is needed for defining a proper method control strategy as required by the ICH Q9 guideline [26]. The working point (0.575 mL/min flow rate, 14.2 min gradient time, and 33 • C column temperature) was further selected from the center of the MODR in order to additionally contribute to method robustness. Figure 5 shows a chromatogram obtained experimentally at the selected working point.
zone of theoretical method robustness may be extracted, pointing out to the limits to w the reaching of the ATP would not be compromised. The whole MODR central fig considered as a "safe zone" from which the working point can be selected rand However, it is recommendable to select the working point from the central part (illus in Figure 4 in yellow) since the blue colored part represents the edges at risk of fallin of the robust region. Setting up these boundaries is needed for defining a proper m control strategy as required by the ICH Q9 guideline [26]. The working point mL/min flow rate, 14.2 min gradient time, and 33 °C column temperature) was furth lected from the center of the MODR in order to additionally contribute to method ro ness. Figure 5 shows a chromatogram obtained experimentally at the selected wo point.

Validation
The final chromatographic method (see Section 2.3) was validated with regard to guideline ICH Q2 (R1) [37]. In the context of this, specificity, linearity and range, accuracy, repeatability, and limit of quantitation (LOQ) were investigated. When using the AQbD approach, an experimental evaluation of robustness is not mandatory as long as one works within the MODR framework (see Section 2.3) [7,10].

Validation
The final chromatographic method (see Section 2.3) was validated with regard to guideline ICH Q2 (R1) [37]. In the context of this, specificity, linearity and range, accuracy, repeatability, and limit of quantitation (LOQ) were investigated. When using the AQbD approach, an experimental evaluation of robustness is not mandatory as long as one works within the MODR framework (see Section 2.3) [7,10].
Based on the signal-to-noise (S/N) ratios obtained at the calibration level of 1 µg/mL, dilutions of the FAs were prepared to obtain solutions with a S/N of 10:1. The LOQs for each analyte were less than 2 ng on the column (see Table 3). This improvement compared to the UPLC method [22] is probably due to the lower flow rate (0.575 vs. 1.5 mL/min) and thus lower level of background noise. As expected, the highest value was determined for myristic acid (1.85 ng/column) as the shortest-chain FA examined. This reduced detector response is also evident from the high value of the correction factor compared to the other FAs (with respect to oleic acid).

Application Examples
To investigate the performance of the HPLC-CAD method developed here for the analysis of non-volatile FAs using the AQbD approach, we selected magnesium stearate as another application example in addition to PS 80.
In the chromatogram overlay of the PS 80 sample with the blank extraction, no interfering peaks with the reagents or other PS 80 components were visible (see Figure 6a), which was considered as evidence of specificity (see above Section 2.4.1). In addition to the peaks of the indicated eight FAs, three unknown peaks were detectable at 3.0 min, 9.3 min, and 12.6 min with an S/N of just above 10. If unknown signals appear in future samples, especially if their signals increase, an MS analysis for structure elucidation can be performed with the proposed method without adjustments. These three unknown peaks had a combined percent area of only 0.62%. As shown in Table 4, the tested batch of PS 80 complies with Ph. Eur. FA compositional requirements [15]. As already discussed by Ilko et al. [21], the pharmacopoeial method does not distinguish between the two C18:1 isomers oleic acid (ω-9) and petroselinic acid (ω-12). However, since the percentage content of this additional unknown FA is higher at 6.35% compared to the PS 80 batches examined at that time (<3%), it could be useful to introduce a specification for quality control in the future. Due to poor solubility in water and pure organic solvents, magnesium stearate was dissolved directly in 100.0 mL of a mixture corresponding to the initial gradient conditions and treated for 10 min in an ultrasound bath before analysis. Specificity was similarly ensured by comparison with the chromatogram of the blank sample (see Figure 6b). In the chromatogram of the test solution, besides a large injection peak and the peaks of palmitic and stearic acid, only one additional unknown peak at the end of the gradient was detected. The high intensity of the injection peak is mainly due to the magnesium ions, which are not retained on the C 18 column used and therefore do not interfere with the determination of the FAs. A quantitative, chromatographic determination of magnesium ions as an alternative to titration has already been demonstrated for magnesium stearate with the nano quantity analyte detector [38] as another type of aerosol−based detector and would theoretically also be possible using the CAD with, e.g., a suitable mixed-mode column [39]. The investigated Mg stearate sample fulfills the requirements of the Ph. Eur. [18] (see Table 4) and the LC-CAD method should also be a simplification for the other monographed salts of stearic acid (i.e., Na, Ca, Zn, and Al). The advantages are not only time savings (no derivatization and shorter analysis time), but also a less error-prone procedure that does not require toxic boron trifluoride as a catalyst.

Preparation of Solutions and Samples
A 0.1 mg/mL test solution was prepared by dissolving 10.0 mg of magnesium stearate in 100.0 mL of the solvent mixture. A reference solution of magnesium acetate × 4 H 2 O (0.05 mg/mL) in water was also analyzed.

HPLC-CAD Conditions and Equipment
All experiments were performed on a Vanquish TM Flex modular chromatographic system (Thermo Fisher Scientific, Germering, Germany) consisting of a dual pump F (two independent ternary solvent blending flow streams in one housing) with an online vacuum degasser, a thermostatted split sampler, a thermostatted column compartment with an active pre-heater, and a diode array detector in-line with a Vanquish TM Horizon CAD. The CAD was supplied with nitrogen gas from a 1010 Corona Nitrogen generator (Peak Scientific Instruments, Inchinnan, UK) connected to the in-house compressed air system. The instrument was controlled, and runs were processed using the Chromeleon TM 7.3 Chromatography Data System (Thermo Fisher Scientific).
The face-centered CCD was applied in this study, as an RSM design, in order to obtain mathematical equations, their coefficients, and related standard errors [30]. Experimental factors and their levels varied in CCD are presented in Table S2, while the plan of experiments obtained according to CCD is displayed in Table 1 (see Section 2.2.3). Components of the mobile phase were degassed in an ultrasonic bath for 15 min prior to use. After each change of LC conditions, the system was equilibrated for at least 10 column volumes, followed by a blank sample (water:ACN = 25:75, v/v).
In order to achieve the best sensitivity, the final tuning of the CAD operating parameters was conducted. The CAD evaporation temperature, filter constant, and data collection rate were varied within following ranges: 20-40 • C, 1-10 s, and 1-10 Hz, respectively.
The final chromatographic conditions consisted of water and ACN each with 0.05% (v/v) formic acid as mobile phase components and the Hypersil Gold C 18 column as stationary phase. The column temperature was set to 33 • C and the injection volume to 10 µL at a flow rate of 0.575 mL/min. The percentage of the organic modifier was linearly increased starting from 70% (v/v) to 80% (v/v) within 14.2 min. This was followed by a re-equilibration step of 2.8 min. The CAD evaporation temperature was set to 25 • C, the filter constant to 3.6 s, and the data collection rate to 5 Hz. The PFV was set to 1.0 and a log-log transformation was used during validation experiments.

Tools Used for Generating RSM and MODR
The sequence of the required experimental runs (Table 1) was obtained by the Design Expert 7.0.0 software (Stat-Ease Inc., Minneapolis, MN, USA); it was also used for fitting mathematical models to the collected results and model quality assessment. In order to determine the zone of theoretical robustness based on developed CCD models, their coefficients, and standard errors MATLAB ® R2018b (MathWorks, Natick, MA, USA) software was used. In particular, MATLAB served for indirect modeling of CMAs, grid point discretization, running MC simulations (statistical tool that takes into account the probability of meeting defined quality criteria) and, finally, generating an MODR graphical presentation.

Conclusions
Using a risk-based systematic AQbD approach, a robust and trustworthy HPLC-CAD method was developed for the analysis of eight FAs. The performance of the method was ensured within the defined MODR, allowing a long-lasting analytical method lifecycle where changes in chromatographic conditions within the MODR do not require regulatory notification and revalidation. The applicability of the method for the analysis of nonvolatile FAs in PS 80 and magnesium stearate was demonstrated. Its main limitation is the applicability in the analysis of only non-volatile FAs due to the basic operating principles of CADs. However, the main advantages of the proposed HPLC-CAD method are its less error-prone and time-saving nature compared to the pharmacopoeial GC method, and the higher sensitivity compared to the already existing HPLC-CAD method. The future perspective is the development of uniform sample preparation procedure for FA analysis. The powerful AQbD approach could be used in the optimization of potentially critical process parameters such as the saponification of the polysorbate and the subsequent extraction of the FAs.
Supplementary Materials: The following supporting information can be downloaded at https://www. mdpi.com/article/10.3390/ph16040478/s1, Figure S1: Chromatograms of the test solution applying different scouting gradients; Table S1: Gradient program for column screening; Table S2: Investigated levels of the CMPs; Table S3: Peak characteristics of the column screening.