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AN0184162980;kua01apr.25;2025Apr03.03:00;v2.2.500
Enhancing quality-by-design through weighted goal programming: a case study on formulation of ultradeformable liposomes
Introduction: Optimization of pharmaceutical formulations requires advanced tools to ensure quality, safety, and efficacy. quality-by-design (QbD), introduced by the FDA, emphasizes understanding and controlling processes early in development. Advanced optimization methods, such as desirability, have surpassed traditional single-objective techniques. Others, such as weighted goal programming (WGP) offers unique advantages by integrating decision-maker preferences, enabling balanced solutions for complex drug delivery systems. This study applies WGP to optimize timolol (TM)-loaded nanoliposomes aligning with QbD principles. Methods: The optimization process followed six steps: identifying factors and responses, developing a Design of Experiments (DoE) plan, defining ideal and anti-ideal points, setting aspiration levels, assigning relative weights, and applying WGP compared to desirability function. Minimized and balanced deviations from aspiration levels served as criteria for selecting the most robust optimization results. Six responses were analyzed: vesicle size ( z 1 ) , polydispersity index ( z 2 ) , zeta potential ( z 3 ) , deformability index ( z 4 ) , phosphorus content ( z 5 ) , and drug entrapment efficiency ( z 6 ). Results: WGP produced a more balanced formulation that simultaneously optimized multiple responses. By incorporating the importance of each response, the WGP approach improved control over size, colloidal stability, and drug entrapment, based on its mathematical formulation. Comparative analysis with the desirability function confirmed that WGP effectively addressed potential tradeoffs without oversimplifying conflicting objectives. Conclusions: This case-study demonstrates WGP potential as an advanced multi-objective optimization tool for pharmaceutical applications, improving upon traditional methods in complex formulations. Its ability to harmonize multiple critical attributes in line with QbD highlights its value in developing high-quality pharmaceutical products.
Keywords: Quality-by-design; goal programming; desirability; multiobjective optimization; ultradeformable liposomes
Introduction
In the pharmaceutical field, optimizing drug formulations becomes crucial for ensuring safety, effectiveness, and quality. This process involves meticulous adjustments to factors influencing therapeutic performance and regulatory [[1]].
A key advancement is the adoption of quality-by-design (QbD), a U.S. FDA framework ensuring quality by addressing critical variables early in development [[2]]. QbD enhances product quality, regulatory compliance, innovation, and cost-efficiency, key elements in the competitive and heavily regulated pharmaceutical industry. This approach also identifies and refines formulation and manufacturing parameters to ensure consistent and high-quality products [[3]].
In the context of nanotechnology-based drug delivery, where nanoscale particles, structures, and behaviors can be sensitive to even small changes in formulation or process, optimizing these systems through QbD is vital to achieving high-quality, reproducible, and effective therapies [[4]].
Historical methods, such as Design of Experiments (DoE), remain foundational. DoE systematically investigates how multiple factors affect critical quality attributes (CQAs). Response surface methodology (RSM), a statistical approach, models and optimizes relationships between variables and outcomes [[6]]. Monte Carlo simulations assess process robustness under varying conditions, offering insights into how input variability impacts final products [[8]].
Nevertheless, traditional methods, while valuable, often fall short when addressing the complexity of modern drug delivery systems. Machine learning (ML) algorithms further advance optimization by identifying intricate patterns in large datasets, particularly useful in complex drug delivery systems [[9]]. Derived techniques like artificial neural networks (ANNs) effectively model nonlinear relationships between formulation variables [[11], [13], [15]]. When combined with genetic algorithms (GAs), they provide robust frameworks for optimizing multivariable systems [[16]]. Bayesian optimization has emerged as a valuable tool that combines statistical modeling with probability theory to select experimental conditions based on prior data [[17]]. This method reduces the number of experiments needed, offering a cost-effective way to optimize intricate formulations [[18]].
Moreover, traditional methods often focus on single-objective optimization, which can be restrictive for complex systems requiring simultaneous optimization of conflicting outcomes [[19]]. The desirability function approach addresses this limitation by consolidating multiple response variables into a single 'desirability' score, ranging from 0 (undesirable) to 1 (fully desirable). This approach balances variables such as drug release rate, stability, and encapsulation efficiency, making it invaluable in drug delivery research [[20]]. However, even the desirability function may oversimplify the conflicting nature of certain responses.
Weighted goal programming (WGP) offers a flexible alternative by allowing researchers to prioritize multiple objectives based on their relative importance [[22]]. WGP incorporates decision-makers' preferences directly into the optimization process, enabling a balanced solution that minimizes tradeoffs between conflicting responses. By assigning different weights to each response, WGP enables prioritization, allowing the optimization process to achieve the most balanced solution.
WGP has been successfully applied to several fields in the pharmaceutical industry. For instance, Alizadeh and Kianfar [[23]] extended the Markowitz model using WGP to manage uncertainty in pharmaceutical portfolio optimization. Similarly, Babaei et al. [[24]] used a two-stage framework integrating WGP in the medical tourism supply chain, while Giri et al. [[25]] applied multi-objective models, including WGP, to improve decision-making processes in pharmaceutical companies. These examples showcase the adaptability of WGP in tackling complex, multi-objective problems across different sectors of the pharmaceutical industry.
Despite WGP has been applied in several fields, its use in optimizing drug delivery systems, and particularly nanotechnology-based drug delivery, remains underexplored.
To the best of our knowledge, this study represents the first application of WGP in the optimization of pharmaceutical formulation. Previous research primarily relied on methods, such as ANNs and regression analysis for single-criterion optimization [[11], [26]]. In contrast, our study introduces WGP to the pharmaceutical formulation domain, emphasizing its novelty and significance. Notably, WGP extends the optimization approach to a multi-objective scenario, effectively balancing multiple relevant outputs. By integrating decision-maker preferences, WGP has demonstrated its ability to achieve optimal formulations with minimal tradeoffs, marking a significant advancement in the field.
To demonstrate WGP potential, a case study was conducted on optimizing timolol (TM)-loaded ultradeformable nanoliposomes, an antiglaucoma formulation. Following QbD principles, the study aimed to achieve high-quality liposomal products aligned with the guidelines of the International Council for Harmonization of Technical Requirements for Pharmaceuticals for Human Use [[27]].
Methodology
In pharmaceutical formulation, carefully balancing essential quality attributes, such as efficacy, stability, and safety, is key. This often requires competing objectives that need to be optimized together. The proposed methodology integrates DoE and WGP to create an organized, multi-objective optimization framework.
The use of DoE enables a systematic approach to understand how formulation factors affect the outcomes, allowing efficient data gathering and analysis. The addition or WGP to this process provides a structured way to prioritize responses by their relative importance, so the optimization stays aligned with the intended performance outcomes of the formulation.
The main stages involved in this methodology are shown in Figure 1.
Graph: Figure 1. Main stages of the multi-objective optimization study.
- Identifying factors and responses. This initial step involves selecting the critical factors (inputs) and responses (outputs) for optimization, such as stability, efficacy, or other quality attributes vital to the formulation success.
- Developing the DoE plan. Here, a DoE plan is established to systematically investigate how each factor influences the responses. This structured experimentation helps efficiently gather and analyze data.
- Setting ideal and anti-ideal points. In this phase, ideal and anti-ideal (nadir) points are defined for each response, marking the best and least favorable outcomes. These benchmarks help measure how close each solution comes to an optimal result.
- Defining aspiration levels. Aspiration levels, or target values, are set for each response, providing clear goals aligned with the formulation performance objectives.
- Assigning relative weights to responses. Here, weights are assigned to each response based on its importance in the formulation. WGP then uses these priorities to target the optimization toward the most critical responses.
- Executing the optimization. Finally, using WGP, the optimization considers both aspiration levels and weights to find a balanced solution across all responses, meeting the formulation goals and priorities for overall performance.
Stage 1: establishing the factors and responses to optimize
Details about the methodology for preparation and characterization of TM-loaded transfersomes are described in González-Rodríguez et al. [[26]]. The starting point was based on the optimization process of formulating these liposomes from the DoE. In that work, five key factors (amount of cholesterol, amount of edge-activator (EA), phase in which TM was added, addition of stearylamine (SA) and type of EA) and six outputs (Vesicle size (z<subs>1</subs>), Polydispersity index (z<subs>2</subs>), Zeta potential (z<subs>3</subs>), Deformability index (z<subs>4</subs>), Phosphorous content (z<subs>5</subs>), and percentage of drug entrapment (z<subs>6</subs>)), were evaluated.
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Therefore,
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Stage 2: DoE plan
The screening stage followed an orthogonal array L16 Taguchi, corresponding to
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Table 1. Measured responses of TM-loaded transfersomes.
<table><thead><tr><td /><td>Variables</td></tr><tr><td /><td>Independent</td><td>Dependent</td></tr><tr><td><italic>N</italic>°</td><td><italic>F</italic><sub>1</sub></td><td><italic>F</italic><sub>2</sub></td><td><italic>F</italic><sub>3</sub></td><td><italic>F</italic><sub>4</sub></td><td><italic>F</italic><sub>5</sub></td><td><italic>z</italic><sub>1</sub></td><td><italic>z</italic><sub>2</sub></td><td><italic>z</italic><sub>3</sub></td><td><italic>z</italic><sub>4</sub></td><td><italic>z</italic><sub>5</sub></td><td><italic>z</italic><sub>6</sub></td></tr></thead><tbody valign="top"><tr><td char=".">1</td><td char=".">−</td><td char=".">−</td><td char=".">−</td><td char=".">−</td><td char=".">−</td><td char=".">181.2 ± 3.5</td><td char=".">0.2 ± 0.01</td><td char=".">7.4 ± 1.2</td><td char=".">0.33 ± 0.013</td><td char=".">30.1 ± 0.2</td><td char=".">3.0 ± 0.05</td></tr><tr><td char=".">2</td><td char=".">−</td><td char=".">−</td><td char=".">−</td><td char=".">+</td><td char=".">+</td><td char=".">159.7 ± 1.3</td><td char=".">0.1 ± 0.01</td><td char=".">−16.0 ± 0.5</td><td char=".">0.26 ± 0.004</td><td char=".">24.7 ± 0.1</td><td char=".">2.9 ± 0.08</td></tr><tr><td char=".">3</td><td char=".">−</td><td char=".">−</td><td char=".">+</td><td char=".">−</td><td char=".">+</td><td char=".">271.5 ± 1.3</td><td char=".">0.2 ± 0.01</td><td char=".">2.4 ± 0.5</td><td char=".">0.74 ± 0.007</td><td char=".">33.7 ± 0.5</td><td char=".">10.7 ± 0.03</td></tr><tr><td char=".">4</td><td char=".">−</td><td char=".">−</td><td char=".">+</td><td char=".">+</td><td char=".">−</td><td char=".">177.9 ± 1.7</td><td char=".">0.2 ± 0.03</td><td char=".">−6.8 ± 0.4</td><td char=".">0.32 ± 0.006</td><td char=".">25.7 ± 0.1</td><td char=".">3.0 ± 0.02</td></tr><tr><td char=".">5</td><td char=".">−</td><td char=".">+</td><td char=".">−</td><td char=".">−</td><td char=".">+</td><td char=".">198.1 ± 1.0</td><td char=".">0.3 ± 0.02</td><td char=".">−1.4 ± 0.4</td><td char=".">0.39 ± 0.004</td><td char=".">25.1 ± 0.4</td><td char=".">9.9 ± 0.02</td></tr><tr><td char=".">6</td><td char=".">−</td><td char=".">+</td><td char=".">−</td><td char=".">+</td><td char=".">−</td><td char=".">185.8 ± 2.0</td><td char=".">0.2 ± 0.02</td><td char=".">−8.1 ± 0.7</td><td char=".">0.35 ± 0.007</td><td char=".">31.1 ± 0.3</td><td char=".">3.2 ± 0.03</td></tr><tr><td char=".">7</td><td char=".">−</td><td char=".">+</td><td char=".">+</td><td char=".">−</td><td char=".">−</td><td char=".">186.3 ± 1.9</td><td char=".">0.2 ± 0.01</td><td char=".">5.4 ± 0.3</td><td char=".">0.35 ± 0.007</td><td char=".">21.5 ± 0.4</td><td char=".">1.6 ± 0.01</td></tr><tr><td char=".">8</td><td char=".">−</td><td char=".">+</td><td char=".">+</td><td char=".">+</td><td char=".">+</td><td char=".">141.2 ± 1.8</td><td char=".">0.1 ± 0.01</td><td char=".">−11.7 ± 0.7</td><td char=".">0.2 ± 0.005</td><td char=".">23.2 ± 0.2</td><td char=".">1.6 ± 0.04</td></tr><tr><td char=".">9</td><td char=".">+</td><td char=".">−</td><td char=".">−</td><td char=".">−</td><td char=".">+</td><td char=".">166.6 ± 2.0</td><td char=".">0.2 ± 0.03</td><td char=".">−0.9 ± 0.4</td><td char=".">0.28 ± 0.007</td><td char=".">24.7 ± 0.1</td><td char=".">1.1 ± 0.01</td></tr><tr><td char=".">10</td><td char=".">+</td><td char=".">−</td><td char=".">−</td><td char=".">+</td><td char=".">−</td><td char=".">178.1 ± 2.2</td><td char=".">0.2 ± 0.01</td><td char=".">−6.4 ± 0.5</td><td char=".">0.32 ± 0.008</td><td char=".">25.4 ± 0.1</td><td char=".">4.0 ± 0.02</td></tr><tr><td char=".">11</td><td char=".">+</td><td char=".">−</td><td char=".">+</td><td char=".">−</td><td char=".">−</td><td char=".">172.8 ± 1.9</td><td char=".">0.2 ± 0.02</td><td char=".">5.6 ± 0.2</td><td char=".">0.30 ± 0.007</td><td char=".">23.6 ± 0.2</td><td char=".">0.9 ± 0.03</td></tr><tr><td char=".">12</td><td char=".">+</td><td char=".">−</td><td char=".">+</td><td char=".">+</td><td char=".">+</td><td char=".">157.9 ± 1.0</td><td char=".">0.2 ± 0.03</td><td char=".">−13.2 ± 0.1</td><td char=".">0.25 ± 0.003</td><td char=".">31.1 ± 0.3</td><td char=".">5.5 ± 0.01</td></tr><tr><td char=".">13</td><td char=".">+</td><td char=".">+</td><td char=".">−</td><td char=".">−</td><td char=".">−</td><td char=".">167.1 ± 1.5</td><td char=".">0.2 ± 0.01</td><td char=".">−13.8 ± 0.8</td><td char=".">0.28 ± 0.005</td><td char=".">33.4 ± 0.3</td><td char=".">5.2 ± 0.02</td></tr><tr><td char=".">14</td><td char=".">+</td><td char=".">+</td><td char=".">−</td><td char=".">+</td><td char=".">+</td><td char=".">163.2 ± 4.4</td><td char=".">0.1 ± 0.01</td><td char=".">−15.4 ± 0.9</td><td char=".">0.27 ± 0.015</td><td char=".">33.5 ± 0.2</td><td char=".">3.1 ± 0.02</td></tr><tr><td char=".">15</td><td char=".">+</td><td char=".">+</td><td char=".">+</td><td char=".">−</td><td char=".">+</td><td char=".">172.3 ± 0.7</td><td char=".">0.2 ± 0.03</td><td char=".">−0.9 ± 0.2</td><td char=".">0.30 ± 0.003</td><td char=".">33.7 ± 0.3</td><td char=".">5.9 ± 0.01</td></tr><tr><td char=".">16</td><td char=".">+</td><td char=".">+</td><td char=".">+</td><td char=".">+</td><td char=".">−</td><td char=".">183.7 ± 2.2</td><td char=".">0.2 ± 0.01</td><td char=".">−5.7 ± 0.3</td><td char=".">0.34 ± 0.008</td><td char=".">33.3 ± 0.3</td><td char=".">5.6 ± 0.03</td></tr><tr><td>Independent variables</td><td>Low (−1)</td><td>High (+1)</td><td /><td /><td /></tr><tr><td>F1: Cholesterol (µmol)</td><td char=".">20</td><td char=".">27</td><td /><td /><td /></tr><tr><td>F2: Edge-activator (EA, mg)</td><td char=".">10</td><td char=".">12</td></tr><tr><td>F3: Phase in which TM was added</td><td>O</td><td>A</td></tr><tr><td>F4: Addition of stearylamine (SA)</td><td>Yes</td><td>No</td></tr><tr><td>F5: Type of edge-activator</td><td>T20</td><td>Deo</td></tr></tbody></table>
1 PDE (%), PC (mg), DI (mL/min). From González-rodríguez et al. [26]. z<subs>1</subs>: vesicle size (size, nm); z<subs>2</subs>: polydispersity index (PdI); z<subs>3</subs>: zeta potential (ZP, mV); z<subs>4</subs>: deformability index (DI, mL/min); z<subs>5</subs>: amount of phosphates PC (PC, mg); z<subs>6</subs>: percentage of drug entrapped (PDE, %).
After the experiments, each response was studied independently. Usually, the expression of the responses is unknown, and the experimental estimation is derived from the equation through regression. A highly adjusted R-squared (the closer to one, the better) is often a good indicator of the fit of this regression. At this stage, a response reduction strategy in line with ICH Q8(R<sups>2</sups>) can simplify the number of responses used. For example, correlation analysis or principal component analysis (PCA) can be applied.
Stage 3: determining the ideal and anti-ideal points
The first step in multi-objective optimization was to optimize each response individually. The optimized value for each response is obtained as well as the value of the factors. LINGO software [[28]] was used for this, since it addresses the optimization of non-linear responses and can deal with continuous and discrete variables.
The ideal point represents the best achievable values in multi-objective optimization, while the nadir point reflects the worst values [[29]]. The nadir serves as a reference to evaluate solutions and estimate the anti-ideal case, which is undesirable.
The Pareto Front depicted in Figure 2 represents the set of optimal solutions in a multi-objective optimization problem, where improving one objective (e.g.
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PHOTO (COLOR): Figure 2. Ideal and nadir points.
Normalizing responses using ideal and nadir points enables balanced multi-response optimization, ensuring an optimal solution aligns with desired criteria. While this approach is common in WGP, it is not well-documented in the context of desirability functions. Although the foundational concepts of desirability functions are well-explained by Derringer and Suich [[19]], and the general methodology of RSM is comprehensively covered by Gunst et al. [[30]], neither source specifically addresses using ideal and nadir points as boundaries in the desirability function. Using these points as boundaries in desirability functions is a novel approach, extending their utility to complex multi-objective optimization.
Stage 4: aspiration levels of each response
Determining the efficient Pareto frontier in multi-objective problems becomes unfeasible. To address this issue, specific response values are defined to study the system within the domain set by the decision-maker. These values are known as aspiration levels, which are denoted as
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In multi-objective optimization, target values can be interpreted in three ways: nominal-the-best (NTB), smaller-the-best (STB), and larger-the-best (LTB). For NTB, the goal is to achieve a specific target value. For STB, the objective is to minimize the response, meaning that lower values than the aspiration level are more desirable. For LTB, the aim is to maximize the response, making higher values than the aspiration level more desirable. Thus, the target values guide the optimization process in achieving specific objectives when feasible.
Stage 5: relative weights of each response
Typically, weights are assigned to each response, reflecting the decision-maker's preferences or priorities for the corresponding objectives. These weights signify the relative importance of achieving or minimizing each objective in the overall decision-making process. By adjusting these weights, decision-makers can effectively influence the optimization process, directing it toward outcomes that align more closely with their desired objectives.
Weights can be obtained in various ways, including the following. Each response in the system is assigned with value, denoted as
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Once the
Graph
<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mrow><mi>r</mi></mrow><mrow><mi>i</mi></mrow></msub></math> values are assigned, the weight
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<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mrow><mi>w</mi></mrow><mrow><mi>i</mi></mrow></msub></math> is calculated as the reciprocal of each
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<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi>r</mi></mrow></math> value
Graph
<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mo stretchy="true">(</mo><mrow><mn>1</mn><mo>/</mo><msub><mrow><mi>r</mi></mrow><mrow><mi>i</mi></mrow></msub></mrow><mo stretchy="true">)</mo></math> , and then normalized.
Graph
<math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mrow><mi>w</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>=</mo><mfrac><mrow><mn>1</mn><mo>/</mo><msub><mrow><mi>r</mi></mrow><mrow><mi>i</mi></mrow></msub></mrow><mrow><mrow><munderover><mo stretchy="false">∑</mo><mrow><mi>j</mi><mo>=</mo><mn>1</mn></mrow><mrow><mi>m</mi></mrow></munderover><mrow><msub><mrow><mn>1</mn><mo>/</mo><mi>r</mi></mrow><mrow><mi>j</mi></mrow></msub></mrow></mrow></mrow></mfrac></math> (1)
Stage 6: optimization phase
The final phase optimizes models using WGP and the desirability function. WGP minimizes weighted deviations from aspiration levels, emphasizing critical objectives. The desirability function converts responses into a composite score for maximization. Both methods balance multiple objectives differently.
The WGP model
Deviation of variables are critical components in WGP, as they measure the extent to which each response deviates from its respective aspiration level. These variables can be categorized into two types:
- Positive deviations (
Graph
<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mrow><mi mathvariant="bold-italic">p</mi></mrow><mrow><mi mathvariant="bold-italic">i</mi></mrow></msub></math> ). These indicate the amount by which a response exceeds its aspiration level. Positive deviations are minimized when the goal is to achieve or fall below a target value.
- Negative deviations (
Graph
<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mrow><mi mathvariant="bold-italic">n</mi></mrow><mrow><mi mathvariant="bold-italic">i</mi></mrow></msub></math> ). These indicate the amount by which a response falls short of its aspiration level. Negative deviations are minimized when the goal is to achieve or exceed a target value.
For this, each response must meet the constraint associated with its aspiration level, for example:
Graph
<math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mrow><mi>z</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>+</mo><msub><mrow><mi>n</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>−</mo><msub><mrow><mi>p</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>=</mo><msub><mrow><mi>T</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>.</mo></math>
For a maximization problem, we are interested in obtaining values above the target value
Graph
<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mrow><mi>T</mi></mrow><mrow><mi>i</mi></mrow></msub></math> . If the aspiration level, for example, is
Graph
<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mrow><mi>T</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>=</mo><mn>1000</mn></math> , we would have,
Graph
<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mrow><mi>z</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>+</mo><msub><mrow><mi>n</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>−</mo><msub><mrow><mi>p</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>=</mo><mn>1000</mn><mo>.</mo></math> In this problem, the term
Graph
<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mrow><mi>n</mi></mrow><mrow><mi>i</mi></mrow></msub></math> contributes to achieving the aspiration level, while
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<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mrow><mi>p</mi></mrow><mrow><mi>i</mi></mrow></msub></math> opposes it. This last variable is referred to as the undesirable deviation variable. In the minimization problem, the opposite occurs, with
Graph
<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mrow><mi>n</mi></mrow><mrow><mi>i</mi></mrow></msub></math> being the undesirable deviation variable. In a response where a specific value is sought, both variables are considered deviation variables. Therefore, in a NTB study, where the criterion is to target T<subs>i</subs>, the slack variables are
Graph
<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mrow><mi>n</mi></mrow><mrow><mi>i</mi></mrow></msub></math> ,
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<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mrow><mi>p</mi></mrow><mrow><mi>i</mi></mrow></msub></math> . For LTB, where the criterion is max z<subs>i</subs>, the slack variable is p<subs>i</subs>, and for STB, to min z<subs>i</subs>, the slack variable is n<subs>i</subs>.
Deviation values are usually normalized by dividing by the aspiration level of each response. This makes the deviations comparable, as they are dimensionless. Therefore, using the previously stated weights
Graph
<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mrow><mi>w</mi></mrow><mrow><mi>i</mi></mrow></msub></math> and aspiration levels
Graph
<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mrow><mi>T</mi></mrow><mrow><mi>i</mi></mrow></msub></math> , the WGP model seeks to minimize the sum of all normalized deviations and can be finally stated as follows:
Graph
<math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><mtext mathvariant="italic">minimize</mtext><mo /><mi>G</mi><mo>=</mo><mrow><munderover><mo stretchy="false">∑</mo><mrow><mo>∀</mo><mi>i</mi><mo>∈</mo><msub><mrow><mi>S</mi></mrow><mrow><mtext mathvariant="italic">Norm</mtext></mrow></msub></mrow><mrow /></munderover><mrow><mfrac><mrow><msub><mrow><mi>w</mi></mrow><mrow><mi>i</mi></mrow></msub></mrow><mrow><msub><mrow><mi>T</mi></mrow><mrow><mi>i</mi></mrow></msub></mrow></mfrac><mo stretchy="true">(</mo><mrow><msub><mrow><mi>n</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>+</mo><msub><mrow><mi>p</mi></mrow><mrow><mi>i</mi></mrow></msub></mrow><mo stretchy="true">)</mo></mrow></mrow><mo>+</mo><mrow><munderover><mo stretchy="false">∑</mo><mrow><mo>∀</mo><mi>i</mi><mo>∈</mo><msub><mrow><mi>S</mi></mrow><mrow><mtext mathvariant="normal">Max</mtext></mrow></msub></mrow><mrow /></munderover><mrow><mfrac><mrow><msub><mrow><mi>w</mi></mrow><mrow><mi>i</mi></mrow></msub></mrow><mrow><msub><mrow><mi>T</mi></mrow><mrow><mi>i</mi></mrow></msub></mrow></mfrac><msub><mrow><mi>n</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>+</mo><mrow><munderover><mo stretchy="false">∑</mo><mrow><mo>∀</mo><mi>i</mi><mo>∈</mo><msub><mrow><mi>S</mi></mrow><mrow><mtext mathvariant="normal">min</mtext></mrow></msub></mrow><mrow /></munderover><mrow><mfrac><mrow><msub><mrow><mi>w</mi></mrow><mrow><mi>i</mi></mrow></msub></mrow><mrow><msub><mrow><mi>T</mi></mrow><mrow><mi>i</mi></mrow></msub></mrow></mfrac><msub><mrow><mi>p</mi></mrow><mrow><mi>i</mi></mrow></msub></mrow></mrow></mrow></mrow></math> (2)
Subjected to:
Graph
<math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mrow><mi>z</mi></mrow><mrow><mi>i</mi></mrow></msub><mo stretchy="true">(</mo><mrow><mi mathvariant="bold-italic">F</mi></mrow><mo stretchy="true">)</mo><mo>+</mo><msub><mrow><mi>n</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>−</mo><msub><mrow><mi>p</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>=</mo><msub><mrow><mi>T</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>∀</mo><mi>i</mi><mo>=</mo><mn>1</mn><mo>,</mo><mo>.</mo><mo>.</mo><mo>,</mo><mi>m</mi></math> (3)
Graph
<math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mrow><mi>n</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>,</mo><msub><mrow><mi>p</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>≥</mo><mn>0</mn><mo>∀</mo><mi>i</mi><mo>∈</mo><mi>m</mi></math> (4)
The desirability function approach
The desirability function is widely used in the pharmaceutical industry to optimize multi-response processes by ensuring quality meets desired limits. If these limits are not met, the quality is considered unacceptable. The method aims to identify the operating conditions or factor values
Graph
<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi>F</mi></mrow></math> that yield the 'most desirable' response values, thus ensuring optimal performance across all quality characteristics.
For each response
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<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mrow><mi>z</mi></mrow><mrow><mi>i</mi></mrow></msub><mo stretchy="true">(</mo><mrow><mi>F</mi></mrow><mo stretchy="true">)</mo></math> , a desirability function
Graph
<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mrow><mi>d</mi></mrow><mrow><mi>i</mi></mrow></msub></math> assigns values between 0 and 1 to the possible outcomes of
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<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mrow><mi>z</mi></mrow><mrow><mi>i</mi></mrow></msub></math> .
Graph
<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mrow><mi>d</mi></mrow><mrow><mi>i</mi></mrow></msub><mo stretchy="true">(</mo><mrow><msub><mrow><mi>z</mi></mrow><mrow><mi>i</mi></mrow></msub></mrow><mo stretchy="true">)</mo><mo>=</mo><mn>0</mn></math> indicates a completely undesirable outcome, and
Graph
<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mrow><mi>d</mi></mrow><mrow><mi>i</mi></mrow></msub><mo stretchy="true">(</mo><mrow><msub><mrow><mi>z</mi></mrow><mrow><mi>i</mi></mrow></msub></mrow><mo stretchy="true">)</mo><mo>=</mo><mn>1</mn></math> represents a completely desirable or ideal outcome. The individual desirability values for the
Graph
<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi>m</mi></mrow></math> responses are then combined using the geometric mean and the previously stated weights
Graph
<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mrow><mi>w</mi></mrow><mrow><mi>i</mi></mrow></msub></math> to obtain the overall desirability
Graph
<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi>D</mi></mrow></math> :
Graph
<math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><mi>D</mi><mo>=</mo><msup><mrow><mo stretchy="true">(</mo><mrow><msubsup><mrow><mi>d</mi></mrow><mrow><mn>1</mn></mrow><mrow><msub><mrow><mi>w</mi></mrow><mrow><mn>1</mn></mrow></msub></mrow></msubsup><mi>​</mi><mo stretchy="true">(</mo><mrow><msub><mrow><mi>z</mi></mrow><mrow><mn>1</mn></mrow></msub><mi>​</mi></mrow><mo stretchy="true">)</mo><msubsup><mrow><mi>d</mi></mrow><mrow><mn>2</mn></mrow><mrow><msub><mrow><mi>w</mi></mrow><mrow><mn>2</mn></mrow></msub></mrow></msubsup><mo /><mi>​</mi><mo stretchy="true">(</mo><mrow><msub><mrow><mi>z</mi></mrow><mrow><mn>2</mn></mrow></msub><mi>​</mi></mrow><mo stretchy="true">)</mo><mo>⋯</mo><msubsup><mrow><mi>d</mi></mrow><mrow><mi>m</mi></mrow><mrow><msub><mrow><mi>w</mi></mrow><mrow><mi>m</mi></mrow></msub></mrow></msubsup><mo /><mi>​</mi><mo stretchy="true">(</mo><mrow><msub><mrow><mi>z</mi></mrow><mrow><mi>m</mi></mrow></msub><mi>​</mi></mrow><mo stretchy="true">)</mo></mrow><mo stretchy="true">)</mo></mrow><mrow><mn>1</mn><mo>/</mo><mi>m</mi></mrow></msup></math> (5)
Depending on whether a particular response
Graph
<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mrow><mi>z</mi></mrow><mrow><mi>i</mi></mrow></msub></math> must be maximized, minimized, or assigned a target value, different desirability functions
Graph
<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mrow><mi>d</mi></mrow><mrow><mi>i</mi></mrow></msub><mo stretchy="true">(</mo><mrow><msub><mrow><mi>z</mi></mrow><mrow><mi>i</mi></mrow></msub></mrow><mo stretchy="true">)</mo></math> can be used. A useful class of desirability functions was proposed by Derringer and Suich [[19]].
The desirability function employs lower and upper boundaries, usually established based on the decision-maker's experience. Therefore,
Graph
<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mrow><mi>L</mi></mrow><mrow><mi>i</mi></mrow></msub></math> ,
Graph
<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mrow><mi>T</mi></mrow><mrow><mi>i</mi></mrow></msub></math> , and
Graph
<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mrow><mi>U</mi></mrow><mrow><mi>i</mi></mrow></msub></math> represent the lower, target, and upper boundaries, respectively, that are desired for the response
Graph
<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mrow><mi>z</mi></mrow><mrow><mi>i</mi></mrow></msub></math> with
Graph
<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mrow><mi>L</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>≤</mo><msub><mrow><mi>T</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>≤</mo><msub><mrow><mi>U</mi></mrow><mrow><mi>i</mi></mrow></msub></math> .
For a response NTB, where the aim is to obtain a nominal or target, the desirability function is defined as:
Graph
<math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mrow><mi>d</mi></mrow><mrow><mi>i</mi></mrow></msub><mo stretchy="true">(</mo><mrow><msub><mrow><mi>z</mi></mrow><mrow><mi>i</mi></mrow></msub></mrow><mo stretchy="true">)</mo><mo>=</mo><mo stretchy="true">{</mo><mrow><mtable><mtr><mtd><mrow><maligngroup /><mtable><mtr><mtd><mn>0</mn></mtd><mtd><mo /><mtext mathvariant="italic">if</mtext></mtd><mtd><msub><mrow><mi>z</mi></mrow><mrow><mi>i</mi></mrow></msub><mo><</mo><msub><mrow><mi>L</mi></mrow><mrow><mi>i</mi></mrow></msub></mtd></mtr></mtable></mrow></mtd></mtr><mtr><mtd><mrow><maligngroup /><mtable><mtr><mtd><msup><mrow><mo stretchy="true">(</mo><mrow><mfrac><mrow><msub><mrow><mi>z</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>−</mo><msub><mrow><mi>L</mi></mrow><mrow><mi>i</mi></mrow></msub></mrow><mrow><msub><mrow><mi>T</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>−</mo><msub><mrow><mi>L</mi></mrow><mrow><mi>i</mi></mrow></msub></mrow></mfrac></mrow><mo stretchy="true">)</mo></mrow><mrow><mi>s</mi></mrow></msup></mtd><mtd><mtext mathvariant="italic">if</mtext></mtd><mtd><msub><mrow><mi>L</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>≤</mo><msub><mrow><mi>z</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>≤</mo><msub><mrow><mi>T</mi></mrow><mrow><mi>i</mi></mrow></msub></mtd></mtr><mtr><mtd><msup><mrow><mo stretchy="true">(</mo><mrow><mfrac><mrow><msub><mrow><mi>z</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>−</mo><msub><mrow><mi>U</mi></mrow><mrow><mi>i</mi></mrow></msub></mrow><mrow><msub><mrow><mi>T</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>−</mo><msub><mrow><mi>U</mi></mrow><mrow><mi>i</mi></mrow></msub></mrow></mfrac></mrow><mo stretchy="true">)</mo></mrow><mrow><mi>t</mi></mrow></msup></mtd><mtd><mtext mathvariant="italic">if</mtext></mtd><mtd><msub><mrow><mi>T</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>≤</mo><msub><mrow><mi>z</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>≤</mo><msub><mrow><mi>U</mi></mrow><mrow><mi>i</mi></mrow></msub></mtd></mtr></mtable></mrow></mtd></mtr><mtr><mtd><mrow><maligngroup /><mtable><mtr><mtd><mn>0</mn></mtd><mtd><mo /><mtext mathvariant="italic">if</mtext></mtd><mtd><msub><mrow><mi>z</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>></mo><msub><mrow><mi>U</mi></mrow><mrow><mi>i</mi></mrow></msub></mtd></mtr></mtable></mrow></mtd></mtr></mtable></mrow></math> (6)
The exponents
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<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi>s</mi></mrow></math> and
Graph
<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi>t</mi></mrow></math> determine the importance of hitting the target value. In this case, the ideal and the anti-ideal points for each response can be used as estimators of
Graph
<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mrow><mi>U</mi></mrow><mrow><mi>i</mi></mrow></msub></math> and
Graph
<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mrow><mi>L</mi></mrow><mrow><mi>i</mi></mrow></msub></math> , respectively, reducing the subjectivity of the decision-maker.
For responses LTB, that need to be maximized, the desirability function is:
Graph
<math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mrow><mi>d</mi></mrow><mrow><mi>i</mi></mrow></msub><mo stretchy="true">(</mo><mrow><msub><mrow><mi>z</mi></mrow><mrow><mi>i</mi></mrow></msub></mrow><mo stretchy="true">)</mo><mo>=</mo><mo stretchy="true">{</mo><mrow><mtable><mtr><mtd><mrow><maligngroup /><mtable><mtr><mtd><mn>0</mn></mtd><mtd><mo /><mtext mathvariant="italic">if</mtext></mtd><mtd><msub><mrow><mi>z</mi></mrow><mrow><mi>i</mi></mrow></msub><mo><</mo><msub><mrow><mi>L</mi></mrow><mrow><mi>i</mi></mrow></msub></mtd></mtr></mtable></mrow></mtd></mtr><mtr><mtd><mrow><maligngroup /><mtable><mtr><mtd><msup><mrow><mo stretchy="true">(</mo><mrow><mfrac><mrow><msub><mrow><mi>z</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>−</mo><msub><mrow><mi>L</mi></mrow><mrow><mi>i</mi></mrow></msub></mrow><mrow><msub><mrow><mi>T</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>−</mo><msub><mrow><mi>L</mi></mrow><mrow><mi>i</mi></mrow></msub></mrow></mfrac></mrow><mo stretchy="true">)</mo></mrow><mrow><mi>s</mi></mrow></msup></mtd><mtd><mtext mathvariant="italic">if</mtext></mtd><mtd><msub><mrow><mi>L</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>≤</mo><msub><mrow><mi>z</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>≤</mo><msub><mrow><mi>T</mi></mrow><mrow><mi>i</mi></mrow></msub></mtd></mtr></mtable></mrow></mtd></mtr><mtr><mtd><mrow><maligngroup /><mtable><mtr><mtd><mn>1.0</mn></mtd><mtd><mo /><mtext mathvariant="italic">if</mtext></mtd><mtd><msub><mrow><mi>z</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>></mo><msub><mrow><mi>T</mi></mrow><mrow><mi>i</mi></mrow></msub></mtd></mtr></mtable></mrow></mtd></mtr></mtable></mrow></math> (7)
As in the previous case occurs, the anti-ideal points for each maximization response can be used as estimators of
Graph
<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mrow><mi>L</mi></mrow><mrow><mi>i</mi></mrow></msub></math> .
To minimize a response STB, the following individual desirability function can be used:
Graph
<math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mrow><mi>d</mi></mrow><mrow><mi>i</mi></mrow></msub><mo stretchy="true">(</mo><mrow><msub><mrow><mi>z</mi></mrow><mrow><mi>i</mi></mrow></msub></mrow><mo stretchy="true">)</mo><mo>=</mo><mo stretchy="true">{</mo><mrow><mtable><mtr><mtd><mrow><maligngroup /><mtable><mtr><mtd><mn>1.0</mn></mtd><mtd><mo /><mtext mathvariant="italic">if</mtext></mtd><mtd><msub><mrow><mi>z</mi></mrow><mrow><mi>i</mi></mrow></msub><mo><</mo><msub><mrow><mi>T</mi></mrow><mrow><mi>i</mi></mrow></msub></mtd></mtr></mtable></mrow></mtd></mtr><mtr><mtd><mrow><maligngroup /><mtable><mtr><mtd><msup><mrow><mo stretchy="true">(</mo><mrow><mfrac><mrow><msub><mrow><mi>U</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>−</mo><msub><mrow><mi>z</mi></mrow><mrow><mi>i</mi></mrow></msub></mrow><mrow><msub><mrow><mi>U</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>−</mo><msub><mrow><mi>T</mi></mrow><mrow><mi>i</mi></mrow></msub></mrow></mfrac></mrow><mo stretchy="true">)</mo></mrow><mrow><mi>s</mi></mrow></msup></mtd><mtd><mtext mathvariant="italic">if</mtext></mtd><mtd><msub><mrow><mi>T</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>≤</mo><msub><mrow><mi>z</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>≤</mo><msub><mrow><mi>U</mi></mrow><mrow><mi>i</mi></mrow></msub></mtd></mtr></mtable></mrow></mtd></mtr><mtr><mtd><mrow><maligngroup /><mtable><mtr><mtd><mn>0</mn></mtd><mtd><mo /><mtext mathvariant="italic">if</mtext></mtd><mtd><msub><mrow><mi>z</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>></mo><msub><mrow><mi>U</mi></mrow><mrow><mi>i</mi></mrow></msub></mtd></mtr></mtable></mrow></mtd></mtr></mtable></mrow></math> (8)
In this case, the anti-ideal points for each minimization response can be used as estimators of
Graph
<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mrow><mi>U</mi></mrow><mrow><mi>i</mi></mrow></msub></math> .
LINGO software was used for solving models due to its ability to optimize non-linear responses and handle various variables, making it suitable for WGP and desirability function approaches. Other software can also be used for similar tasks.
Results
Establishing the factors and responses
The starting point was the study realized by González-Rodríguez et al. [[26]] based on the optimization process of TM-loaded nanoliposomes from DoE. In that work, a DoE study was presented, detailing their nomenclature and respective levels in Table 1. It includes five key factors: the amount of cholesterol (µmol), amount of EA (mg), the phase in which TM was added, the addition of SA, and the type of EA. Each factor is assigned an alias (
Graph
<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mrow><mi>F</mi></mrow><mrow><mn>1</mn></mrow></msub></math> to
Graph
<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mrow><mi>F</mi></mrow><mrow><mn>5</mn></mrow></msub></math> ), respectively, for easy reference. The levels for each factor are specified as lower (−) and upper (+) levels. For example, the amount of cholesterol ranges from 20 to 27 µmol, and the amount of EA ranges from 10 to 12 mg. The phase of TM addition is either the organic phase (O) or aqueous phase (A). The presence of SA is noted as either 'Yes' or 'No,' and the type of EA used is either Tween<sups>®</sups> 20 (T20) or sodium deoxycholate (Deo). Therefore,
Graph
<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mrow><mi>F</mi></mrow><mrow><mn>1</mn></mrow></msub></math> and
Graph
<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mrow><mi>F</mi></mrow><mrow><mn>2</mn></mrow></msub></math> are continuous variables while
Graph
<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mrow><mi>F</mi></mrow><mrow><mn>3</mn></mrow></msub></math> ,
Graph
<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mrow><mi>F</mi></mrow><mrow><mn>4</mn></mrow></msub></math> , and
Graph
<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mrow><mi>F</mi></mrow><mrow><mn>5</mn></mrow></msub></math> are discrete variables.
The study aims to optimize several response criteria: minimizing vesicle size
Graph
<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mo stretchy="false">(</mo><msub><mrow><mi>z</mi></mrow><mrow><mn>1</mn></mrow></msub><mo stretchy="false">)</mo></math> and polydispersity index
Graph
<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mo stretchy="false">(</mo><msub><mrow><mi>z</mi></mrow><mrow><mn>2</mn></mrow></msub><mo stretchy="false">)</mo></math> , while maximizing zeta potential
Graph
<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mo stretchy="false">(</mo><msub><mrow><mi>z</mi></mrow><mrow><mn>3</mn></mrow></msub><mo stretchy="false">)</mo></math> , deformability index
Graph
<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mo stretchy="false">(</mo><msub><mrow><mi>z</mi></mrow><mrow><mn>4</mn></mrow></msub><mo stretchy="false">)</mo></math> , phosphorous content
Graph
<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mo stretchy="true">(</mo><mrow><msub><mrow><mi>z</mi></mrow><mrow><mn>5</mn></mrow></msub></mrow><mo stretchy="true">)</mo></math> , and percentage of drug entrapment
Graph
<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mo stretchy="false">(</mo><msub><mrow><mi>z</mi></mrow><mrow><mn>6</mn></mrow></msub><mo stretchy="false">)</mo></math> .
According to the response criteria, the set of minimization responses is
Graph
<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mrow><mi>S</mi></mrow><mrow><mtext mathvariant="normal">min</mtext></mrow></msub><mo>=</mo><msub><mrow><mo stretchy="false">(</mo><mi>z</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>z</mi></mrow><mrow><mn>2</mn></mrow></msub><mo stretchy="false">)</mo></math> , while the maximization set is formed by
Graph
<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mrow><mi>S</mi></mrow><mrow><mtext mathvariant="normal">Max</mtext></mrow></msub><mo>=</mo><msub><mrow><mo stretchy="false">(</mo><mi>z</mi></mrow><mrow><mn>3</mn></mrow></msub><mo>,</mo><msub><mrow><mi>z</mi></mrow><mrow><mn>4</mn></mrow></msub><mo>,</mo><msub><mrow><mi>z</mi></mrow><mrow><mn>5</mn></mrow></msub><mo>,</mo><msub><mrow><mi>z</mi></mrow><mrow><mn>6</mn></mrow></msub><mo stretchy="false">)</mo></math> . The response
Graph
<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mrow><mi>z</mi></mrow><mrow><mn>2</mn></mrow></msub></math> was revised, omitting the
Graph
<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mrow><mi>F</mi></mrow><mrow><mn>3</mn></mrow></msub></math> term and obtaining a better adjusted
Graph
<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msup><mrow><mi>R</mi></mrow><mrow><mn>2</mn></mrow></msup></math> .
Graph
<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mrow><mi>z</mi></mrow><mrow><mn>3</mn></mrow></msub></math> is reviewed and improved, achieving a better adjusted
Graph
<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msup><mrow><mi>R</mi></mrow><mrow><mn>2</mn></mrow></msup></math> . The variables are encoded for values ranging from 0 to 1. The following regression curves for coded variables between 0 and 1 are used:
Graph
<math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><mtable><mtr><mtd><msub><mrow><mi mathvariant="bold-italic">z</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>=</mo><mn>10</mn><mo>̂</mo><mo>(</mo><mn>2.25</mn><mo>−</mo><mn>0.0179</mn><mo>·</mo><msub><mrow><mi>F</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>−</mo><mn>0.00826</mn><mo>·</mo><msub><mrow><mi>F</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>+</mo><mn>0.00655</mn><mo>·</mo><msub><mrow><mi>F</mi></mrow><mrow><mn>3</mn></mrow></msub><mo>−</mo><mn>0.0237</mn><mo>·</mo><msub><mrow><mi>F</mi></mrow><mrow><mn>4</mn></mrow></msub><mo>−</mo><mn>0.00438</mn><mo>·</mo><msub><mrow><mi>F</mi></mrow><mrow><mn>5</mn></mrow></msub><mo>+</mo></mtd></mtr><mtr><mtd><mn>0.0117</mn><mo>·</mo><msub><mrow><mi>F</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>·</mo><msub><mrow><mi>F</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>−</mo><mn>0.00301</mn><mo>·</mo><msub><mrow><mi>F</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>·</mo><msub><mrow><mi>F</mi></mrow><mrow><mn>3</mn></mrow></msub><mo>−</mo><mn>0.0167</mn><mo>·</mo><msub><mrow><mi>F</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>·</mo><msub><mrow><mi>F</mi></mrow><mrow><mn>3</mn></mrow></msub><mo>+</mo><mn>0.0246</mn><mo>·</mo><msub><mrow><mi>F</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>·</mo><msub><mrow><mi>F</mi></mrow><mrow><mn>4</mn></mrow></msub><mo>+</mo></mtd></mtr><mtr><mtd><mtable><mtr><mtd><mn>0.00735</mn><mo>·</mo><msub><mrow><mi>F</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>·</mo><msub><mrow><mi>F</mi></mrow><mrow><mn>4</mn></mrow></msub><mo>−</mo><mn>0.0157</mn><mo>·</mo><msub><mrow><mi>F</mi></mrow><mrow><mn>3</mn></mrow></msub><mo>·</mo><msub><mrow><mi>F</mi></mrow><mrow><mn>4</mn></mrow></msub><mo>−</mo><mn>0.00889</mn><mo>·</mo><msub><mrow><mi>F</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>·</mo><msub><mrow><mi>F</mi></mrow><mrow><mn>5</mn></mrow></msub><mo>−</mo><mn>0.0119</mn><mo>·</mo><msub><mrow><mi>F</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>·</mo><msub><mrow><mi>F</mi></mrow><mrow><mn>5</mn></mrow></msub><mo>+</mo></mtd></mtr><mtr><mtd><mn>0.0039</mn><mo>·</mo><msub><mrow><mi>F</mi></mrow><mrow><mn>3</mn></mrow></msub><mo>·</mo><msub><mrow><mi>F</mi></mrow><mrow><mn>5</mn></mrow></msub><mo>−</mo><mn>0.0294</mn><mo>·</mo><msub><mrow><mi>F</mi></mrow><mrow><mn>4</mn></mrow></msub><mo>·</mo><msub><mrow><mi>F</mi></mrow><mrow><mn>5</mn></mrow></msub><mo>)</mo></mtd></mtr></mtable></mtd></mtr></mtable></math> (9)
Graph
<math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><mtable><mtr><mtd><msub><mrow><mi mathvariant="bold-italic">z</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>=</mo><mn>0.1895</mn><mo>−</mo><mn>0.005521</mn><mo>·</mo><msub><mrow><mi>F</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>−</mo><mn>0.003896</mn><mo>·</mo><msub><mrow><mi>F</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>−</mo><mn>0.02144</mn><mo>·</mo><msub><mrow><mi>F</mi></mrow><mrow><mn>4</mn></mrow></msub><mo>+</mo><mn>0.002812</mn><mo>·</mo><msub><mrow><mi>F</mi></mrow><mrow><mn>5</mn></mrow></msub><mo>+</mo></mtd></mtr><mtr><mtd><mn>0.01106</mn><mo>·</mo><msub><mrow><mi>F</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>·</mo><msub><mrow><mi>F</mi></mrow><mrow><mn>3</mn></mrow></msub><mo>−</mo><mn>0.009396</mn><mo>·</mo><msub><mrow><mi>F</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>·</mo><msub><mrow><mi>F</mi></mrow><mrow><mn>3</mn></mrow></msub><mo>+</mo><mn>0.009396</mn><mo>·</mo><msub><mrow><mi>F</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>·</mo><msub><mrow><mi>F</mi></mrow><mrow><mn>4</mn></mrow></msub><mo>+</mo></mtd></mtr><mtr><mtd><mn>0.005979</mn><mo>·</mo><msub><mrow><mi>F</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>·</mo><msub><mrow><mi>F</mi></mrow><mrow><mn>5</mn></mrow></msub><mo>−</mo><mn>0.02227</mn><mo>·</mo><msub><mrow><mi>F</mi></mrow><mrow><mn>4</mn></mrow></msub><mo>·</mo><msub><mrow><mi>F</mi></mrow><mrow><mn>5</mn></mrow></msub></mtd></mtr></mtable></math> (10)
Graph
<math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><mtable><mtr><mtd><msub><mrow><mi mathvariant="bold-italic">z</mi></mrow><mrow><mn>3</mn></mrow></msub><mo>=</mo><mo>|</mo><mo>−</mo><mn>4.97</mn><mo>−</mo><mn>1.38</mn><mo>·</mo><msub><mrow><mi>F</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>−</mo><mn>1.49</mn><mo>·</mo><msub><mrow><mi>F</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>+</mo><mn>1.86</mn><mo>·</mo><msub><mrow><mi>F</mi></mrow><mrow><mn>3</mn></mrow></msub><mo>−</mo><mn>5.45</mn><mo>·</mo><msub><mrow><mi>F</mi></mrow><mrow><mn>4</mn></mrow></msub><mo>−</mo><mn>2.15</mn><mo>·</mo><msub><mrow><mi>F</mi></mrow><mrow><mn>5</mn></mrow></msub><mo>−</mo><mn>1.14</mn><mo>·</mo><msub><mrow><mi>F</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>·</mo><msub><mrow><mi>F</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>+</mo></mtd></mtr><mtr><mtd><mn>0.928</mn><mo>·</mo><msub><mrow><mi>F</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>·</mo><msub><mrow><mi>F</mi></mrow><mrow><mn>3</mn></mrow></msub><mo>+</mo><mn>1.37</mn><mo>·</mo><msub><mrow><mi>F</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>·</mo><msub><mrow><mi>F</mi></mrow><mrow><mn>3</mn></mrow></msub><mo>+</mo><mn>1.61</mn><mo>·</mo><msub><mrow><mi>F</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>·</mo><msub><mrow><mi>F</mi></mrow><mrow><mn>4</mn></mrow></msub><mo>+</mo><mn>1.67</mn><mo>·</mo><msub><mrow><mi>F</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>·</mo><msub><mrow><mi>F</mi></mrow><mrow><mn>4</mn></mrow></msub><mo>−</mo><mn>0.789</mn><mo>·</mo><msub><mrow><mi>F</mi></mrow><mrow><mn>3</mn></mrow></msub><mo>·</mo><msub><mrow><mi>F</mi></mrow><mrow><mn>4</mn></mrow></msub><mo>+</mo></mtd></mtr><mtr><mtd><mn>0.909</mn><mo>·</mo><msub><mrow><mi>F</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>·</mo><msub><mrow><mi>F</mi></mrow><mrow><mn>5</mn></mrow></msub><mo>+</mo><mn>1.26</mn><mo>·</mo><msub><mrow><mi>F</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>·</mo><msub><mrow><mi>F</mi></mrow><mrow><mn>5</mn></mrow></msub><mo>−</mo><mn>0.569</mn><mo>·</mo><msub><mrow><mi>F</mi></mrow><mrow><mn>3</mn></mrow></msub><mo>·</mo><msub><mrow><mi>F</mi></mrow><mrow><mn>5</mn></mrow></msub><mo>−</mo><mn>1.49</mn><mo>·</mo><msub><mrow><mi>F</mi></mrow><mrow><mn>4</mn></mrow></msub><mo>·</mo><msub><mrow><mi>F</mi></mrow><mrow><mn>5</mn></mrow></msub><mo>|</mo></mtd></mtr></mtable></math> (11)
Graph
<math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><mtable><mtr><mtd><msub><mrow><mi mathvariant="bold-italic">z</mi></mrow><mrow><mn>4</mn></mrow></msub><mo>=</mo><mn>0.328</mn><mo>−</mo><mn>0.0381</mn><mo>·</mo><msub><mrow><mi>F</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>−</mo><mn>0.0206</mn><mo>·</mo><msub><mrow><mi>F</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>+</mo><mn>0.0194</mn><mo>·</mo><msub><mrow><mi>F</mi></mrow><mrow><mn>3</mn></mrow></msub><mo>−</mo><mn>0.0425</mn><mo>·</mo><msub><mrow><mi>F</mi></mrow><mrow><mn>4</mn></mrow></msub><mo>+</mo><mn>0.00687</mn><mo>·</mo><msub><mrow><mi>F</mi></mrow><mrow><mn>5</mn></mrow></msub><mo>+</mo></mtd></mtr><mtr><mtd><mn>0.0238</mn><mo>·</mo><msub><mrow><mi>F</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>·</mo><msub><mrow><mi>F</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>−</mo><mn>0.0137</mn><mo>·</mo><msub><mrow><mi>F</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>·</mo><msub><mrow><mi>F</mi></mrow><mrow><mn>3</mn></mrow></msub><mo>−</mo><mn>0.0325</mn><mo>·</mo><msub><mrow /><mrow><mn>2</mn></mrow></msub><mo>·</mo><msub><mrow><mi>F</mi></mrow><mrow><mn>3</mn></mrow></msub><mo>+</mo><mn>0.0431</mn><mo>·</mo><msub><mrow><mi>F</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>·</mo><msub><mrow><mi>F</mi></mrow><mrow><mn>4</mn></mrow></msub><mo>+</mo><mn>0.0206</mn><mo>·</mo><msub><mrow><mi>F</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>·</mo><msub><mrow><mi>F</mi></mrow><mrow><mn>4</mn></mrow></msub><mo>−</mo></mtd></mtr><mtr><mtd><mn>0.0294</mn><mo>·</mo><msub><mrow><mi>F</mi></mrow><mrow><mn>3</mn></mrow></msub><mo>·</mo><msub><mrow><mi>F</mi></mrow><mrow><mn>4</mn></mrow></msub><mo>−</mo><mn>0.025</mn><mo>·</mo><msub><mrow><mi>F</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>·</mo><msub><mrow><mi>F</mi></mrow><mrow><mn>5</mn></mrow></msub><mo>−</mo><mn>0.0238</mn><mo>·</mo><msub><mrow><mi>F</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>·</mo><msub><mrow><mi>F</mi></mrow><mrow><mn>5</mn></mrow></msub><mo>+</mo><mn>0.0188</mn><mo>·</mo><msub><mrow><mi>F</mi></mrow><mrow><mn>3</mn></mrow></msub><mo>·</mo><msub><mrow><mi>F</mi></mrow><mrow><mn>5</mn></mrow></msub><mo>−</mo><mn>0.049</mn><mo>·</mo><msub><mrow><mi>F</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>·</mo><msub><mrow><mi>F</mi></mrow><mrow><mn>5</mn></mrow></msub></mtd></mtr></mtable></math> (12)
Graph
<math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><mtable><mtr><mtd><msub><mrow><mi mathvariant="bold-italic">z</mi></mrow><mrow><mn>5</mn></mrow></msub><mo>=</mo><mn>28.3</mn><mo>+</mo><mn>1.59</mn><mo>·</mo><msub><mrow><mi>F</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>+</mo><mn>1.02</mn><mo>·</mo><msub><mrow><mi>F</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>−</mo><mn>0.0571</mn><mo>·</mo><msub><mrow><mi>F</mi></mrow><mrow><mn>3</mn></mrow></msub><mo>+</mo><mn>0.143</mn><mo>·</mo><msub><mrow><mi>F</mi></mrow><mrow><mn>4</mn></mrow></msub><mo>+</mo><mn>0.428</mn><mo>·</mo><msub><mrow><mi>F</mi></mrow><mrow><mn>5</mn></mrow></msub><mo>+</mo></mtd></mtr><mtr><mtd><mn>2.61</mn><mo>·</mo><msub><mrow><mi>F</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>·</mo><msub><mrow><mi>F</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>+</mo><mn>0.696</mn><mo>·</mo><msub><mrow><mi>F</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>·</mo><msub><mrow><mi>F</mi></mrow><mrow><mn>3</mn></mrow></msub><mo>−</mo><mn>1.38</mn><mo>·</mo><msub><mrow><mi>F</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>·</mo><msub><mrow><mi>F</mi></mrow><mrow><mn>3</mn></mrow></msub><mo>+</mo><mn>0.811</mn><mo>·</mo><msub><mrow><mi>F</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>·</mo><msub><mrow><mi>F</mi></mrow><mrow><mn>4</mn></mrow></msub><mo>+</mo></mtd></mtr><mtr><mtd><mn>0.735</mn><mo>·</mo><msub><mrow><mi>F</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>·</mo><msub><mrow><mi>F</mi></mrow><mrow><mn>4</mn></mrow></msub><mo>+</mo><mn>0.449</mn><mo>·</mo><msub><mrow><mi>F</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>·</mo><msub><mrow><mi>F</mi></mrow><mrow><mn>5</mn></mrow></msub><mo>−</mo><mn>0.871</mn><mo>·</mo><msub><mrow><mi>F</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>·</mo><msub><mrow><mi>F</mi></mrow><mrow><mn>5</mn></mrow></msub><mo>+</mo><mn>1.79</mn><mo>·</mo><msub><mrow><mi>F</mi></mrow><mrow><mn>3</mn></mrow></msub><mo>·</mo><msub><mrow><mi>F</mi></mrow><mrow><mn>5</mn></mrow></msub><mo>−</mo><mn>0.82</mn><mo>·</mo><msub><mrow><mi>F</mi></mrow><mrow><mn>4</mn></mrow></msub><mo>·</mo><msub><mrow><mi>F</mi></mrow><mrow><mn>5</mn></mrow></msub></mtd></mtr></mtable></math> (13)
Graph
<math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><mtable><mtr><mtd><msub><mrow><mi mathvariant="bold-italic">z</mi></mrow><mrow><mn>6</mn></mrow></msub><mo>=</mo><mn>4.21</mn><mo>−</mo><mn>0.279</mn><mo>·</mo><msub><mrow><mi>F</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>+</mo><mn>0.311</mn><mo>·</mo><msub><mrow><mi>F</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>+</mo><mn>0.152</mn><mo>·</mo><msub><mrow><mi>F</mi></mrow><mrow><mn>3</mn></mrow></msub><mo>−</mo><mn>0.595</mn><mo>·</mo><msub><mrow><mi>F</mi></mrow><mrow><mn>4</mn></mrow></msub><mo>+</mo><mn>0.878</mn><mo>·</mo><msub><mrow><mi>F</mi></mrow><mrow><mn>5</mn></mrow></msub><mo>+</mo></mtd></mtr><mtr><mtd><mn>0.726</mn><mo>·</mo><msub><mrow><mi>F</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>·</mo><msub><mrow><mi>F</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>+</mo><mn>0.417</mn><mo>·</mo><msub><mrow><mi>F</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>·</mo><msub><mrow><mi>F</mi></mrow><mrow><mn>3</mn></mrow></msub><mo>−</mo><mn>0.973</mn><mo>·</mo><msub><mrow><mi>F</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>·</mo><msub><mrow><mi>F</mi></mrow><mrow><mn>3</mn></mrow></msub><mo>+</mo><mn>1.21</mn><mo>·</mo><msub><mrow><mi>F</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>·</mo><msub><mrow><mi>F</mi></mrow><mrow><mn>4</mn></mrow></msub><mo>−</mo></mtd></mtr><mtr><mtd><mtable><mtr><mtd><mn>0.543</mn><mo>·</mo><msub><mrow><mi>F</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>·</mo><msub><mrow><mi>F</mi></mrow><mrow><mn>4</mn></mrow></msub><mo>+</mo><mn>0.169</mn><mo>·</mo><msub><mrow><mi>F</mi></mrow><mrow><mn>3</mn></mrow></msub><mo>·</mo><msub><mrow><mi>F</mi></mrow><mrow><mn>4</mn></mrow></msub><mo>−</mo><mn>0.898</mn><mo>·</mo><msub><mrow><mi>F</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>·</mo><msub><mrow><mi>F</mi></mrow><mrow><mn>5</mn></mrow></msub><mo>−</mo><mn>0.269</mn><mo>·</mo><msub><mrow><mi>F</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>·</mo><msub><mrow><mi>F</mi></mrow><mrow><mn>5</mn></mrow></msub><mo>+</mo></mtd></mtr><mtr><mtd><mn>0.682</mn><mo>·</mo><msub><mrow><mi>F</mi></mrow><mrow><mn>3</mn></mrow></msub><mo>·</mo><msub><mrow><mi>F</mi></mrow><mrow><mn>5</mn></mrow></msub><mo>−</mo><mn>1.22</mn><mo>·</mo><msub><mrow><mi>F</mi></mrow><mrow><mn>4</mn></mrow></msub><mo>·</mo><msub><mrow><mi>F</mi></mrow><mrow><mn>5</mn></mrow></msub></mtd></mtr></mtable></mtd></mtr></mtable></math> (14)
Table 2 summarizes the fitting results of response curves, including the mean squared error (MSE) and adjusted
Graph
<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msup><mrow><mi>R</mi></mrow><mrow><mn>2</mn></mrow></msup></math> values for each response.
Table 2. Summary of fitting results of the regression curves.
<table><thead><tr><td>Response</td><td>Alias</td><td>Criterion</td><td><p><graphic href="iddi_a_2470397_ilm0163.gif" content-type="Graph" /><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow xmlns=""><mtext mathvariant="italic">MSE</mtext></mrow></math></p></td><td><p><graphic href="iddi_a_2470397_ilm0164.gif" content-type="Graph" /><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msubsup xmlns=""><mrow><mi>R</mi></mrow><mrow><mtext mathvariant="italic">adj</mtext></mrow><mrow><mn>2</mn></mrow></msubsup></math></p></td></tr></thead><tbody valign="top"><tr><td>Vesicle size</td><td><p><graphic href="iddi_a_2470397_ilm0165.gif" content-type="Graph" /><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub xmlns=""><mrow><mi>z</mi></mrow><mrow><mn>1</mn></mrow></msub></math></p></td><td><p><graphic href="iddi_a_2470397_ilm0166.gif" content-type="Graph" /><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow xmlns=""><mtext mathvariant="normal">min</mtext></mrow></math></p></td><td char=".">0.2784e-5</td><td char=".">0.9921</td></tr><tr><td>Polydispersity index</td><td><p><graphic href="iddi_a_2470397_ilm0167.gif" content-type="Graph" /><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub xmlns=""><mrow><mi>z</mi></mrow><mrow><mn>2</mn></mrow></msub></math></p></td><td><p><graphic href="iddi_a_2470397_ilm0168.gif" content-type="Graph" /><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow xmlns=""><mtext mathvariant="normal">min</mtext></mrow></math></p></td><td char=".">0.0001</td><td char=".">0.8033</td></tr><tr><td>Zeta potential</td><td><p><graphic href="iddi_a_2470397_ilm0169.gif" content-type="Graph" /><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub xmlns=""><mrow><mi>z</mi></mrow><mrow><mn>3</mn></mrow></msub></math></p></td><td><p><graphic href="iddi_a_2470397_ilm0170.gif" content-type="Graph" /><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow xmlns=""><mtext mathvariant="normal">max</mtext></mrow></math></p></td><td char=".">0.4000</td><td char=".">0.9932</td></tr><tr><td>Deformability index</td><td><p><graphic href="iddi_a_2470397_ilm0171.gif" content-type="Graph" /><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub xmlns=""><mrow><mi>z</mi></mrow><mrow><mn>4</mn></mrow></msub></math></p></td><td><p><graphic href="iddi_a_2470397_ilm0172.gif" content-type="Graph" /><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow xmlns=""><mtext mathvariant="normal">max</mtext></mrow></math></p></td><td char=".">0.0004</td><td char=".">0.9981</td></tr><tr><td>Phosphorous content</td><td><p><graphic href="iddi_a_2470397_ilm0173.gif" content-type="Graph" /><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub xmlns=""><mrow><mi>z</mi></mrow><mrow><mn>5</mn></mrow></msub></math></p></td><td><p><graphic href="iddi_a_2470397_ilm0174.gif" content-type="Graph" /><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow xmlns=""><mtext mathvariant="normal">max</mtext></mrow></math></p></td><td char=".">0.5480</td><td char=".">0.9916</td></tr><tr><td>Percentage of drug entrapment</td><td><p><graphic href="iddi_a_2470397_ilm0175.gif" content-type="Graph" /><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub xmlns=""><mrow><mi>z</mi></mrow><mrow><mn>6</mn></mrow></msub></math></p></td><td><p><graphic href="iddi_a_2470397_ilm0176.gif" content-type="Graph" /><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow xmlns=""><mtext mathvariant="normal">max</mtext></mrow></math></p></td><td char=".">0.0000</td><td char=".">0.9990</td></tr></tbody></table>
ANOVA analysis (Supplementary Material: 'ANOVA') confirms that significant factors and interactions align with the regression models. This consistency reinforces model robustness, ensuring key contributors are appropriately captured and the optimization process remains reliable.
A correlation analysis (provided in the Supplementary Material, 'Response Correlation Matrix') identified a strong correlation between vesicle size (
Graph
<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mrow><mi>z</mi></mrow><mrow><mn>1</mn></mrow></msub></math> ) and deformability index (
Graph
<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mrow><mi>z</mi></mrow><mrow><mn>4</mn></mrow></msub></math> ), while the remaining responses showed weaker correlations. Despite this, all six responses were retained, as each represents a distinct CQA essential to formulation integrity. Furthermore, given that our optimization approach relies on WGP, omitting any response would exclude its aspiration level, limiting the ability to balance all objectives comprehensively.
The ideal and anti-ideal points
Following the methodology, the tradeoff matrix through the individual optimization of each response using LINGO software was obtained. This allows us to determine the ideal and anti-ideal (or nadir) points. Resulted values are shown in Table 3.
Table 3. Tradeoff matrix for individual optimization through LINGO.
<table><thead><tr><td /><td /><td>Response values</td><td>Factors</td></tr><tr><td /><td /><td><p><graphic href="iddi_a_2470397_ilm0177.gif" content-type="Graph" /><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub xmlns=""><mrow><mi>z</mi></mrow><mrow><mn>1</mn></mrow></msub></math></p></td><td><p><graphic href="iddi_a_2470397_ilm0178.gif" content-type="Graph" /><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub xmlns=""><mrow><mi>z</mi></mrow><mrow><mn>2</mn></mrow></msub></math></p></td><td><p><graphic href="iddi_a_2470397_ilm0179.gif" content-type="Graph" /><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub xmlns=""><mrow><mi>z</mi></mrow><mrow><mn>3</mn></mrow></msub></math></p></td><td><p><graphic href="iddi_a_2470397_ilm0180.gif" content-type="Graph" /><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub xmlns=""><mrow><mi>z</mi></mrow><mrow><mn>4</mn></mrow></msub></math></p></td><td><p><graphic href="iddi_a_2470397_ilm0181.gif" content-type="Graph" /><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub xmlns=""><mrow><mi>z</mi></mrow><mrow><mn>5</mn></mrow></msub></math></p></td><td><p><graphic href="iddi_a_2470397_ilm0182.gif" content-type="Graph" /><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub xmlns=""><mrow><mi>z</mi></mrow><mrow><mn>6</mn></mrow></msub></math></p></td><td><p><graphic href="iddi_a_2470397_ilm0183.gif" content-type="Graph" /><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub xmlns=""><mrow><mi>F</mi></mrow><mrow><mn>1</mn></mrow></msub></math></p></td><td><p><graphic href="iddi_a_2470397_ilm0184.gif" content-type="Graph" /><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub xmlns=""><mrow><mi>F</mi></mrow><mrow><mn>2</mn></mrow></msub></math></p></td><td><p><graphic href="iddi_a_2470397_ilm0185.gif" content-type="Graph" /><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub xmlns=""><mrow><mi>F</mi></mrow><mrow><mn>3</mn></mrow></msub></math></p></td><td><p><graphic href="iddi_a_2470397_ilm0186.gif" content-type="Graph" /><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub xmlns=""><mrow><mi>F</mi></mrow><mrow><mn>4</mn></mrow></msub></math></p></td><td><p><graphic href="iddi_a_2470397_ilm0187.gif" content-type="Graph" /><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub xmlns=""><mrow><mi>F</mi></mrow><mrow><mn>5</mn></mrow></msub></math></p></td></tr></thead><tbody valign="top"><tr><td>Optimized response</td><td><p><graphic href="iddi_a_2470397_ilm0188.gif" content-type="Graph" /><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow xmlns=""><mrow><mtext mathvariant="normal">min</mtext></mrow><mo /><mrow><msub><mrow><mi>z</mi></mrow><mrow><mn>1</mn></mrow></msub></mrow></mrow></math></p></td><td char="."><bold>143.800</bold></td><td char=".">0.135</td><td char=".">10.748</td><td char=".">0.196</td><td char=".">29.288</td><td char=".">2.802</td><td char=".">0.0</td><td char=".">1.0</td><td char=".">1</td><td char=".">1</td><td char=".">1</td></tr><tr><td><p><graphic href="iddi_a_2470397_ilm0189.gif" content-type="Graph" /><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow xmlns=""><mrow><mtext mathvariant="normal">min</mtext></mrow><mo /><mrow><msub><mrow><mi>z</mi></mrow><mrow><mn>2</mn></mrow></msub></mrow></mrow></math></p></td><td char=".">143.800</td><td char="."><bold>0.135</bold></td><td char=".">10.748</td><td char=".">0.196</td><td char=".">29.288</td><td char=".">2.802</td><td char=".">0.0</td><td char=".">1.0</td><td char=".">1</td><td char=".">1</td><td char=".">1</td></tr><tr><td><p><graphic href="iddi_a_2470397_ilm0190.gif" content-type="Graph" /><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow xmlns=""><mrow><mtext mathvariant="normal">max</mtext></mrow><mo /><mrow><msub><mrow><mi>z</mi></mrow><mrow><mn>3</mn></mrow></msub></mrow></mrow></math></p></td><td char=".">155.783</td><td char=".">0.149</td><td char="."><bold>14.060</bold></td><td char=".">0.292</td><td char=".">28.051</td><td char=".">3.273</td><td char=".">0.0</td><td char=".">0.0</td><td char=".">0</td><td char=".">1</td><td char=".">1</td></tr><tr><td><p><graphic href="iddi_a_2470397_ilm0191.gif" content-type="Graph" /><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow xmlns=""><mrow><mtext mathvariant="normal">max</mtext></mrow><mo /><mrow><msub><mrow><mi>z</mi></mrow><mrow><mn>4</mn></mrow></msub></mrow></mrow></math></p></td><td char=".">180.331</td><td char=".">0.192</td><td char=".">5.829</td><td char="."><bold>0.373</bold></td><td char=".">30.461</td><td char=".">5.922</td><td char=".">0.0</td><td char=".">0.0</td><td char=".">1</td><td char=".">0</td><td char=".">1</td></tr><tr><td><p><graphic href="iddi_a_2470397_ilm0192.gif" content-type="Graph" /><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow xmlns=""><mrow><mtext mathvariant="normal">max</mtext></mrow><mo /><mrow><msub><mrow><mi>z</mi></mrow><mrow><mn>5</mn></mrow></msub></mrow></mrow></math></p></td><td char=".">145.969</td><td char=".">0.156</td><td char=".">9.821</td><td char=".">0.186</td><td char="."><bold>35.444</bold></td><td char=".">3.978</td><td char=".">1.0</td><td char=".">1.0</td><td char=".">1</td><td char=".">1</td><td char=".">1</td></tr><tr><td><p><graphic href="iddi_a_2470397_ilm0193.gif" content-type="Graph" /><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow xmlns=""><mrow><mtext mathvariant="normal">max</mtext></mrow><mo /><mrow><msub><mrow><mi>z</mi></mrow><mrow><mn>6</mn></mrow></msub></mrow></mrow></math></p></td><td char=".">180.331</td><td char=".">0.192</td><td char=".">5.829</td><td char=".">0.373</td><td char=".">30.461</td><td char="."><bold>5.922</bold></td><td char=".">0.0</td><td char=".">0.0</td><td char=".">1</td><td char=".">0</td><td char=".">1</td></tr><tr><td>Ideal point</td><td char="."><bold>143.800</bold></td><td char="."><bold>0.135</bold></td><td char="."><bold>14.060</bold></td><td char="."><bold>0.373</bold></td><td char="."><bold>35.444</bold></td><td char="."><bold>5.922</bold></td><td /><td /><td /><td /><td /></tr><tr><td>Anti-ideal point</td><td char="."><bold>180.331</bold></td><td char="."><bold>0.1923</bold></td><td char="."><bold>5.829</bold></td><td char="."><bold>0.186</bold></td><td char="."><bold>28.051</bold></td><td char="."><bold>2.802</bold></td><td /><td /><td /><td /><td /></tr></tbody></table>
The aspiration levels of each response
Deformable liposomes have opened a new strategy for efficient ocular drug delivery as they can enhance the drug half-life on the eye surface, increase drug solubility, and improve drug transport across the eye layers. Liposome components, surface modification, and some other significant properties can affect some CQAs, such as vesicle size, polydispersity index, surface charge, deformability index, integrity of vesicle by phosphorous measurement, and encapsulation efficiency [[31]].
Once introduced the responses selected in the study and following the methodology, we will take into account the aspiration levels
Graph
<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mrow><mi>T</mi></mrow><mrow><mi>i</mi></mrow></msub></math> and the relative importance
Graph
<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mrow><mi>r</mi></mrow><mrow><mi>i</mi></mrow></msub></math> of each response, based on the decision-maker's preferences, as outlined in Table 4.
Table 4. Aspiration levels and decision preferences.
<table><thead><tr><td>Response</td><td>Aspiration form</td><td>Aspiration level <p><graphic href="iddi_a_2470397_ilm0194.gif" content-type="Graph" /><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mo stretchy="true" xmlns="">(</mo><mrow xmlns=""><msub><mrow><mi>T</mi></mrow><mrow><mi>i</mi></mrow></msub></mrow><mo stretchy="true" xmlns="">)</mo></math></p></td><td>Deviation variable</td><td>Importance <p><graphic href="iddi_a_2470397_ilm0195.gif" content-type="Graph" /><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mo stretchy="true" xmlns="">(</mo><mrow xmlns=""><msub><mrow><mi>r</mi></mrow><mrow><mi>i</mi></mrow></msub></mrow><mo stretchy="true" xmlns="">)</mo></math></p></td><td>Weight <p><graphic href="iddi_a_2470397_ilm0196.gif" content-type="Graph" /><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mo stretchy="true" xmlns="">(</mo><mrow xmlns=""><msub><mrow><mi>w</mi></mrow><mrow><mi>i</mi></mrow></msub></mrow><mo stretchy="true" xmlns="">)</mo></math></p></td></tr></thead><tbody valign="top"><tr><td>Vesicle size (nm)</td><td><p><graphic href="iddi_a_2470397_ilm0197.gif" content-type="Graph" /><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub xmlns=""><mrow><mi>z</mi></mrow><mrow><mn>1</mn></mrow></msub><mo xmlns="">≤</mo></math></p></td><td><p><graphic href="iddi_a_2470397_ilm0198.gif" content-type="Graph" /><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow xmlns=""><mn>160.00</mn></mrow></math></p></td><td><p><graphic href="iddi_a_2470397_ilm0199.gif" content-type="Graph" /><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub xmlns=""><mrow><mi>p</mi></mrow><mrow><mn>1</mn></mrow></msub></math></p></td><td char=".">2</td><td char=".">0.15625</td></tr><tr><td>Polydispersity index</td><td><p><graphic href="iddi_a_2470397_ilm0200.gif" content-type="Graph" /><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub xmlns=""><mrow><mi>z</mi></mrow><mrow><mn>2</mn></mrow></msub><mo xmlns="">≤</mo></math></p></td><td><p><graphic href="iddi_a_2470397_ilm0201.gif" content-type="Graph" /><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow xmlns=""><mn>0.155</mn></mrow></math></p></td><td><p><graphic href="iddi_a_2470397_ilm0202.gif" content-type="Graph" /><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub xmlns=""><mrow><mi>p</mi></mrow><mrow><mn>2</mn></mrow></msub></math></p></td><td char=".">3</td><td char=".">0.10417</td></tr><tr><td>Zeta potential (mV)</td><td><p><graphic href="iddi_a_2470397_ilm0203.gif" content-type="Graph" /><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub xmlns=""><mrow><mi>z</mi></mrow><mrow><mn>3</mn></mrow></msub><mo xmlns="">≥</mo></math></p></td><td><p><graphic href="iddi_a_2470397_ilm0204.gif" content-type="Graph" /><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow xmlns=""><mn>12.00</mn></mrow></math></p></td><td><p><graphic href="iddi_a_2470397_ilm0205.gif" content-type="Graph" /><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub xmlns=""><mrow><mi>n</mi></mrow><mrow><mn>3</mn></mrow></msub></math></p></td><td char=".">1</td><td char=".">0.31250</td></tr><tr><td>Deformability index (mL/min)</td><td><p><graphic href="iddi_a_2470397_ilm0206.gif" content-type="Graph" /><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub xmlns=""><mrow><mi>z</mi></mrow><mrow><mn>4</mn></mrow></msub><mo xmlns="">≥</mo></math></p></td><td><p><graphic href="iddi_a_2470397_ilm0207.gif" content-type="Graph" /><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow xmlns=""><mn>0.320</mn></mrow></math></p></td><td><p><graphic href="iddi_a_2470397_ilm0208.gif" content-type="Graph" /><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub xmlns=""><mrow><mi>n</mi></mrow><mrow><mn>4</mn></mrow></msub></math></p></td><td char=".">5</td><td char=".">0.06250</td></tr><tr><td>Phosphorous content (mg)</td><td><p><graphic href="iddi_a_2470397_ilm0209.gif" content-type="Graph" /><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub xmlns=""><mrow><mi>z</mi></mrow><mrow><mn>5</mn></mrow></msub><mo xmlns="">≥</mo></math></p></td><td><p><graphic href="iddi_a_2470397_ilm0210.gif" content-type="Graph" /><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow xmlns=""><mn>30.00</mn></mrow></math></p></td><td><p><graphic href="iddi_a_2470397_ilm0211.gif" content-type="Graph" /><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub xmlns=""><mrow><mi>n</mi></mrow><mrow><mn>5</mn></mrow></msub></math></p></td><td char=".">6</td><td char=".">0.05208</td></tr><tr><td>Percentage of drug entrapment (%)</td><td><p><graphic href="iddi_a_2470397_ilm0212.gif" content-type="Graph" /><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub xmlns=""><mrow><mi>z</mi></mrow><mrow><mn>6</mn></mrow></msub><mo xmlns="">≥</mo></math></p></td><td><p><graphic href="iddi_a_2470397_ilm0213.gif" content-type="Graph" /><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow xmlns=""><mn>5.00</mn></mrow></math></p></td><td><p><graphic href="iddi_a_2470397_ilm0214.gif" content-type="Graph" /><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub xmlns=""><mrow><mi>n</mi></mrow><mrow><mn>6</mn></mrow></msub></math></p></td><td char=".">1</td><td char=".">0.31250</td></tr></tbody></table>
The right-hand column shows the weights assigned to each response, derived from
Graph
<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mrow><mi>r</mi></mrow><mrow><mi>i</mi></mrow></msub></math> .
Optimization phase
The WGP model
Following the described method, the objective function to be optimized for the WGP approach is as follows:
Graph
<math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mrow><mtext mathvariant="normal">minimize</mtext></mrow><mo>⁡</mo><mrow><mo /><mi>G</mi></mrow></mrow><mo>=</mo><mfrac><mrow><msub><mrow><mi>w</mi></mrow><mrow><mn>1</mn></mrow></msub></mrow><mrow><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub></mrow></mfrac><mo>·</mo><msub><mrow><mi>p</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>+</mo><mfrac><mrow><msub><mrow><mi>w</mi></mrow><mrow><mn>2</mn></mrow></msub></mrow><mrow><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub></mrow></mfrac><mo>·</mo><msub><mrow><mi>p</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>+</mo><mfrac><mrow><msub><mrow><mi>w</mi></mrow><mrow><mn>3</mn></mrow></msub></mrow><mrow><msub><mrow><mi>T</mi></mrow><mrow><mn>3</mn></mrow></msub></mrow></mfrac><mo>·</mo><msub><mrow><mi>n</mi></mrow><mrow><mn>3</mn></mrow></msub><mo>+</mo><mfrac><mrow><msub><mrow><mi>w</mi></mrow><mrow><mn>4</mn></mrow></msub></mrow><mrow><msub><mrow><mi>T</mi></mrow><mrow><mn>4</mn></mrow></msub></mrow></mfrac><mo>·</mo><msub><mrow><mi>n</mi></mrow><mrow><mn>4</mn></mrow></msub><mo>+</mo><mfrac><mrow><msub><mrow><mi>w</mi></mrow><mrow><mn>5</mn></mrow></msub></mrow><mrow><msub><mrow><mi>T</mi></mrow><mrow><mn>5</mn></mrow></msub></mrow></mfrac><mo>·</mo><msub><mrow><mi>n</mi></mrow><mrow><mn>5</mn></mrow></msub><mo>+</mo><mfrac><mrow><msub><mrow><mi>w</mi></mrow><mrow><mn>6</mn></mrow></msub></mrow><mrow><msub><mrow><mi>T</mi></mrow><mrow><mn>6</mn></mrow></msub></mrow></mfrac><mo>·</mo><msub><mrow><mi>n</mi></mrow><mrow><mn>6</mn></mrow></msub></math> (15)
The following constraints are considered:
Graph
<math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mrow><mi>z</mi></mrow><mrow><mi>i</mi></mrow></msub><mo stretchy="true">(</mo><mrow><mi mathvariant="bold-italic">F</mi></mrow><mo stretchy="true">)</mo><mo>+</mo><msub><mrow><mi>n</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>−</mo><msub><mrow><mi>p</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>=</mo><msub><mrow><mi>T</mi></mrow><mrow><mi>i</mi><mo /></mrow></msub><mo>∀</mo><mi>i</mi><mo stretchy="true">(</mo><mrow><mn>1</mn><mo>,</mo><mo>.</mo><mo>.</mo><mi>m</mi></mrow><mo stretchy="true">)</mo></math> (16)
Graph
<math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mrow><mi>n</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>,</mo><msub><mrow><mi>p</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>≥</mo><mn>0</mn><mo>∀</mo><mi>i</mi><mo /><mo stretchy="true">(</mo><mrow><mn>1</mn><mo>,</mo><mo>.</mo><mo>.</mo><mi>m</mi></mrow><mo stretchy="true">)</mo></math> (17)
Graph
<math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mrow><mi>F</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>F</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>∈</mo><mi>R</mi></math> (18)
Graph
<math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mrow><mi>F</mi></mrow><mrow><mn>3</mn></mrow></msub><mo>,</mo><msub><mrow><mi>F</mi></mrow><mrow><mn>4</mn></mrow></msub><mo>,</mo><msub><mrow><mi>F</mi></mrow><mrow><mn>5</mn></mrow></msub><mo>,</mo><msub><mrow><mi>F</mi></mrow><mrow><mn>6</mn></mrow></msub><mo>∈</mo><mo stretchy="true">(</mo><mrow><mn>0</mn><mo>,</mo><mn>1</mn></mrow><mo stretchy="true">)</mo></math> (19)
It is important to note that responses of NTB type are missing in the case study. Therefore, the objective function in our case lacks NTB terms and only considers STB for the first two responses and LTB for the last four responses. The problem is solved by obtaining the optimal solution using LINGO, where the Global Solver option has been selected.
The
Graph
<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi>F</mi></mrow></math> optimal solution for the problem has been obtained for the following values of the coded design factors: 0.753749
Graph
<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mo stretchy="true">(</mo><mrow><msub><mrow><mi>F</mi></mrow><mrow><mn>1</mn></mrow></msub></mrow><mo stretchy="true">)</mo></math> , 0.0
Graph
<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mo stretchy="true">(</mo><mrow><msub><mrow><mi>F</mi></mrow><mrow><mn>2</mn></mrow></msub></mrow><mo stretchy="true">)</mo></math> , 1
Graph
<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mo stretchy="true">(</mo><mrow><msub><mrow><mi>F</mi></mrow><mrow><mn>3</mn></mrow></msub></mrow><mo stretchy="true">)</mo></math> , 1
Graph
<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mo stretchy="true">(</mo><mrow><msub><mrow><mi>F</mi></mrow><mrow><mn>4</mn></mrow></msub></mrow><mo stretchy="true">)</mo></math> , and 1
Graph
<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mo stretchy="true">(</mo><mrow><msub><mrow><mi>F</mi></mrow><mrow><mn>5</mn></mrow></msub></mrow><mo stretchy="true">)</mo></math> . The optimized uncoded factors values correspond to, amount of cholesterol (
Graph
<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mrow><mi>F</mi></mrow><mrow><mn>1</mn></mrow></msub></math> ) 25.28 µmol, amount of EA (
Graph
<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mrow><mi>F</mi></mrow><mrow><mn>2</mn></mrow></msub></math> ) 10.00 mg, the phase in which TM was added (
Graph
<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mrow><mi>F</mi></mrow><mrow><mn>3</mn></mrow></msub></math> ) is Ph A, SA (
Graph
<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mrow><mi>F</mi></mrow><mrow><mn>4</mn></mrow></msub></math> ) is excluded (No), and the type of EA (
Graph
<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mrow><mi>F</mi></mrow><mrow><mn>5</mn></mrow></msub></math> ) is Deo-Na.
It can be observed that the value of
Graph
<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mrow><mi>F</mi></mrow><mrow><mn>1</mn></mrow></msub></math> takes a real value, which contrasts with the individual optimization of the other responses where the optimum for each individual response was obtained for extreme values.
Table 5 shows the details of the obtained solution. Aspiration level is shown as a reference as well as the slacks or deviations (
Graph
<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mrow><mi>n</mi></mrow><mrow><mi>i</mi></mrow></msub></math> and
Graph
<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mrow><mi>p</mi></mrow><mrow><mi>i</mi></mrow></msub></math> ).
Table 5. Optimal response and slacks for the WGP approach.
<table><thead><tr><td>Response</td><td>Alias</td><td>Optimal value</td><td>Aspiration level <p><graphic href="iddi_a_2470397_ilm0215.gif" content-type="Graph" /><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mo stretchy="true" xmlns="">(</mo><mrow xmlns=""><msub><mrow><mi>T</mi></mrow><mrow><mi>i</mi></mrow></msub></mrow><mo stretchy="true" xmlns="">)</mo></math></p></td><td><p><graphic href="iddi_a_2470397_ilm0216.gif" content-type="Graph" /><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub xmlns=""><mrow><mi>n</mi></mrow><mrow><mi>i</mi></mrow></msub></math></p></td><td><p><graphic href="iddi_a_2470397_ilm0217.gif" content-type="Graph" /><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub xmlns=""><mrow><mi>p</mi></mrow><mrow><mi>i</mi></mrow></msub></math></p></td></tr></thead><tbody valign="top"><tr><td>Vesicle size (nm)</td><td><p><graphic href="iddi_a_2470397_ilm0218.gif" content-type="Graph" /><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub xmlns=""><mrow><mi>z</mi></mrow><mrow><mn>1</mn></mrow></msub></math></p></td><td char=".">152.53</td><td char=".">160.00</td><td char=".">7.471674</td><td char=".">0</td></tr><tr><td>Polydispersity index</td><td><p><graphic href="iddi_a_2470397_ilm0219.gif" content-type="Graph" /><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub xmlns=""><mrow><mi>z</mi></mrow><mrow><mn>2</mn></mrow></msub></math></p></td><td char=".">0.16</td><td char=".">0.155</td><td char=".">0</td><td char=".">0.00936</td></tr><tr><td>Zeta potential (mV)</td><td><p><graphic href="iddi_a_2470397_ilm0220.gif" content-type="Graph" /><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub xmlns=""><mrow><mi>z</mi></mrow><mrow><mn>3</mn></mrow></msub></math></p></td><td char=".">12.00</td><td char=".">12.00</td><td char=".">0</td><td char=".">0</td></tr><tr><td>Deformability index (mL/min)</td><td><p><graphic href="iddi_a_2470397_ilm0221.gif" content-type="Graph" /><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub xmlns=""><mrow><mi>z</mi></mrow><mrow><mn>4</mn></mrow></msub></math></p></td><td char=".">0.28</td><td char=".">0.320</td><td char=".">0.04423</td><td char=".">0</td></tr><tr><td>Phosphorous content (mg)</td><td><p><graphic href="iddi_a_2470397_ilm0222.gif" content-type="Graph" /><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub xmlns=""><mrow><mi>z</mi></mrow><mrow><mn>5</mn></mrow></msub></math></p></td><td char=".">32.46</td><td char=".">30.00</td><td char=".">0</td><td char=".">2.45670</td></tr><tr><td>Percentage of drug entrapped (%)</td><td><p><graphic href="iddi_a_2470397_ilm0223.gif" content-type="Graph" /><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub xmlns=""><mrow><mi>z</mi></mrow><mrow><mn>6</mn></mrow></msub></math></p></td><td char=".">4.62</td><td char=".">5.00</td><td char=".">0.3848128</td><td char=".">0</td></tr></tbody></table>
Graphical analysis
Upon completing the optimal analysis using the WGP approach, the following graph displays contour plots for each individual response (Figure 3). These plots consider only the variation of factors
Graph
<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mrow><mi>F</mi></mrow><mrow><mn>1</mn></mrow></msub></math> (Amount of Cholesterol) and
Graph
<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mrow><mi>F</mi></mrow><mrow><mn>2</mn></mrow></msub></math> (Amount of Edge Activator), with factors
Graph
<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mrow><mi>F</mi></mrow><mrow><mn>3</mn></mrow></msub><mo>,</mo><msub><mrow><mi>F</mi></mrow><mrow><mn>4</mn></mrow></msub></math> , and
Graph
<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mrow><mi>F</mi></mrow><mrow><mn>5</mn></mrow></msub></math> fixed at their optimal values as determined by WGP according to preferences.
Graph
<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mrow><mi>F</mi></mrow><mrow><mn>1</mn></mrow></msub></math> and
Graph
<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mrow><mi>F</mi></mrow><mrow><mn>2</mn></mrow></msub></math> were selected since they have continuous values, and a more detailed evolution may be visualized.
PHOTO (COLOR): Figure 3. Sensitivity analysis of the responses in the optimum.
The desirability function approach
The desirability functions applied to each response are as follows:
For vesicle size (minimization):
Graph
<math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mrow><mi>d</mi></mrow><mrow><mn>1</mn></mrow></msub><mo stretchy="true">(</mo><mrow><msub><mrow><mi>z</mi></mrow><mrow><mn>1</mn></mrow></msub></mrow><mo stretchy="true">)</mo><mo>=</mo><mo stretchy="true">{</mo><mrow><mtable><mtr><mtd><mrow><maligngroup /><mtable><mtr><mtd><mn>1.0</mn></mtd><mtd><mo /><mtext mathvariant="italic">if</mtext></mtd><mtd><msub><mrow><mi>z</mi></mrow><mrow><mn>1</mn></mrow></msub><mo><</mo><mn>160.00</mn><mo /></mtd></mtr></mtable></mrow></mtd></mtr><mtr><mtd><mrow><maligngroup /><mtable><mtr><mtd><msup><mrow><mo stretchy="true">(</mo><mrow><mfrac><mrow><mn>180.331</mn><mo>−</mo><msub><mrow><mi>z</mi></mrow><mrow><mn>1</mn></mrow></msub></mrow><mrow><mn>180.331</mn><mo>−</mo><mn>160.00</mn></mrow></mfrac></mrow><mo stretchy="true">)</mo></mrow><mrow><mi>s</mi></mrow></msup></mtd><mtd><mtext mathvariant="italic">if</mtext></mtd><mtd><mn>160.00</mn><mo>≤</mo><msub><mrow><mi>z</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>≤</mo><mn>180.331</mn><mo /></mtd></mtr></mtable></mrow></mtd></mtr><mtr><mtd><mrow><maligngroup /><mtable><mtr><mtd><mn>0</mn></mtd><mtd><mo /><mtext mathvariant="italic">if</mtext></mtd><mtd><msub><mrow><mi>z</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>></mo><mn>180.331</mn><mo /></mtd></mtr></mtable></mrow></mtd></mtr></mtable></mrow></math> (20)
For polydispersity index (minimization):
Graph
<math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mrow><mi>d</mi></mrow><mrow><mn>2</mn></mrow></msub><mo stretchy="true">(</mo><mrow><msub><mrow><mi>z</mi></mrow><mrow><mn>2</mn></mrow></msub></mrow><mo stretchy="true">)</mo><mo>=</mo><mo stretchy="true">{</mo><mrow><mtable><mtr><mtd><mrow><maligngroup /><mtable><mtr><mtd><mo /><mn>1.0</mn></mtd><mtd><mo /><mtext mathvariant="italic">if</mtext></mtd><mtd><msub><mrow><mi>z</mi></mrow><mrow><mn>2</mn></mrow></msub><mo><</mo><mn>0.155</mn><mo /></mtd></mtr></mtable><mo /></mrow></mtd></mtr><mtr><mtd><mrow><maligngroup /><mtable><mtr><mtd><msup><mrow><mo stretchy="true">(</mo><mrow><mfrac><mrow><mn>0.1923</mn><mo>−</mo><msub><mrow><mi>z</mi></mrow><mrow><mn>2</mn></mrow></msub></mrow><mrow><mn>0.1923</mn><mo>−</mo><mn>0.155</mn></mrow></mfrac></mrow><mo stretchy="true">)</mo></mrow><mrow><mi>s</mi></mrow></msup></mtd><mtd><mo /><mtext mathvariant="italic">if</mtext></mtd><mtd><mn>0.155</mn><mo>≤</mo><msub><mrow><mi>z</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>≤</mo><mn>0.1923</mn><mo /></mtd></mtr></mtable></mrow></mtd></mtr><mtr><mtd><mrow><maligngroup /><mtable><mtr><mtd><mn>0</mn><mo /></mtd><mtd><mo /><mtext mathvariant="italic">if</mtext></mtd><mtd><msub><mrow><mi>z</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>></mo><mn>0.1923</mn><mo /></mtd></mtr></mtable></mrow></mtd></mtr></mtable></mrow></math> (21)
For zeta potential (maximization):
Graph
<math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mrow><mi>d</mi></mrow><mrow><mn>3</mn></mrow></msub><mo stretchy="true">(</mo><mrow><msub><mrow><mi>z</mi></mrow><mrow><mn>3</mn></mrow></msub></mrow><mo stretchy="true">)</mo><mo>=</mo><mo stretchy="true">{</mo><mrow><mtable><mtr><mtd><mrow><maligngroup /><mtable><mtr><mtd><mn>0</mn><mo /></mtd><mtd><mo /><mtext mathvariant="italic">if</mtext></mtd><mtd><msub><mrow><mi>z</mi></mrow><mrow><mn>3</mn></mrow></msub><mo><</mo><mn>5.829</mn></mtd></mtr></mtable></mrow></mtd></mtr><mtr><mtd><mrow><maligngroup /><mtable><mtr><mtd><msup><mrow><mo stretchy="true">(</mo><mrow><mfrac><mrow><msub><mrow><mi>z</mi></mrow><mrow><mn>3</mn></mrow></msub><mo>−</mo><mn>5.829</mn></mrow><mrow><mn>12.00</mn><mo>−</mo><mn>5.829</mn></mrow></mfrac></mrow><mo stretchy="true">)</mo></mrow><mrow><mi>s</mi></mrow></msup></mtd><mtd><mo /><mtext mathvariant="italic">if</mtext></mtd><mtd><mn>5.829</mn><mo>≤</mo><msub><mrow><mi>z</mi></mrow><mrow><mn>3</mn></mrow></msub><mo>≤</mo><mn>12.00</mn><mo /></mtd></mtr></mtable></mrow></mtd></mtr><mtr><mtd><mrow><maligngroup /><mtable><mtr><mtd><mn>1.0</mn></mtd><mtd><mo /><mtext mathvariant="italic">if</mtext></mtd><mtd><msub><mrow><mi>z</mi></mrow><mrow><mn>3</mn></mrow></msub><mo>></mo><mn>12.00</mn><mo /></mtd></mtr></mtable></mrow></mtd></mtr></mtable></mrow></math> (22)
For deformability index (maximization):
Graph
<math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mrow><mi>d</mi></mrow><mrow><mn>4</mn></mrow></msub><mo stretchy="true">(</mo><mrow><msub><mrow><mi>z</mi></mrow><mrow><mn>4</mn></mrow></msub></mrow><mo stretchy="true">)</mo><mo>=</mo><mo stretchy="true">{</mo><mrow><mtable><mtr><mtd><mrow><maligngroup /><mtable><mtr><mtd><mn>0</mn><mo /></mtd><mtd><mo /><mtext mathvariant="italic">if</mtext></mtd><mtd><msub><mrow><mi>z</mi></mrow><mrow><mn>4</mn></mrow></msub><mo><</mo><mn>0.186</mn></mtd></mtr></mtable></mrow></mtd></mtr><mtr><mtd><mrow><maligngroup /><mtable><mtr><mtd><msup><mrow><mo stretchy="true">(</mo><mrow><mfrac><mrow><msub><mrow><mi>z</mi></mrow><mrow><mn>4</mn></mrow></msub><mo>−</mo><mn>0.186</mn></mrow><mrow><mn>0.320</mn><mo>−</mo><mn>0.186</mn></mrow></mfrac></mrow><mo stretchy="true">)</mo></mrow><mrow><mi>s</mi></mrow></msup></mtd><mtd><mo /><mtext mathvariant="italic">if</mtext></mtd><mtd><mn>0.186</mn><mo>≤</mo><msub><mrow><mi>z</mi></mrow><mrow><mn>4</mn></mrow></msub><mo>≤</mo><mn>0.320</mn><mo /></mtd></mtr></mtable></mrow></mtd></mtr><mtr><mtd><mrow><maligngroup /><mtable><mtr><mtd><mn>1.0</mn></mtd><mtd><mo /><mtext mathvariant="italic">if</mtext></mtd><mtd><msub><mrow><mi>z</mi></mrow><mrow><mn>4</mn></mrow></msub><mo>></mo><mn>0.320</mn><mo /></mtd></mtr></mtable></mrow></mtd></mtr></mtable></mrow></math> (23)
For phosphorous content (maximization):
Graph
<math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mrow><mi>d</mi></mrow><mrow><mn>5</mn></mrow></msub><mo stretchy="true">(</mo><mrow><msub><mrow><mi>z</mi></mrow><mrow><mn>5</mn></mrow></msub></mrow><mo stretchy="true">)</mo><mo>=</mo><mo stretchy="true">{</mo><mrow><mtable><mtr><mtd><mrow><maligngroup /><mtable><mtr><mtd><mn>0</mn><mo /></mtd><mtd><mo /><mtext mathvariant="italic">if</mtext></mtd><mtd><msub><mrow><mi>z</mi></mrow><mrow><mn>5</mn></mrow></msub><mo><</mo><mn>28.051</mn></mtd></mtr></mtable></mrow></mtd></mtr><mtr><mtd><mrow><maligngroup /><mtable><mtr><mtd><msup><mrow><mo stretchy="true">(</mo><mrow><mfrac><mrow><msub><mrow><mi>z</mi></mrow><mrow><mn>5</mn></mrow></msub><mo>−</mo><mn>28.051</mn></mrow><mrow><mn>30.00</mn><mo>−</mo><mn>28.051</mn></mrow></mfrac></mrow><mo stretchy="true">)</mo></mrow><mrow><mi>s</mi></mrow></msup></mtd><mtd><mo /><mtext mathvariant="italic">if</mtext></mtd><mtd><mn>28.051</mn><mo>≤</mo><msub><mrow><mi>z</mi></mrow><mrow><mn>5</mn></mrow></msub><mo>≤</mo><mn>30.00</mn><mo /></mtd></mtr></mtable></mrow></mtd></mtr><mtr><mtd><mrow><maligngroup /><mtable><mtr><mtd><mn>1.0</mn></mtd><mtd><mo /><mtext mathvariant="italic">if</mtext></mtd><mtd><msub><mrow><mi>z</mi></mrow><mrow><mn>5</mn></mrow></msub><mo>></mo><mn>30.00</mn><mo /></mtd></mtr></mtable></mrow></mtd></mtr></mtable></mrow></math> (24)
For percentage of drug entrapment (maximization):
Graph
<math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mrow><mi>d</mi></mrow><mrow><mn>6</mn></mrow></msub><mo stretchy="true">(</mo><mrow><msub><mrow><mi>z</mi></mrow><mrow><mn>6</mn></mrow></msub></mrow><mo stretchy="true">)</mo><mo>=</mo><mo stretchy="true">{</mo><mrow><mtable><mtr><mtd><mrow><maligngroup /><mtable><mtr><mtd><mn>0</mn><mo /></mtd><mtd><mo /><mtext mathvariant="italic">if</mtext></mtd><mtd><msub><mrow><mi>z</mi></mrow><mrow><mn>6</mn></mrow></msub><mo><</mo><mn>2.802</mn></mtd></mtr></mtable></mrow></mtd></mtr><mtr><mtd><mrow><maligngroup /><mtable><mtr><mtd><msup><mrow><mo stretchy="true">(</mo><mrow><mfrac><mrow><msub><mrow><mi>z</mi></mrow><mrow><mn>6</mn></mrow></msub><mo>−</mo><mn>2.802</mn></mrow><mrow><mn>5.00</mn><mo>−</mo><mn>2.802</mn></mrow></mfrac></mrow><mo stretchy="true">)</mo></mrow><mrow><mi>s</mi></mrow></msup></mtd><mtd><mo /><mtext mathvariant="italic">if</mtext></mtd><mtd><mn>2.802</mn><mo>≤</mo><msub><mrow><mi>z</mi></mrow><mrow><mn>6</mn></mrow></msub><mo>≤</mo><mn>5.00</mn><mo /></mtd></mtr></mtable></mrow></mtd></mtr><mtr><mtd><mrow><maligngroup /><mtable><mtr><mtd><mn>1.0</mn></mtd><mtd><mo /><mtext mathvariant="italic">if</mtext></mtd><mtd><msub><mrow><mi>z</mi></mrow><mrow><mn>6</mn></mrow></msub><mo>></mo><mn>5.00</mn><mo /></mtd></mtr></mtable></mrow></mtd></mtr></mtable></mrow></math> (25)
The choice of
Graph
<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi>s</mi></mrow></math> in the desirability function can influence results, so additional experimentation was conducted with
Graph
<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi>s</mi></mrow></math> values of 0.25, 0.50, 1, and 2. The analysis showed no significant impact on factor values, with differences limited to the numerical desirability score. As
Graph
<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>s</mi><mo>=</mo><mn>1</mn></math> is the most commonly used starting point for its balanced approach, it was selected for presenting the results. This behavior appears case-specific and may not be generalizable, warranting further analysis in other studies.
Table 6 displays the values of the factors
Graph
<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi>F</mi></mrow></math> for which the desirability function
Graph
<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi>D</mi></mrow></math> has been optimized, as well as the value of each response. For comparison, it includes the optimum obtained through WGP, as well as the aspiration levels
Graph
<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mrow><mi>T</mi></mrow><mrow><mi>i</mi></mrow></msub></math> for each case.
Table 6. Optimal responses using WGP vs. desirability approach.
<table><thead><tr><td>Response</td><td>Alias</td><td><p><graphic href="iddi_a_2470397_ilm0224.gif" content-type="Graph" /><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub xmlns=""><mrow><mi>T</mi></mrow><mrow><mi>i</mi></mrow></msub></math></p></td><td>Criterion</td><td>WGP optimal</td><td>Desirab. optimal</td></tr></thead><tbody valign="top"><tr><td>Vesicle size</td><td><p><graphic href="iddi_a_2470397_ilm0225.gif" content-type="Graph" /><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub xmlns=""><mrow><mi>z</mi></mrow><mrow><mn>1</mn></mrow></msub></math></p></td><td char=".">160.00</td><td><italic>min</italic></td><td char=".">152.528325</td><td char=".">153.103442</td></tr><tr><td>Polydispersity index</td><td><p><graphic href="iddi_a_2470397_ilm0226.gif" content-type="Graph" /><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub xmlns=""><mrow><mi>z</mi></mrow><mrow><mn>2</mn></mrow></msub></math></p></td><td char=".">0.155</td><td><italic>min</italic></td><td char=".">0.1643659</td><td char=".">0.1577922</td></tr><tr><td>Zeta potential</td><td><p><graphic href="iddi_a_2470397_ilm0227.gif" content-type="Graph" /><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub xmlns=""><mrow><mi>z</mi></mrow><mrow><mn>3</mn></mrow></msub></math></p></td><td char=".">12.00</td><td><italic>max</italic></td><td char=".">12.000000</td><td char=".">12.6496944</td></tr><tr><td>Deformability index</td><td><p><graphic href="iddi_a_2470397_ilm0228.gif" content-type="Graph" /><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub xmlns=""><mrow><mi>z</mi></mrow><mrow><mn>4</mn></mrow></msub></math></p></td><td char=".">0.320</td><td><italic>max</italic></td><td char=".">0.275768</td><td char=".">0.2863611</td></tr><tr><td>Phosphorous content</td><td><p><graphic href="iddi_a_2470397_ilm0229.gif" content-type="Graph" /><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub xmlns=""><mrow><mi>z</mi></mrow><mrow><mn>5</mn></mrow></msub></math></p></td><td char=".">30.00</td><td><italic>max</italic></td><td char=".">32.456695</td><td char=".">31.3421252</td></tr><tr><td>Percentage of drug entrapment</td><td><p><graphic href="iddi_a_2470397_ilm0230.gif" content-type="Graph" /><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub xmlns=""><mrow><mi>z</mi></mrow><mrow><mn>6</mn></mrow></msub></math></p></td><td char=".">5.00</td><td><italic>max</italic></td><td char=".">4.615187</td><td char=".">4.4737443</td></tr><tr><td /><td>Factors</td><td>Uncoded</td><td>Coded</td><td>Uncoded</td><td>Coded</td></tr><tr><td /><td><p><graphic href="iddi_a_2470397_ilm0231.gif" content-type="Graph" /><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub xmlns=""><mrow><mi>F</mi></mrow><mrow><mn>1</mn></mrow></msub></math></p></td><td char=".">25.28 µmol</td><td char=".">0.7537494</td><td char=".">23.08 µmol</td><td char=".">0.4394318</td></tr><tr><td /><td><p><graphic href="iddi_a_2470397_ilm0232.gif" content-type="Graph" /><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub xmlns=""><mrow><mi>F</mi></mrow><mrow><mn>2</mn></mrow></msub></math></p></td><td char=".">10.00 mg</td><td char=".">0.0</td><td char=".">10.00 mg</td><td char=".">0.0</td></tr><tr><td /><td><p><graphic href="iddi_a_2470397_ilm0233.gif" content-type="Graph" /><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub xmlns=""><mrow><mi>F</mi></mrow><mrow><mn>3</mn></mrow></msub></math></p></td><td>Ph A</td><td char=".">1</td><td>Ph A</td><td char=".">1</td></tr><tr><td /><td><p><graphic href="iddi_a_2470397_ilm0234.gif" content-type="Graph" /><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub xmlns=""><mrow><mi>F</mi></mrow><mrow><mn>4</mn></mrow></msub></math></p></td><td>No</td><td char=".">1</td><td>No</td><td char=".">1</td></tr><tr><td /><td><p><graphic href="iddi_a_2470397_ilm0235.gif" content-type="Graph" /><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub xmlns=""><mrow><mi>F</mi></mrow><mrow><mn>5</mn></mrow></msub></math></p></td><td>Deo-Na</td><td char=".">1</td><td>Deo-Na</td><td char=".">1</td></tr><tr><td>Objective function</td><td /><td char=".">0.038984</td><td /><td char=".">0.9815561</td></tr></tbody></table>
To compare the solutions obtained using the described methodologies, the following graph presents contour plots of the objective functions for each approach: the minimization of
Graph
<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi>z</mi></mrow></math> in WGP and the maximization of
Graph
<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi>D</mi></mrow></math> in the Desirability function. This graph considers only the variation of factors
Graph
<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mrow><mi>F</mi></mrow><mrow><mn>1</mn></mrow></msub></math> (Amount of Cholesterol) and
Graph
<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mrow><mi>F</mi></mrow><mrow><mn>2</mn></mrow></msub></math> (Amount of Edge Activator), with factors
Graph
<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mrow><mi>F</mi></mrow><mrow><mn>3</mn></mrow></msub><mo>,</mo><msub><mrow><mi>F</mi></mrow><mrow><mn>4</mn></mrow></msub></math> , and
Graph
<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mrow><mi>F</mi></mrow><mrow><mn>5</mn></mrow></msub></math> fixed at their optimal values, as both approaches reached the same conclusions regarding these factors.
Discussion
The development of efficient drug delivery systems, such as ultradeformable liposomes, requires a meticulous balance of multiple factors influencing their stability, efficacy, and safety. The application of advanced optimization methods plays a pivotal role in achieving this balance, addressing the complexities inherent to pharmaceutical formulations. By employing WGP, this study showcases a structured approach to multi-objective optimization, surpassing traditional methods in flexibility and precision.
The experiments utilized a systematic DoE approach, based on an orthogonal array of a previous study by González-Rodríguez et al. [[26]] in which a regression analysis was employed to model each response individually. Regression is crucial in this context as it quantifies the relationship between formulation inputs and outputs, providing a mathematical foundation to predict and optimize each response. By fitting experimental data to regression models, the study was able to determine ideal and anti-ideal points for each response, serving as benchmarks in the subsequent multi-objective optimization phase. Table 3 shows the values of the tradeoff matrix between each response with other in the sense of maximizing or minimizing them. The maximum (
Graph
<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mrow><mi>z</mi></mrow><mrow><mn>3</mn></mrow></msub><mo>,</mo><msub><mrow><mi>z</mi></mrow><mrow><mn>4</mn></mrow></msub></math> , or
Graph
<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mrow><mi>z</mi></mrow><mrow><mn>5</mn></mrow></msub></math> ) or minimum (
Graph
<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mrow><mi>z</mi></mrow><mrow><mn>1</mn></mrow></msub></math> or
Graph
<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mrow><mi>z</mi></mrow><mrow><mn>2</mn></mrow></msub></math> ) values appear in bold. In the last columns, the values of factors that optimize each individual response are displayed in the Factors columns of the table and are indicated as 0 and 1. The last two rows display the ideal and anti-ideal values for each response, which are subsequently used according to the methodology.
The establishment of accurate aspiration levels and their relative importance forms the foundation of multi-objective optimization, particularly in pharmaceutical formulation. For practitioners, these values represent the desired outcomes, carefully balanced to reflect their significance in achieving the formulation objectives. As outlined in Table 4, the researcher determines the importance of each response, assigning priority based on therapeutic and regulatory goals. These importance values are subsequently translated into corresponding weights, which guide the optimization process and ensure a balanced approach to competing objectives.
Zeta potential shows a pivotal influence on liposome resistance time onto the eye, prolonging the TM physiologic effect. Our study involves three main components affecting this parameter and conditioning the stability of the vesicles, positive-charged TM and SA (when added), and negatively charged when present of sodium deoxycholate as EA. According to results obtained experimentally [[26]], with a charge-neutralizing effect and from literature, vesicles showing their zeta potential higher than 12 mV are believed to have good colloidal stability [[32]]. Moreover, these values are intended to interact with the negatively charged mucin on the eye surface, but not reducing their preocular clearance through drainage and permeability.
The appearance of liposomal product is affected by free drug and liposomal entrapped drug. Our previous study started with formulations having PDE ranged between 16.7 and 86.4% before extrusion. However, after extruding to be suitable for further in vivo applications (size < 150 nm and PdI 0.1), the PDE decreased to 0.9 − 10.7%. The location of partial dose of TM at the surface may cause its loss after the extrusion. This hypothesis has been studied by Guimaraes et al.'s [[33]] <sups>1</sups>HNMR approach to liposome-encapsulated methotrexate. In some instances, free drug (drug that is not entrapped in liposomes) can cause severe side effects. Therefore, it is important to define the drug encapsulation of the liposomal system since TM, belonging to class beta-blocker, can cause side effects in glaucoma treatment. It is the most significant factor linked to nonadherence which could lead to treatment failure. Serious cardiovascular side effects including bradycardia can suffer the patients because the systemic bioavailability of ophthalmic TM is around 78% compared with oral TM which is around 61% due to first pass metabolism [[34]]. Local adverse effects include periorbital dermatitis [[35]] and burning/stinging, among other effects. Therefore, the entrapped TM into a protective vehicle (such as vesicles) has a great relevance to avoid or minimize the appearance of these side-effects. Yoon et al. suggested that the drug vehicle significantly impacts the safety of ophthalmic TM [[36]].
The size has been shown a significant impact because in the non-corneal pathway, in addition to the transcellular pathway, the intercellular pathway can be used. Additionally, in general, the smaller the size, the more contact the particles have with the surface, resulting in more retention. It has been proved that, in the case of drug penetration, the size threshold is around 200 nm for the anterior compartment of the eye. Soni and Saini developed brimonidine tartrate-loaded cationic-charged liposomes having 150.4 nm and 0.203 PdI (indicating a homogeneous liposomal population with a narrow size distribution), allowing enhanced trans-corneal drug permeation, prolonged corneal residence, and sustained drug release [[37]]. However, size can also affect the loading capacity since more vesicles have a higher volume-to-surface ratio as size decreases and also the volume of the interior space that holds the drug. This fact is exceptionally truthful for hydrophilic drugs as TM located in the internal compartment of the liposome.
Deformability index is a really interesting parameter for vesicles having higher sizes than liposomes in our study. For this, it was included with lower importance than size. Adding EA to liposomes (Tween<sups>®</sups>20 or Na-Deo in our study) makes them more flexible and higher adaptable, because they disrupt the densely packed bilayer structure. This causes liposomes to transport more easily through eye barriers, primarily facilitating the intercellular passage and then, bypassing the efflux proteins or intracellular enzymes. However, to study the deformability capability of transfersomes, we applied extrusion and the demixing of constituents of membrane under external stress was produced. The redistribution of surfactant after removing the stress and reconstructing the vesicle caused TM loss and hence, the PDE can be reduced.
Total phosphorous is the sum of all the phospholipids in the formulation. This parameter impacts the liposome properties and drug encapsulation, contributing the higher values of this response on the higher values of PDE.
With respect to the optimization, the results obtained using WGP and the Desirability function exhibit a high degree of similarity (see Table 6). This can be attributed to two key factors: first, refinements from the WGP approach, such as shared weighting preferences and the incorporation of lower and upper limits derived from the WGP tradeoff matrix, were integrated into the Desirability approach. Second, the alignment of results validates the less widely known WGP methodology through its consistency with a well-established tool like the Desirability function.
Being close to each target value indicates that the CQAs (e.g. vesicle size, polydispersity index, and drug loading) are all maintained within their optimal ranges. In practical terms, this translates into a more uniform and stable product: vesicles remain at the desired size for improved tissue penetration, the formulation exhibits sufficient zeta potential for effective retention, and drug entrapment is optimized for consistent therapeutic delivery. Consequently, patients benefit from a formulation that consistently meets safety and efficacy requirements.
The results show that for
Graph
<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mrow><mi>z</mi></mrow><mrow><mn>1</mn></mrow></msub></math> (Vesicle Size) and
Graph
<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mrow><mi>z</mi></mrow><mrow><mn>2</mn></mrow></msub></math> (Polydispersity Index), where the objective is to minimize, both approaches improve upon the target values (160.00 and 0.155, respectively), with WGP slightly outperforming Desirability for
Graph
<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mrow><mi>z</mi></mrow><mrow><mn>1</mn></mrow></msub></math> (152.53 vs. 153.10) and Desirability achieving a marginally better result for
Graph
<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mrow><mi>z</mi></mrow><mrow><mn>2</mn></mrow></msub></math> (0.1570.157 vs. 0.164). For
Graph
<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mrow><mi>z</mi></mrow><mrow><mn>3</mn></mrow></msub></math> (Zeta Potential), a maximization objective, WGP matches the target (12.00), while Desirability produces a slightly higher value (12.65). Both methods fall short of the target for
Graph
<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mrow><mi>z</mi></mrow><mrow><mn>4</mn></mrow></msub></math> (Deformability Index, 0.320), with Desirability (0.2864) marginally outperforming WGP (0.2758). For
Graph
<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mrow><mi>z</mi></mrow><mrow><mn>5</mn></mrow></msub></math> (Phosphorous Content), WGP exceeds the target (32.46 vs. 30.00) and outperforms Desirability (31.34). Lastly, for
Graph
<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mrow><mi>z</mi></mrow><mrow><mn>6</mn></mrow></msub></math> (Percentage of Drug Entrapment), neither approach achieves the target (5.00), though WGP provides a slightly better result (4.615 vs. 4.474). Overall, WGP demonstrates a slight advantage in achieving a mathematically balanced solution across these objectives.
A sensitivity analysis was performed alongside WGP optimization, offering valuable insights. The optimal point was plotted, with arrows indicating improvement directions based on gradient direction – red for maximization and blue for minimization – highlighting response sensitivity to factor changes.
Regarding the response vesicle size
Graph
<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mo stretchy="true">(</mo><mrow><msub><mrow><mi>z</mi></mrow><mrow><mn>1</mn></mrow></msub></mrow><mo stretchy="true">)</mo></math> and deformability index
Graph
<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mo stretchy="true">(</mo><mrow><msub><mrow><mi>z</mi></mrow><mrow><mn>4</mn></mrow></msub></mrow><mo stretchy="true">)</mo></math> , it can be observed that it is hardly affected by the increase or decrease of
Graph
<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mrow><mi>F</mi></mrow><mrow><mn>2</mn></mrow></msub></math> (y-axis), as the line is very vertical. However, it is less sensitive to the variation of
Graph
<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mrow><mi>F</mi></mrow><mrow><mn>1</mn></mrow></msub></math> (x-axis). However, the effect is precisely the opposite. While an increase in
Graph
<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mrow><mi>F</mi></mrow><mrow><mn>2</mn></mrow></msub></math> leads to an improvement in
Graph
<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mrow><mi>z</mi></mrow><mrow><mn>1</mn></mrow></msub></math> by decreasing its value, a worse response is obtained in
Graph
<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mrow><mi>z</mi></mrow><mrow><mn>4</mn></mrow></msub></math> , decreasing the response. Conversely,
Graph
<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mrow><mi>z</mi></mrow><mrow><mn>5</mn></mrow></msub></math> is clearly affected by the values of
Graph
<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mrow><mi>F</mi></mrow><mrow><mn>1</mn></mrow></msub></math> , being less sensitive to variations in
Graph
<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mrow><mi>F</mi></mrow><mrow><mn>1</mn></mrow></msub></math> . The remaining responses are clearly influenced by both variables, making their joint interpretation challenging. It can be observed that a decrease in
Graph
<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mrow><mi>F</mi></mrow><mrow><mn>1</mn></mrow></msub></math> results in an improvement in the response
Graph
<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mrow><mi>z</mi></mrow><mrow><mn>2</mn></mrow></msub></math> and
Graph
<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mrow><mi>z</mi></mrow><mrow><mn>3</mn></mrow></msub></math> , leading to a decrease in the PdI and an increase in the Zeta Potential. However, there is a deterioration in the response
Graph
<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mrow><mi>z</mi></mrow><mrow><mn>6</mn></mrow></msub></math> , referring to the PDE. A decrease in
Graph
<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mrow><mi>F</mi></mrow><mrow><mn>2</mn></mrow></msub></math> would worsen
Graph
<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mrow><mi>z</mi></mrow><mrow><mn>2</mn></mrow></msub></math> (PdI), yet improve
Graph
<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mrow><mi>z</mi></mrow><mrow><mn>3</mn></mrow></msub></math> (Zeta Potential) and
Graph
<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mrow><mi>z</mi></mrow><mrow><mn>6</mn></mrow></msub></math> (PDE).
In summary, it is important to emphasize that although only three of the studied responses have met their aspiration levels, and the other three have not reached their designated targets, the model has successfully achieved a mathematically equidistant point among all responses while minimizing the sum of weighted deviations.
The WGP approach proves effective in aligning responses closer to their ideal targets while maintaining balance across all objectives, making it a robust tool for multi-objective optimization. Additionally, its mathematically precise framework provides a clear advantage over the inherent subjectivity of decisions influenced by the Desirability function.
The overlayed contour plot (Figure 4) highlights the sensitivity of the Desirability function and WGP outcomes across the two primary factors (
Graph
<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mrow><mi>F</mi></mrow><mrow><mn>1</mn></mrow></msub></math> and
Graph
<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mrow><mi>F</mi></mrow><mrow><mn>2</mn></mrow></msub></math> ). The white contour lines represent levels of the desirability score, clearly showing how desirability increases as the factors approach the optimal region. The maximal desirability region aligns with the optimal region identified by WGP, reinforcing that the Desirability function can achieve comparable optimization outcomes. The color scale reflects the
Graph
<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi>z</mi></mrow></math> -value obtained from WGP optimization, with the lowest value (indicating optimality in WGP) closely corresponding to regions of high desirability, further demonstrating the alignment between the two approaches. Both methods exhibit a similar tradeoff structure between (
Graph
<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mrow><mi>F</mi></mrow><mrow><mn>1</mn></mrow></msub></math> and
Graph
<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mrow><mi>F</mi></mrow><mrow><mn>2</mn></mrow></msub></math> ), converging the solution space. This consistency is expected, as the Desirability function was parameterized using the same weights and limits derived from WGP. However, while the results may appear similar, the Desirability approach struggles to efficiently balance the responses, making WGP a more effective and reliable method for achieving a well-balanced solution across all objectives.
PHOTO (COLOR): Figure 4. WGP and desirability response surface.
This makes the WGP methodology a very useful tool in experimental practice, given its methodological precision and good results, avoiding the subjectivity of the researcher.
Conclusions
The study demonstrates how modern optimization frameworks, such as WGP, can enhance QbD in complex drug delivery contexts. The use of these advanced optimization methods provides a structured and effective way to achieve this balance, surpassing traditional methods in flexibility and precision. As a concise conclusion, four phases may be remarked in this methodology: Phase 1 (Factors & Responses) where five key formulation factors for TM-loaded ultradeformable liposomes were identified and six CQAs were targeted. Each response was mathematically modeled via regression equations to predict how changes in factors would influence the liposome performance attributes; Phase 2 (Ideal & Anti-Ideal), optimizing each response separately to find best (ideal) and worst (nadir) values, creating a tradeoff matrix; Phase 3 (Targets & Weights) in which target (aspiration) values were specified and relative importance (weights) was assigned to each response, guided by therapeutic and formulation priorities; Phase 4 (WGP Optimization) demonstrated that WGP minimizes the sum of weighted deviations from each response target, producing an optimal 'balanced' solution, satisfying some targets exactly (e.g. zeta potential) and coming close to others (e.g. polydispersity index, entrapment efficiency). A sensitivity analysis (contour plots) highlighted how each response changes if one factor is altered while the others remain fixed at the WGP optimum. Phase 5 (Desirability Comparison) was added for fair comparison to WGP, establishing the same conditions, validating the results obtained by the WGP approach. Consequently, WGP emerges as a more precise and less subjective alternative, offering a robust and effective approach to pharmaceutical formulation optimization.
Author contributions statement
Sonia Valverde Cabeza: Data curation, Formal analysis, Investigation, Methodology, Writing – original draft.
Pedro Luis González-R: Conceptualization, Data curation, Formal analysis, Investigation, Methodology, Software, Supervision, Writing – original draft, Writing – review & editing.
María Luisa González-Rodríguez: Conceptualization, Data curation, Formal analysis, Investigation, Methodology, Supervision, Writing – original draft, Writing – review & editing.
All authors agree to be accountable for all aspects of the work.
<h31 id="AN0184162980-23">Disclosure statement</h31>
No potential conflict of interest was reported by the author(s).
Footnotes
1
Supplemental data for this article can be accessed online at https://doi.org/10.1080/03639045.2025.2470397.
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By Sonia Valverde Cabeza; Pedro Luis González-R and María Luisa González-Rodríguez
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