Trade-offs in Encapsulation Efficiency and Drug Release: A Multi-Objective Optimization Approach

This study was based on Response Surface Methodology (RSM) models (Equations (1) and (2)) to optimize the formulation of candesartan cilexetil microencapsulation using biodegradable polymers [7]. The mixed factorial design was used to develop mathematical models that correlate encapsulation efficiency and cumulative drug release at 12 h with the levels of various formulation factors. The study used two primary biodegradable polymers: polylactic acid (PLA) and polyvinylpyrrolidone (PVP K30). PLA is a biodegradable polyester with excellent chemical and physical properties, suitable for drug delivery systems. The concentrations of poly(lactic acid) (PLA) were evaluated within the range of 100 mg to 300 mg. The addition of PLA concentrations enhanced encapsulation efficiency by facilitating the formation of a more compact polymer network. However, an excessive amount of PLA may hinder drug release, as the dense matrix restricts the diffusion of the drug into the surrounding medium. PVP K30, a hydrophilic polymer, plays a significant role in modulating the drug release profile. As a pore-forming agent, increasing concentrations of PVP K30 (ranging from 0 mg to 100 mg) facilitate the penetration of the dissolution medium into the microcapsule, resulting in an accelerated drug release. Consequently, PVP K30 is recognized as a valuable component for enhancing the bioavailability of poorly soluble drugs. The interaction between PLA and PVP K30 is critical, as achieving an optimal balance between these components ensures effective drug encapsulation while preserving desirable release rates. Furthermore, the volume of PVA (V-PVA) solution (1% w/v), which varies between 50 mL and 75 mL, significantly impacts the stability and size of the microcapsules. The encapsulation duration, ranging from 0 to 12 h, plays a crucial role in the degree of polymer solidification. This, in turn, impacts both the structural integrity of the microcapsules and the efficiency of the encapsulation process. The study found that the ratio of polymer to active ingredient influences the encapsulation efficiency of candesartan cilexetil. Higher polymer ratios increase efficiency but may hinder drug release. Formulations with a PLA: drug ratio of 6:1 showed maximum efficiencies of 68%–71%. However, formulations with higher polymer content may reduce bioavailability. The study suggested optimizing formulations balances encapsulation efficiency and drug release to improve overall bioavailability [7]. The procedure of performing MOO can be summarized in the following five steps: 1. The use of the RSM models as problem definition for each objective (Equations (1) and (2)): where Y1 represents the Encapsulation efficiency, and Y2 represents the percentage of drug release at 12 h. PLA, VPVA, and PVP represent polylactic acid amount, volume of 1% polyvinyl alcohol and polyvinylpyrrolidone K30 amount, respectively. 2. The utilization of five multi-objective algorithms, including the Reference point based Non-dominated Sorting Genetic Algorithm III (NSGA-III) [8], a Multi-objective Evolutionary Algorithm Based on Decomposition (MOEAD) [9], a Reference Vector guided Evolutionary Algorithm for many-objective optimization (RVEA) [10], a Two-Archive Evolutionary Algorithm for Constrained Multi-objective Optimization (C-TAEA) [11], and an Adaptive Geometry Estimation based Many-Objective Evolutionary Algorithm (AGE-MOEA) [12]. The NSGA-III is predicated on a reference point methodology that facilitates concurrently managing multiple reference points, thereby enabling users to concentrate on particular areas of the Pareto-optimal front. Its advantages are evident in its capacity to adeptly navigate conflicting objectives through the application of guided reference directions, which aids in the identification of diverse trade-off solutions. This approach significantly enhances decision-making by offering a well-distributed array of near-Pareto-optimal solutions, even within high-dimensional objective spaces. This algorithm upholds a systematic approach to the distribution of solutions, thereby ensuring a well-converged and diverse array of solutions along the Pareto-optimal front. Furthermore, the normalization process employed by NSGA-III, along with its methodical assignment of reference directions, significantly enhances its efficiency and effectiveness in navigating complex objective spaces [8]. The MOEA/D method decomposes a multi-objective optimization problem into multiple scalar optimization subproblems, which are simultaneously optimized using a population of solutions. The algorithm’s strengths are evident in its ability to effectively manage conflicting objectives by leveraging neighborhood relationships among the subproblems. This approach enables a more thorough exploration of the solution space and encourages a more uniform distribution of optimal solutions. Furthermore, MOEA/D is capable of integrating various decomposition strategies, thereby increasing its adaptability to a diverse range of multi-objective problems. The Multi-Objective Evolutionary Algorithm based on Decomposition (MOEA/D) presents several advantages in comparison to alternative algorithms. Notably, it exhibits reduced computational complexity at each generation, facilitating expedited processing. Additionally, MOEA/D has been shown to achieve higher solution quality in multi-objective knapsack problems and continuous test instances, particularly when employing advanced decomposition techniques. Moreover, MOEA/D adeptly addresses challenges associated with fitness assignment and diversity maintenance, thereby enhancing its efficiency in optimizing scalar problems relative to non-decomposition approaches [9]. The Reference Vector Guided Evolutionary Algorithm (RVEA) is specifically designed to address many-objective optimization problems. RVEA integrates the principles of dominance and decomposition, effectively leveraging the advantages of both methodologies to manage multiple conflicting objectives. It introduces a scalarization technique known as Angle Penalized Distance, which dynamically balances convergence and diversity within high-dimensional objective spaces, thereby addressing the inherent challenges associated with many-objective problems. The algorithm employs reference vectors to direct the search process toward preferred regions of the Pareto front, facilitating a more focused exploration of the solution space in accordance with user preferences. Furthermore, RVEA incorporates mechanisms to preserve population diversity, a critical factor in high-dimensional contexts where solutions may become sparse. This feature enhances the algorithm’s ability to approximate the Pareto front more effectively. The performance of RVEA has been empirically validated against several state-of-the-art multi-objective evolutionary algorithms across a variety of benchmark problems, demonstrating its robust capabilities [10]. The C-TAEA algorithm, on the other hand, uses two archives—one for feasible solutions (CA) and another for infeasible solutions (DA)—enabling it to maintain diversity while effectively exploring both feasible and infeasible regions of the solution space. By concurrently managing both archives, C-TAEA strikes a balance between the search for feasible solutions and the preservation of diversity within the population, which is essential when addressing conflicting objectives. The algorithm employs various constraint-handling techniques, such as prioritizing feasible solutions and applying a constrained dominance relation that combines both objective and constraint rankings for solution evaluation. Additionally, it employs a modified version of nondominated sorting that considers both objective values and constraints in ranking solutions, thereby enhancing the search for optimal outcomes. The efficacy of C-TAEA has been empirically validated against leading constrained evolutionary multi-objective optimizers and a range of engineering and real-world problems, demonstrating its competitiveness across various benchmark scenarios [11]. The AGE-MOEA is based on evolutionary algorithms specifically designed to tackle multi-objective and many-objective optimization problems. This algorithm approximates the geometry of the generated Pareto front through a computationally efficient process, which is characterized by a complexity of O(M × N), where M signifies the number of objectives and N represents the population size. This efficiency level allows the algorithm to adjust its strategies in alignment with the specific geometric characteristics of the problem being addressed. AGE-MOEA modifies diversity and proximity metrics in accordance with the estimated geometry of the problem space. This adaptation implies that the selection of solutions for subsequent generations is informed by their contributions to sustaining a well-diversified set of solutions while also ensuring a close approximation to the Pareto front, specifically tailored to the unique characteristics of the problem at hand. By emphasizing both diversity—ensuring a broad distribution of solutions—and proximity—aiming to position solutions near the Pareto optimal front—the algorithm adeptly navigates the trade-offs associated with conflicting objectives. This dual emphasis facilitates the generation of a more comprehensive array of solutions that more accurately reflect the trade-offs between these conflicting objectives. Empirical evaluations have demonstrated that AGE-MOEA significantly surpasses several established multi-objective evolutionary algorithms across a variety of benchmark test problems characterized by differing geometries and numbers of objectives. The findings indicate that AGE-MOEA achieves lower Inverted Generational Distance (IGD) values, which serve as a metric for assessing the quality of the non-dominated front produced [12]. 3. Maximize the entrapment efficiency and drug release objectives and generate comparable Pareto fronts, with a termination criterion set at a maximum of 100 generations. 4. Evaluate the Pareto-optimal set using different performance indicators (Equations (3)–(8)) [4]:

• The hypervolume metric, denoted as $$Hv\ (S,r)$$, quantifies the size of the space covered by the Pareto-optimal set S in relation to a reference point $$r\in\mathbb{R}^m$$.

where $$\mathcal{L}(·)$$ is the Lebesgue measure, D and $$D^\prime$$ refer to dominated and non-dominated solutions, respectively, and $$r\in\mathbb{R}^m$$ is a reference point.

• The generational distance (GD) metric measures the convergence of the Pareto-optimal set to the true Pareto front:

where n represents the number of solutions on the true Pareto front, and $$d_i$$ represents the Euclidean distance of each optimal solution to its nearest solution on the true Pareto-optimal front.

• The inverted generational distance (IGD) metric complements GD by measuring how well the true Pareto front is represented by the generated Pareto set:

where the $${d_i}^\ast$$ represents the Euclidean distance of each solution on the true Pareto-optimal front to its nearest optimal solution, which is the inverse regarding $$d_i$$.

• The spacing metric (SP) assesses the uniformity of the distribution of solutions in the Pareto set:

where N is the number of solutions in the Pareto set, $$d_n$$ is the Euclidean distance of each solution n from the mean of the solutions, and $$\overline{d}$$ is the mean of the distances.

• The maximum spread metric (MaxSP) evaluates the extent of the objective space covered by the Pareto set:

where o denotes the number of objectives, $$a_i$$ and $$b_i$$ are the maximum and minimum values of the i^th objective over all solutions, respectively.

• The spread metric (SM) evaluates both the extent of coverage and the uniformity of the Pareto front:

where M signifies the number of extreme points, {$$e_1,\ e_2,\ \ldots,\ e_m$$} represent the extreme solutions for m number objectives, $$d\left(e_m,\ S\right)$$ denotes the Euclidean distance between the extreme solutions and the true Pareto-optimal front, and $$\overline{d}$$ is the average of all $$d_i$$. 5. The selection of a single optimal formulation based on the WSM (Equation (9)): where $${Score}_{WSM}$$ is the scores for a given Pareto-optimal solution, $$a_i$$ is the value of the i^th objective for an Pareto-optimal solution, $$w_i$$ is the weight assigned to the i^th objective, n is the total number of objectives. This method allows for prioritizing certain objectives based on their relative importance, facilitating the selection of a solution that aligns with specific performance criteria [13].