Modeling Evaluation Research Articles

Modeling construct change over time amidst potential changes in construct measurement: A longitudinal moderated factor analysis approach.

In analyzing longitudinal data with growth curve models, a critical assumption is that changes in the observed measures reflect construct changes and not changes in the manifestation of the construct over time. However, growth curve models are often fit to a repeated measure constructed as a sum or mean of scale items, making an implicit assumption of constancy of measurement. This practice risks confounding actual construct change with changes in measurement (i.e., differential item functioning [DIF]), threatening the validity of conclusions. An improved method that avoids such confounding is the second-order growth curve (SGC) model. It specifies a measurement model at each occasion of measurement that can be evaluated for invariance over time. The applicability of the SGC model is hindered by key limitations: (a) the SGC model treats time as continuous when modeling construct growth but as discrete when modeling measurement, reducing interpretability and parsimony; (b) the evaluation of DIF becomes increasingly error-prone given multiple timepoints and groups; (c) DIF associated with continuous covariates is difficult to incorporate. Drawing on moderated nonlinear factor analysis, we propose an alternative approach that provides a parsimonious framework for including many time points and DIF from different types of covariates. We implement this model through Bayesian estimation, allowing for incorporation of regularizing priors to facilitate efficient evaluation of DIF. We demonstrate a two-step workflow of measurement evaluation and growth modeling, with an empirical example examining changes in adolescent delinquency over time. (PsycInfo Database Record (c) 2024 APA, all rights reserved).

Knowledge embedding synchronous surrogate modeling for multi-objective operational reliability evaluation of complex mechanical systems

To effectively assess the multi-objective operational reliability of complex mechanical system, the concept of Knowledge Embedding Surrogate Modeling (KeSM) is proposed by integrating Knowledge-based Decomposition-Coordination (K-DC) strategy, Knowledge-based Parameter Decomposition (K-PD) strategy, and the surrogate model. In this concept, the K-DC strategy is used to decompose a complex problem into several smaller sub problems, coordinating outcomes from the bottom-up; the K-PD is applied to decompose complex parameters into two parts: associated trends and local fluctuations; the surrogate model is adopted to establish the mapping relationship between input and output. Under this concept, the Knowledge Embedding Synchronous Surrogate Modeling (S-KeSM) multi-objective operational reliability evaluation method is developed by combining linkage important sampling technique and synchronous modeling thought with KeSM concept. In addition, an engineering case (civil aircraft braking system temperature with multi-failures) is employed to validate the proposed method. Through comparison and validation with multiple methods, the results show that the proposed method has significant advantages in analysis accuracy and efficiency. This study can provide theoretical references for the multi-objective operational reliability evaluation of complex mechanical systems.

An accessible and intuitive approach to 3D data modeling for morphometric surface evaluation in pig farming

The recording of pig surfaces allows for the estimation of growth-related body size dimensions from which the animals’ condition can be inferred. Furthermore, for purposes such as animal breeding, it is of particular interest to describe the shape variations within the population. Therefore, we built a Statistical Shape Model (SSM) that enables a quantitative and visual exploration of variation in size, form and posture in an intuitive way.We included 4315 images from 582animals with a body weight ranging from 50 to 140 kg. After data processing, i.e. the removal of data points corresponding to the head and tail, the 3D point clouds were reduced to an equal number of reference points. These reference points were arranged so that the pig surface could be reconstructed via Bézier surface patches. Then, Procrustes Analysis was performed to align the reference points from all shapes. A mean shape and its main modes of variation were derived, some of which were used to correct the posture of the animals.A linear regression for weight prediction resulted in a mean absolute error of 5.06 kg before and 2.84 kg after data processing. The median error of surface reconstruction was 2.5 mm taking eight surface patches with 45 reference points and 1.3 mm taking 32 surface patches with 153 reference points into account. The first six modes of variation from the SSM indicated differences in size and posture. Interestingly, regions prone to noise were also revealed.In conclusion, we present a novel approach that can be used to receive a compact number of corresponding data points, providing data that is easier and more efficient to handle while exposing shape-specific attributes. Our approach is helpful to visualize and compare shapes, including the collection of body size measurements as well as unsupervised and supervised clustering for more objective data-driven decision-making.

Novel anti-inflammatory agents featuring phenoxy acetic acid moiety as a pharmacophore for selective COX-2 inhibitors: Synthesis, biological evaluation, histopathological examination and molecular modeling investigation

Inflammation management presents a critical challenge in modern medicine, with nonsteroidal anti-inflammatory drugs (NSAIDs) being a widely used therapeutic option. However, their efficacy is often accompanied by significant gastrointestinal adverse effects, necessitating the exploration of safer alternatives, particularly through the investigation of cyclooxygenase-2 (COX-2) inhibitors. This study endeavors to address this imperative through the synthesis and evaluation of pyrazoline-phenoxyacetic acid derivatives. Among the synthesized compounds, 6a and 6c emerged as promising candidates, demonstrating potent COX-2 inhibition with IC50 values of 0.03 µM for both and selectivity index = 365.4 and 196.9, respectively. Furthermore, these compounds exhibited efficacy in mitigating formalin-induced edema in male Wistar rats, accompanied by favorable safety profiles upon histological examination of vital organs. Comprehensive safety assessments, including evaluation of creatinine, AST, and ALT enzymatic as well as troponin T and creatine kinase-MB levels, further reinforce the promising attributes of the synthetic candidates. Molecular docking studies endorsed by molecular dynamic simulations corroborate the biological findings, elucidating significant protein–ligand interactions at COX-2 active sites indicative of therapeutic potential.

Evaluating Cost-Effectiveness of Antiretroviral Therapy over Time: A Cohort and Cost-Effectiveness Study.

To estimate the costs and cost-effectiveness of introducing highly active antiretroviral therapy (HAART) in Denmark based on real-world evidence for the three treatment eras pre-HAART (1985-1995), early HAART (1996-2005), and late HAART (2006-2017). We performed a cohort study using Danish clinical and administrative registries to estimate costs, quality-adjusted life-years (QALYs), and life-years (LY) gained per person living with human immunodeficiency virus (PLHIV) in three treatment eras. The study utilized Markov modeling for a health economic evaluation, which summarized inputs from real-world evidence and estimated the cost-effectiveness in 2017 prices of the introduction of HAART in Denmark. We performed deterministic and probabilistic sensitivity analyses to assess the robustness of the results. The total annual costs per PLHIV increased with the introduction of HAART for the index year but decreased in the incremental years and the last year of life. The total lifetime discounted (and undiscounted) cost for an average PLHIV was €91,010 (€128,981) in pre-HAART, €103,130 (€199,062) in early HAART, and €126,317 (€254,964) in late HAART. The estimated incremental cost-effectiveness ratios showed that early HAART was cost-effective compared with pre-HAART with an incremental cost-effectiveness ratio (ICER) of €1378 per QALY, and that late HAART was cost-effective compared with early HAART with an ICER of €7385 per QALY. Sensitivity analyses confirmed cost-effectiveness in all scenarios. The introduction and implementation of HAART in Danish healthcare was cost-effective, and in some scenarios, even disruptive, i.e., led to both cheaper and more effective care of PLHIV.

An experiential account with recommendations for the design, installation, operation and maintenance of a farm-scale soil moisture sensing and mapping system

Context The research explores the benefits of real time tracking of soil moisture for various land management contexts and the importance of spatio-temporal modelling and mapping to gain clear and visual understanding of soil moisture fluxes across a farm. Aims This research aims to outline the key processes required for building an operational on-farm soil moisture monitoring system where the product is highly granular daily soil moisture maps depicting variations temporally, spatially and vertically. Methods We describe processes of capacitance soil moisture probe installation, data collection infrastructure, sensor calibration, spatio-temporal modelling, and mapping. Key results An out-of-bag soil moisture evaluation modelling system was tested for nearly 2 years. We found a model accuracy (RMSE) estimate of 0.002 cm−3 cm−3 and concordance of 0.96 were found. This result is averaged over this period but fluctuated daily, and related to rainfall patterns across the target farm, which were not directly incorporated into the modelling framework. As expected, incorporating prior estimates of soil moisture into the modelling framework contributed to very accurate estimates of real time available soil moisture. Conclusions This research promotes the importance of iterative improvements to the soil moisture monitoring system, particularly in areas of sensor recalibration and spatio-temporal modelling. We stress the need for a longer-term view and plan for ongoing maintenance and improvement of such systems in the emerging digital farming ecosystem. Implications The results of this research will be useful for researchers and practitioners involved in the design and implementation of on-farm soil monitoring systems.

Thermal Conductivity of AlSi12/Al2O3-Graded Composites Consolidated by Hot Pressing and Spark Plasma Sintering: Experimental Evaluation and Numerical Modeling

Functionally graded metal matrix composites have attracted the attention of various industries as materials with tailorable properties due to spatially varying composition of constituents. This research work was inspired by an application, such as automotive brake disks, which requires advanced materials with improved wear resistance on the outer surface as combined with effective heat flux dissipation of the graded system. To this end, graded AlSi12/Al2O3 composites (FGMs) with a stepwise gradient in the volume fraction of alumina reinforcement were produced by hot pressing and spark plasma sintering techniques. The thermal conductivities of the individual composite layers and the FGMs were evaluated experimentally and simulated numerically using 3D finite element (FE) models based on micro-computed X-ray tomography (micro-XCT) images of actual AlSi12/Al2O3 microstructures. The numerical models incorporated the effects of porosity of the fabricated AlSi12/Al2O3 composites, thermal resistance, and imperfect interfaces between the AlSi12 matrix and the alumina particles. The obtained experimental data and the results of the numerical models are in good agreement, the relative error being in the range of 4 to 6 pct for different compositions and FGM structure. The predictive capability of the proposed micro-XCT-based FE model suggests that this model can be applied to similar types of composites and different composition gradients.

Practical evaluation and theoretical modeling of charging process for Lead Acid batteries in PV systems

Practical evaluation and theoretical modeling of charging process for Lead Acid batteries in PV systems

Development of a Novel Soft Tissue Measurement Device for Individualized Finite Element Modeling in Custom-Fit CPAP Mask Evaluation.

Individual facial soft tissue properties are necessary for creating individualized finite element (FE) models to evaluate medical devices such as continuous positive airway pressure (CPAP) masks. There are no standard tools available to measure facial soft tissue elastic moduli, and techniques in literature require advanced equipment or custom parts to replicate. We propose a simple and inexpensive soft tissue measurement (STM) indenter device to estimate facial soft tissue elasticity at five sites: chin, cheek near lip, below cheekbone, cheekbone, and cheek. The STM device consists of a probe with a linear actuator and force sensor, an adjustment system for probe orientation, a head support frame, and a controller. The device was validated on six ballistics gel samples and then tested on 28 subjects. Soft tissue thickness was also collected for each subject using ultrasound. Thickness and elastic modulus measurements were successfully collected for all subjects. The mean elastic modulus for each site is Ec = 53.04 ± 20.97kPa for the chin, El = 16.33 ± 8.37kPa for the cheek near lip, Ebc = 27.09 ± 11.38kPa for below cheekbone, Ecb = 64.79 ± 17.12kPa for the cheekbone, and Ech = 16.20 ± 5.09kPa for the cheek. The thickness and elastic modulus values are in the range of previously reported values. One subject's measured soft tissue elastic moduli and thickness were used to evaluate custom-fit CPAP mask fit in comparison to a model of that subject with arbitrary elastic moduli and thickness. The model with measured values more closely resembles in vivo leakage results. Overall, the STM provides a first estimate of facial soft tissue elasticity and is affordable and easy to build with mostly off-the-shelf parts. These values can be used to create personalized FE models to evaluate custom-fit CPAP masks.

Design, synthesis, biological evaluation, and molecular modeling simulations of new phthalazine-1,4-dione derivatives as anti-Alzheimer's agents.

The development of targeted phthalazine-1,4-dione acetylcholinesterase (AChE) inhibitors for treating Alzheimer's disease involved the synthesis of 32 compounds via a multistage process. Various analytical techniques confirmed the compounds' identities. Thirteen compounds were found to inhibit AChE by more than 50% without affecting butyrylcholinesterase (BChE). Among these, three compounds, 8m, 8n, and 8p, exhibited extraordinary activity similar to donepezil, a reference AChE inhibitor. During enzyme kinetic studies, compound 8n, displaying the highest AChE inhibitory activity, underwent evaluation at three concentrations (2 × IC50, IC50, and IC50/2). Lineweaver-Burk plots indicated mixed inhibition activity for compound 8n against AChE, suggesting a combination of competitive and noncompetitive characteristics. Additionally, effective derivatives 8m, 8n, and 8p exhibited high blood-brain barrier (BBB) permeability in in vitro parallel artificial membrane permeability assay tests. Molecular docking studies revealed that these compounds bind to the enzyme's active site residues in a position similar to donepezil. Molecular dynamic simulations confirmed the stability of the protein-ligand system, and the chemical reactivity characteristics of the compounds were investigated using density functional theory. The compounds' wide energy gaps suggest stability and therapeutic potential. This research represents a significant step toward finding a potential cure for Alzheimer's disease. However, further research and testing are required to determine the compounds' safety and efficacy. The unique structure of phthalazine derivatives makes them suitable for various biological activities, and these compounds show promise for developing effective drugs for treating Alzheimer's disease. Overall, the development of these targeted compounds is a crucial advancement in the search for an effective treatment for Alzheimer's disease.

Synthesis, biological evaluation, and molecular modelling of substituted thiazolyl thiourea derivatives: A new class of prolyl oligopeptidase inhibitors

Prolyl oligopeptidase (POP) is a compelling therapeutic target associated with aging and neurodegenerative disorders due to its pivotal role in neuropeptide processing. Despite initial promise demonstrated by early-stage POP inhibitors, their progress in clinical trials has been halted at Phase I or II. This impediment has prompted the pursuit of novel inhibitors. The current study seeks to contribute to the identification of efficacious POP inhibitors through the design, synthesis, and comprehensive evaluation (both in vitro and in silico) of thiazolyl thiourea derivatives (5a-r). In vitro experimentation exhibited that the compounds displayed significant higher potency as POP inhibitors. Compound 5e demonstrated an IC50 value of 16.47 ± 0.54 μM, representing a remarkable potency. A meticulous examination of the structure-activity relationship indicated that halogen and methoxy substituents were the most efficacious. In silico investigations delved into induced fit docking, pharmacokinetics, and molecular dynamics simulations to elucidate the intricate interactions, orientation, and conformational changes of these compounds within the active site of the enzyme. Moreover, our pharmacokinetic assessments confirmed that the majority of the synthesized compounds possess attributes conducive to potential drug development.

Packaging performance evaluation and freshness intelligent prediction modeling in grape transportation

Vibrations during transportation inevitably lead to mechanical damage, endangering grape freshness and directly impacting their economic worth. While adequate packaging serves as a viable solution, current studies on packaging efficacy lack depth. Moreover, conventional methods for forecasting fruit freshness fail to accommodate the varying freshness levels of grapes across different packaging techniques. Consequently, a novel approach for predicting fruit freshness leveraging multi-sensing technology and machine learning algorithms is introduced. By reasonably evaluating packaging performance, the automation, intelligence, and accuracy of fruit freshness prediction are enhanced. Initially, critical control points in grape supply chain logistics were scrutinized using the HACCP method to identify key environmental parameters (vibration, temperature, and humidity) and their interaction with grape freshness. Subsequently, an environmental monitoring platform was devised for the grape supply chain, facilitating environmental surveillance under distinct packaging types (corrugated carton, foam box, plastic box, and inflatable package). Through a blend of environmental monitoring outcomes and physical-chemical indicators, the protective efficacy of diverse transport packaging was meticulously analyzed and appraised alongside finite element analysis. Notably, environmental data proved capable of characterizing grape freshness in lieu of quality data, with vibration metrics exhibiting strong correlations with quality metrics. Machine learning models were developed to predict grape freshness based on environmental cues, yielding prediction accuracies of 92.512% (SVM) and 94.334% (GA-ANN). The automated, non-destructive data acquisition and novel machine learning approaches offer a fresh avenue for evaluating packaging, predicting freshness, and managing food quality within grape logistics operations.

Automatic Specification and Analysis of Software Reliability Using Alloy: Selecting the Best Use Case Scenario in Terms of Reliability

Software reliability modeling, as a non-functional characteristic of software quality to achieve high reliability, is of particular importance. To obtain software systems with high reliability, it is important to select the best scenario, from the various scenarios produced, in a short time during the software design stage. Since the scenario includes interactions between the user and use case, producing various scenarios and selecting the best among them in terms of reliability is a difficult and time-consuming task for software designers. Therefore, it is of great importance to use a modeling language that has a tool-based evaluation modeling approach and does the analysis automatically. In this paper, Alloy modeling language has been used to model and analyze software reliability. Compared to other lightweight modeling languages, Alloy has such advantages as being more concise, accurate and abstract. The primary objective of this paper aims to present a modeling framework based on the Alloy modeling language with which reliability of target systems can be efficiently examined. The results indicate that the use of Alloy analyzer facilitates the immediate production of various scenarios and allows for the selection of the best one in terms of reliability.

Pharmacokinetics-Pharmacodynamics Modeling for Evaluating Drug-Drug Interactions in Polypharmacy: Development and Challenges.

Polypharmacy is commonly employed in clinical settings. The potential risks of drug-drug interactions (DDIs) can compromise efficacy and pose serious health hazards. Integrating pharmacokinetics (PK) and pharmacodynamics (PD) models into DDIs research provides a reliable method for evaluating and optimizing drug regimens. With advancements in our comprehension of both individual drug mechanisms and DDIs, conventional models have begun to evolve towards more detailed and precise directions, especially in terms of the simulation and analysis of physiological mechanisms. Selecting appropriate models is crucial for an accurate assessment of DDIs. This review details the theoretical frameworks and quantitative benchmarks of PK and PD modeling in DDI evaluation, highlighting the establishment of PK/PD modeling against a backdrop of complex DDIs and physiological conditions, and further showcases the potential of quantitative systems pharmacology (QSP) in this field. Furthermore, it explores the current advancements and challenges in DDI evaluation based on models, emphasizing the role of emerging in vitro detection systems, high-throughput screening technologies, and advanced computational resources in improving prediction accuracy.

(5S)-5-Benzylswainsonines as potent and selective inhibitors of Golgi α-mannosidase II: synthesis, enzyme evaluation and molecular modelling

Development of novel anti-cancer therapeutics based on Golgi α-mannosidase II (GMII) inhibition is considerably impeded by an undesired co-inhibition of lysosomal α-mannosidase leading to severe side-effects. In this contribution, we describe a fully stereoselective synthesis of (5S)-5-[4-(halo)benzyl]swainsonines as highly potent and selective inhibitors of GMII. The synthesis starts from a previously reported aldehyde readily available from l-ribose, and the key features include an intramolecular reductive amination with substrate-controlled stereoselectivity and a late-stage derivatisation of the benzyl group via ipso-substitution. These novel swainsonine analogues were found to be nanomolar inhibitors of the Golgi-type α-mannosidase AMAN-2 (Ki = 23–75 nM) with excellent selectivity (selectivity index = 205–870) over the lysosomal-type Jack bean α-mannosidase. Finally, molecular docking and pKa calculations were performed to provide more insight into the structure of the inhibitor:enzyme complexes, and a pair interaction energy analysis (FMO-PIEDA) was carried out to rationalise the observed potency and selectivity of the inhibitors.

Aggregate interlock and effective strain of FRP-strengthened flanged RC members subjected to torsion: Experimental evaluation and analytical modeling

The effective strain of externally bonded FRP strengthening systems and the loss of concrete aggregate interlock are critical factors for predicting strength and developing design models for structural strengthening. Current research on FRP strengthening of reinforced concrete (RC) members primarily focuses on flexural, shear, and axial loading, neglecting torsional systems. This study introduces an innovative analytical-experimental model/formulation predicting FRP effective strain and concrete strain, based on Digital Image Correlation (DIC) analysis outcomes of an experimental study on RC beams strengthened with FRP sheets under torsion. Eight flanged RC beams, strengthened with various FRP ratios, configurations, and installation methods were also systematically tested. The findings emphasize the need for reevaluation of strain limits in existing guidelines. It points out that the shear integrity of concrete is influenced by concrete compressive strength and strengthening scheme. The DIC results suggest that the strain limit for concrete surfaces between FRP strips should be in the range of 0.0055–0.008, as opposed to the limit of 0.004 set by ACI 440.2R-17. The use of the grooving method in certain systems also requires an increase factor of 1.65 and 1.45 for the effective strain of transverse and longitudinal FRP sheets.

Integrated environmental-economic modelling for cross sectoral water policy evaluation

The Water Framework Directive (WFD) has set a deadline for 2027 to reach at least good ecological status (GES) in coastal waters in the EU. As nutrient pollution (eutrophication) is one of the main pressures in most EU coastal waters, and Danish waters in particular, significant nutrient reductions are required. In this paper, we take an integrated environmental-economic modelling approach to assess alternative strategies to mitigate non-point source nutrient pollution. A spatially explicit optimization model, TargetEconN, is implemented at the Danish national scale and extended to include mussel production as a marine water quality improvement measure. Different eutrophication mitigation strategies investigated in the model are characterized by whether nitrogen emissions are reduced at the source, between the source and the recipient e.g., by establishing wetlands, or in the recipient itself. We run scenarios exploring the uncertainty in baseline load assumptions and the effects of mussel farming. The results show that the potential for marine measures depends on the baseline load assumptions and that marine measures have a limited impact on the overall costs of achieving GES. The results also show that including marine measures has a significant indirect impact through the influence on the spatial distribution of land-based measures. We conclude that including mussel farming in policy initiatives to meet WFD targets has potential, but that the distributional effects across sectors and spillover effects to other policy targets should be a central part of the ex-ante policy discussions. We argue therefore that spatially explicit integrated modelling, as the model developed for this paper, can offer useful insights to manage the unescapable trade-offs in effective policy design to meet the WFD.

Energy efficiency evaluation and optimization for wastewater treatment plant

Wastewater treatment plants (WWTPs) are considered as energy-intensive industries. A comprehensive assessment of energy efficiency in sewage treatment reveals issues of energy waste, offers insights into the energy consumption structure, fosters optimization of energy management, and enhances overall energy utilization. However, the mathematical modeling for energy efficiency evaluation becomes challenging due to the ambiguity and uncertainty of the parameters regarding energy consumption and various indicators. In this paper, the newly constructed evaluation method was used to quantify the energy efficiency. This method employs dimensional reduction, projecting the numerical values of unit energy consumption for various pollutants onto a lower-dimensional space, and then computing the projected eigenvalues. Larger eigenvalues indicate higher energy efficiency. Based on the evaluation results, the back propagation (BP) neural network is used to simulate the sewage treatment process. The results demonstrated that when the inflow load is small and the concentration of pollutants is high, the energy efficiency of sewage treatment plants performs well. When the inflow load is high and the concentration of pollutants is low, the energy efficiency of sewage treatment plants is poor. Meanwhile, the energy consumption could be decreased by 11.61 %, 14 %, and 15.26 % respectively in different scenarios.

Distribution characteristics and integrated ecological risks evaluation modelling of microplastics and heavy metals in geological high background soil

The microplastics (MPs), a novel pollutant, and heavy metals (HMs) significantly affect soil ecology. The study investigated HMs and MPs in Qianxi's high geological background soil, established a model for risk evaluation with MPs types and shapes, and proposed a two-dimensional comprehensive index model for MPs-HMs combined pollution and risk evaluation criterion. The results revealed a high soil Cd concentration, with a mean value of 0.38 mg·kg−1. Additionally, soils from soybean–wheat intercropping–potato–corn rotation (SWI-PCR) exhibited significantly higher concentrations of Hg, As, and Pb compared with those from soybean–wheat intercropping-corn rotation (SWI-CR). Moreover, the soil exhibited a high abundance of MPs (8667.66 ± 3864.26 items·kg−1), mainly characterized by PS and fiber. The mean of adjusted ecological risk index (ARI) for MPs in soil was 525.27, indicating a grade 3 risk. The two-dimensional combined index (TPI) was used to assess the ecological risk of MPs–HMs combined pollution, exhibiting an exceedance rate of 56 % with a mean of 445.07. The risk level of the combined pollution was graded as 6, indicating high risk. The microplastic risk evaluation model and the comprehensive evaluation method of combined pollution established in this study provide a reference for the future risk evaluation of multi-pollutant combined pollution.

Accurate analytical evaluation of the generalized logarithmic and double Fermi–Dirac and Bose–Einstein functions

AbstractThe accurate definition and powerful evaluation modeling of the various generalized Fermi–Dirac and Bose–Einstein functions remain a challenging problem in various areas of physics. In this study, we develop a general analytical technique for accurately calculating logarithmic and double Fermi–Dirac and Bose–Einstein functions. The obtaining analytical formulae are established by considering the binomial expansion theorem. The obtained expressions are valid in chemical potential values between ‐∞ <μ <0 and have been designated as explicit form features, high precision, and less computing time. The calculation results are tabularly illustrated to show the consistency of the analytical relations analysis under the effect of parameters. Based on a comprehensive analysis of the results, they are potentially useful in applications to evaluate thermionic emission and astrophysics problems.