Radiomorphometric indices of the mandible as indicators of decreased bone mineral density and osteoporosis - meta-analysis and systematic review.

This review aims to evaluate the accuracy of various mandibular radiomorphometric indices in comparison with DEXA BMD measurements in the diagnosis of osteopenia and osteoporosis based on a meta-analysis of the sensitivity and specificity of the indices. PRISMA statement was followed. The materials for analysis were collected in August 2023 by searching three databases: PubMed Central, Web of Science, and Scopus. The selection of studies consisted of three selection stages, and 64 articles were finally obtained. Quality assessment was performed with the QUADAS-2 tool, and the general methodological quality of retrieved studies was low. Statistical analysis was performed based on 2 × 2 tables and estimated sensitivity and specificity were obtained using SROC curves. The most used indices were MCI, MCW and PMI. The best results in detecting reduced BMD obtained for MCW ≤ 3 mm, estimated sensitivity and specificity were 0.712 (95% CI, 0.477-0.870) and 0.804 (95% CI, 0.589-0.921), respectively. The most prone to the risk of bias is the MCI due to the examiner's subjectivism. Radiomorphometric indices of the mandible can be useful as a screening tool to identify patients with low BMD, but should not be used as a diagnostic method. Further research needs to focus on analysing the ability of the indices to detect osteoporosis and also in combination the indices with clinical parameters.

Age at peak height velocity in Polish adolescents: Effect of socioeconomic factors.

Age at peak height velocity (APHV) is an indicator of maturity timing which is applicable to both sexes, and which is influenced by environmental factors. The objective of this study was to assess variation in APHV associated with several indicators of socioeconomic status (SES) in a longitudinal sample of Polish adolescents. The sample included 739 boys born in 1983 and followed annually from 12 to 16 years, and 597 girls born in 1985 and followed annually from 9 to 13 years. The height records were fitted with the SITAR model to estimate APHV. SES was estimated using principal component analysis of indicators of familial status based on parental education, family size, living conditions and household possessions. Statistical analyses included analysis of variance (one-way for general SES and three-way for parental education and family size) and Tukey post-hoc tests for unequal samples. General SES (p <.001) and family size (p < .05) significantly influenced APHV among boys, while only maternal education (p < .05) significantly influenced APHV among girls. Among youth from families of higher SES, as defined by the respective indicators, APHV was attained significantly earlier, on average, than in peers from families of lower SES. Overall, the results showed a sex-dependent effect of SES on APHV, and highlighted the influence of favorable socioeconomic conditions for optimal growth and maturation during adolescence.

The change in handgrip strength, after physical exercise, is moderated by digit ratio (2D:4D): A study among the young adults in Poland.

BACKGROUND: The digit ratio (2D:4D), the ratio of the lengths of second (2D) to the fourth (4D) fingers, is a proxy indicator of prenatal androgen exposure. On average, males display lower 2D:4D than females. Previous studies have shown that lower 2D:4D ratios were associated with better sports and physical abilities. AIM: To assess whether a challenge condition, imposed by intense exercise, could increase handgrip strength (HGS) associated with 2D:4D. METHODS: This cross-sectional experimental study included 90 healthy young Polish adults (40 males, 50 females). They underwent intense physical exercise, before (7 days) and after which they were measured for HGS and 2D:4D. Height and weight were also measured. Analyses of Covariance were employed to delineate associations. RESULTS: 2D:4D had significant predictive effects on the differences in HGS (DHGS) measured in two occasions, without and after, physical exercises. The lower was the 2D:4D, the higher the DHGS, particularly, for the left hand. CONCLUSION: The results reconfirmed that the link between prenatal testosterone exposure (indicated by 2D:4D) and physical strength depends on the context, such as a challenged condition.

Height and integration in proximity networks among Tanzanian Hadza men.

In recent years there has been much interest in investigating the extent to which social status or prestige are related to an individual's degree of integration in social networks. It has been shown that, among hunter-gatherers, social characteristics of an individual based on social status or prestige, such foraging reputation, friendship popularity, and pro-social reputation, can influence the extent to which an individual is embedded in a social network. However, little is known regarding the extent to which height, a physical trait that in Western societies is often associated with social status, is associated in integration in social networks among small-scale hunter gatherers. Here, we investigated the relationship between height and a position an individual occupies in proximity networks among Hadza men (n = 30), hunter-gatherers living in Northern Tanzania. The results of our study show that height is not related to the position an individual maintains in proximity networks. We argue that, in a relatively egalitarian small-scale hunter-gatherer societies such as the Hadza, social interactions driving proximity networks might be influenced by social traits, such as popularity and hunting reputation, rather than physical traits, such as height.

Exploring the Beam Squint Effects on Reflectarray Performance: A Comprehensive Analysis of the Specular and Scattered Reflection of the Unit Cell.

In this article, the phenomena of beam deviation in reflectarray is discussed. The radiation pattern of the unit cell, which plays a vital role in shaping the beam of the reflectarray, is analyzed by considering undesired specular and scattered reflections. These unwanted reflections adversely affect the pattern of the single unit cell, thereby reducing the overall performance of the reflectarray. To conduct our investigations, three cases of reflectarray-i.e., (i) a center-fed with broadside beam (Case-I), (ii) a center-fed with the beam at 30° (Case-II), and (iii) off-center-fed with the beam at 30° reciprocal to feed position with reference to the broadside direction (Case-III)-are simulated. Different degrees of beam deviation are analyzed in each reflectarray by assessing the radiation pattern of a single element. The simulation results shows that maximum of 0°, 3.4°, and 0.54° beam squint across the bandwidth found in Case-I, Case-II, and Case-III, respectively; this leads to aperture efficiencies of 31.2%, 11.9%, and 31.2%, respectively. The significance of specular reflections is further confirmed by half (left half and right half) aperture analysis of Case-II. This involves simulating the half-plane aperture illuminated by horn antenna, resulting in a distinct beam angle at the same frequency. However, deviations of -4.71 to +4.1 for the left half aperture and -1.82 to +1.1 for the right half aperture are noticed. Although the analysis specifically focuses on the three cases of the reflectarray, the proposed methodology is applicable to any type of reflectarray. The study presented in this work provides an important insight into the practical aspects of reflectarray operation, particularly in terms of quantifying undesirable effects that are normally overlooked in the design of this class of arrays. To achieve a good performance, a new design of the dielectric loaded horn feed is proposed. This design approach is both simple and applicable to any reflectarray, with the added benefit of maintaining a low profile for the RA. Moreover, this work holds significant potential for remote sensing satellite systems as beam deviation can adversely impact data collection accuracy and compromise observation precision, resulting in distorted images, reduced data quality, and overall hindrance to the system's performance in capturing reliable information.

Highly Sensitive Microwave Sensors Based on Open Complementary Square Split-Ring Resonator for Sensing Liquid Materials.

This paper presents high-sensitivity sensors based on an open complementary square split-ring resonator and a modified open complementary split-ring resonator operating at 4.5 GHz and 3.4 GHz, respectively. The sensors are designed for the detection of multiple liquid materials, including distilled water, methanol, and ethanol. The liquid under test is filled in a glass container loaded using a pipette. Compared to the conventional OCSSRR, the modified OCSSRR with multiple rings exhibits a higher frequency shift of 1200 MHz, 1270 MHz, and 1520 MHz for ethanol, methanol, and distilled water, respectively. The modified sensor also demonstrates a high sensitivity of 308 MHz/RIU for ethanol concentration which is the highest among the existing microwave sensors. The sensors in this manuscript are suitable for multiple liquid-material-sensing applications.

Ages at peak height velocity in male soccer players 11-16 years: relationships with skeletal age and comparisons among longitudinal studies.

Estimated ages at take-off (TO) and at peak height velocity (PHV) based on two models and maturity status based upon age at PHV and skeletal age (SA) were compared in a longitudinal sample of male soccer players. In addition, estimated ages at PHV in 13 longitudinal samples of soccer players were compared. The longitudinal height records of 58 players of European ancestry, measured annually on four or five occasions between 11 and 16 years, were modeled with Superimposition by Translation and Rotation (SITAR) and Functional Principal Component Analysis (FPCA) to estimate ages at TO and PHV. SAs were assessed with the Fels method. Ages at PHV in 13 longitudinal samples of soccer players (Europe 7, Japan 6) were evaluated with meta-analysis. Estimated ages at TO, 11.2 ± 0.8 (SITAR) and 11.0 ± 0.8 (FCPA) years, and at PHV, 13.6 ± 0.9 (SITAR) and 13.7 ± 0.0 (FCPA) years, were similar. An earlier age at PHV was associated with advanced skeletal maturity status (rho = -0.77 at ~14 years). Ages at PHV among European players indicated a north (later) - south (earlier) gradient, and were later than ages at PHV among Japanese players. In summary, ages at TO and PHV were similar with SITAR and FPCA, and ages at PHV were most strongly correlated with SA at ~14 years. Mean ages at PHV showed a north-south gradient among European samples, and were later compared to Japanese samples.

Is 2D:4D digit ratio related to avowed religion? A study among Mongolian young males from Ulaanbaatar.

BACKGROUND: The ratio of the second- (2D) to fourth (4D) digit lengths of hand (2D:4D) is a proxy marker of the relative testosterone and estrogen concentration during a relatively narrow period of fetal development that might affect behavioral and personality characteristics. AIM: To estimate the differences in 2D:4D between different religious groups among a sample of young adult males in Mongolia. METHODS: Two hundred and sixty-five Mongolian male students with mean age of 20.5 (SD = 1.7) years from different universities in Ulan Bator, were included in the study. Information on age, religious affiliation, marital status and parental education were obtained directly from each study participant. Digit lengths were measured from scanned images by using the ImageJ software 1.53 K. One-way analysis of variance was employed to evaluate whether they were significant differences in 2D:4D ratio between groups, along with Scheffe's post hoc comparison. RESULTS: Study participants significantly differed in 2D:4D across religion. Left 2D:4D, but not the right, showed a significant difference between religions, with Muslims exhibiting the highest 2D:4D mean and the lowest DR-L . CONCLUSION: Our study suggests that the 2D:4D ratio is related to the participants' religion. However, the distinctiveness of the Muslim students from study participants belonging to other religious groups in this study could also be related to ethnic differences, given that the students were Kazakhs. This is, to our knowledge, the only study looking at the relationship between the 2D:4D ratio and religious affiliation and, thus, further research is needed to confirm its results.

Fine and gross motor skills in 7-10 years old Indian children exposed to a natural disaster during early development.

Fetal life and infancy are extremely sensitive to adverse environmental conditions. This study aimed to assess the effect of exposure to a natural disaster (cyclone Aila) in utero or during infancy on fine and gross motor functions in preadolescent Indian children. The study was conducted in West Bengal, India, and included approximately 700 children (7-10 years old) who were prenatally or postnatally exposed to cyclone Aila and a nonaffected group. Anthropometric measures included height, weight, and birthweight. Socioeconomic status was based on parental education, family size, and income. Motor functions were assessed using the short form of Bruininks-Oseretsky Test of Motor Proficiency (BOT-2). Statistical analyses included, for example, generalized linear models. There were no differences in motor functions relative to the timing of the exposure (trimester) during pregnancy. Compared to the controls, prenatal Aila exposure resulted in poorer performance in all BOT-2 subtests, except for fine motor precision, strength, and balance (the last in boys), while postnatal Aila exposure, compared to the controls, resulted in poorer performance in manual dexterity, bilateral coordination, balance (girls only), and speed and agility. Early life exposure to a natural disaster has long-term adverse effect on motor proficiency in children. By inference, the welfare of pregnant women and infants should be of particular concern for emergency and health services during an environmental cataclysm.

Rapid Design Optimization and Calibration of Microwave Sensors Based on Equivalent Complementary Resonators for High Sensitivity and Low Fabrication Tolerance.

This paper presents the design, optimization, and calibration of multivariable resonators for microwave dielectric sensors. An optimization technique for the circular complementary split ring resonator (CC-SRR) and square complementary split ring resonator (SC-SRR) is presented to achieve the required transmission response in a precise manner. The optimized resonators are manufactured using a standard photolithographic technique and measured for fabrication tolerance. The fabricated sensor is presented for the high-resolution characterization of dielectric substrates and oil samples. A three-dimensional dielectric container is attached to the sensor and acts as a pool for the sample under test (SUT). In the presented technique, the dielectric substrates and oil samples can interact directly with the electromagnetic (EM) field emitted from the resonator. For the sake of sensor calibration, a relation between the relative permittivity of the dielectric samples and the resonant frequency of the sensor is established in the form of an inverse regression model. Comparisons with state-of-the-art sensors indicate the superiority of the presented design in terms of oil characterization reliability. The significant technical contributions of this work include the employment of the rigorous optimization of geometry parameters of the sensor, leading to its superior performance, and the development and application of the inverse-model-based calibration procedure.

New Complementary Resonator for Permittivity- and Thickness-Based Dielectric Characterization.

The design of high-performance complementary meta-resonators for microwave sensors featuring high sensitivity and consistent evaluation of dielectric materials is challenging. This paper presents the design and implementation of a novel complementary resonator with high sensitivity for dielectric substrate characterization based on permittivity and thickness. A complementary crossed arrow resonator (CCAR) is proposed and integrated with a fifty-ohm microstrip transmission line. The CCAR's distinct geometry, which consists of crossed arrow-shaped components, allows for the implementation of a resonator with exceptional sensitivity to changes in permittivity and thickness of the material under test (MUT). The CCAR's geometrical parameters are optimized to resonate at 15 GHz. The CCAR sensor's working principle is explained using a lumped-element equivalent circuit. The optimized CCAR sensor is fabricated using an LPKF protolaser on a 0.762-mm thick dielectric substrate AD250C. The MUTs with dielectric permittivity ranging from 2.5 to 10.2 and thickness ranging from 0.5 mm to 1.9 mm are used to investigate the properties and calibrate the proposed CCAR sensor. A two-dimensional calibration surface is developed using an inverse regression modelling approach to ensure precise and reliable measurements. The proposed CCAR sensor is distinguished by its high sensitivity of 5.74%, low fabrication cost, and enhanced performance compared to state-of-the-art designs, making it a versatile instrument for dielectric characterization.

Bandwidth-Controllable Third-Order Band Pass Filter Using Substrate-Integrated Full- and Semi-Circular Cavities.

The article presents a novel circular substrate-integrated waveguide (SIW) bandpass filter (BPF) with controllable bandwidth. The proposed BPF was configured using two microstrip feed lines, semi-circular SIW cavities, capacitive slots, and inductive vias. The circular cavity was divided into two halves, and the two copies were cascaded. The resulting bisected and cascaded structures were then connected back-to-back. Finally, by introducing two inductive vias to the circular center cavity, a transmission zero was generated. In order to examine the design concept, a coupling matrix was generated. To demonstrate the theory, a third-order BPF was realized, fabricated, and experimentally validated. The BPF prototype features a wide passband of 8.7%, a low insertion loss of 1.1 dB, and a stopband of 1.5 f0 with a rejection level better than 20 dB, which makes it a potential candidate for microwave sensing and communication industries.

Mutual Coupling Reduction in Antenna Arrays Using Artificial Intelligence Approach and Inverse Neural Network Surrogates.

This paper presents a novel approach to reducing undesirable coupling in antenna arrays using custom-designed resonators and inverse surrogate modeling. To illustrate the concept, two standard patch antenna cells with 0.07λ edge-to-edge distance were designed and fabricated to operate at 2.45 GHz. A stepped-impedance resonator was applied between the antennas to suppress their mutual coupling. For the first time, the optimum values of the resonator geometry parameters were obtained using the proposed inverse artificial neural network (ANN) model, constructed from the sampled EM-simulation data of the system, and trained using the particle swarm optimization (PSO) algorithm. The inverse ANN surrogate directly yields the optimum resonator dimensions based on the target values of its S-parameters being the input parameters of the model. The involvement of surrogate modeling also contributes to the acceleration of the design process, as the array does not need to undergo direct EM-driven optimization. The obtained results indicate a remarkable cancellation of the surface currents between two antennas at their operating frequency, which translates into isolation as high as -46.2 dB at 2.45 GHz, corresponding to over 37 dB improvement as compared to the conventional setup.

Frequency-Selective Surface-Based MIMO Antenna Array for 5G Millimeter-Wave Applications.

In this paper, a radiating element consisting of a modified circular patch is proposed for MIMO arrays for 5G millimeter-wave applications. The radiating elements in the proposed 2 × 2 MIMO antenna array are orthogonally configured relative to each other to mitigate mutual coupling that would otherwise degrade the performance of the MIMO system. The MIMO array was fabricated on Rogers RT/Duroid high-frequency substrate with a dielectric constant of 2.2, a thickness of 0.8 mm, and a loss tangent of 0.0009. The individual antenna in the array has a measured impedance bandwidth of 1.6 GHz from 27.25 to 28.85 GHz for S11 ≤ -10 dB, and the MIMO array has a gain of 7.2 dBi at 28 GHz with inter radiator isolation greater than 26 dB. The gain of the MIMO array was increased by introducing frequency-selective surface (FSS) consisting of 7 × 7 array of unit cells comprising rectangular C-shaped resonators, with one embedded inside the other with a central crisscross slotted patch. With the FSS, the gain of the MIMO array increased to 8.6 dBi at 28 GHz. The radiation from the array is directional and perpendicular to the plain of the MIMO array. Owing to the low coupling between the radiating elements in the MIMO array, its Envelope Correlation Coefficient (ECC) is less than 0.002, and its diversity gain (DG) is better than 9.99 dB in the 5G operating band centered at 28 GHz between 26.5 GHz and 29.5 GHz.

Maternal genetic origin of the late and final Neolithic human populations from present-day Poland.

OBJECTIVE: We aim to identify maternal genetic affinities between the Middle to Final Neolithic (3850-2300 BC) populations from present-day Poland and possible genetic influences from the Pontic steppe. MATERIALS AND METHODS: We conducted ancient DNA studies from populations associated with Zlota, Globular Amphora, Funnel Beaker, and Corded Ware cultures (CWC). We sequenced genomic libraries on Illumina platform to generate 86 complete ancient mitochondrial genomes. Some of the samples were enriched for mitochondrial DNA using hybridization capture. RESULTS: The maternal genetic composition found in Zlota-associated individuals resembled that found in people associated with the Globular Amphora culture which indicates that both groups likely originated from the same maternal genetic background. Further, these two groups were closely related to the Funnel Beaker culture-associated population. None of these groups shared a close affinity to CWC-associated people. Haplogroup U4 was present only in the CWC group and absent in Zlota group, Globular Amphora, and Funnel Beaker cultures. DISCUSSION: The prevalence of mitochondrial haplogroups of Neolithic farmer origin identified in Early, Middle and Late Neolithic populations suggests a genetic continuity of these maternal lineages in the studied area. Although overlapping in time - and to some extent - in cultural expressions, none of the studied groups (Zlota, Globular Amphora, Funnel Beaker), shared a close genetic affinity to CWC-associated people, indicating a larger extent of cultural influence from the Pontic steppe than genetic exchange. The higher frequency of haplogroup U5b found in populations associated with Funnel Beaker, Globular Amphora, and Zlota cultures suggest a gradual maternal genetic influx from Mesolithic hunter-gatherers. Moreover, presence of haplogroup U4 in Corded Ware groups is most likely associated with the migrations from the Pontic steppe at the end of the Neolithic and supports the observed genetic distances.

Prediction of maturity offset and age at peak height velocity in a longitudinal series of boys and girls.

BACKGROUND: Predicted maturity offset, defined as time before peak height velocity (PHV) is increasingly used as an indicator of maturity status in studies of physical activity, fitness, and sport. OBJECTIVE: To validate maturity offset prediction equations in longitudinal samples of boys and girls. METHODS: The original and modified maturity offset prediction equations were applied to serial data for 266 boys (8-17 years) and 147 girls (8-16 years) from the Cracow Growth Study. Actual age at PHV for each youngster was estimated with the SITAR protocol. In addition to maturity offset, the difference between CA at prediction and maturity offset provided an estimate of predicted age at PHV. RESULTS: Predicted maturity offset and age at PHV increased, on average, with CA at prediction. Variation in predictions was reduced compared to that in observed ages at offset and at PHV, and was more apparent with the modified equations. Relatively few predicted ages at PHV approximated observed age at PHV in early and late maturing youth of both sexes; predictions were later than observed among the former, and earlier than observed among the latter. CONCLUSION: Predicted maturity offset and ages at PHV with the original and modified equations increase with CA at prediction, have reduced variation, and have major limitations with early and late maturing boys and girls.

Trends in growth and developmental tempo in boys aged 7 to 18 years between 1966 and 2012 in Poland.

OBJECTIVES: To assess trends in growth in different developmental periods and trends in developmental tempo in Polish boys between 1966 and 2012. METHODS: Data on 34 828 boys aged 7 to 18 years were collected during Polish Anthropological Surveys conducted in 1966, 1978, 1988, and 2012. Biological parameters, related to onset of adolescent growth spurt (OGS) and peak height velocity (PHV), were derived from a Preece-Baines 1 model (PB1). Childhood (height at 7 years of age), pre-adolescent (height at OGS) and adolescent growth (adult height minus height at OGS) were identified. RESULTS: Positive secular trend between 1966 and 2012 in adult height accounted for, on average, 1.5 cm/decade, with varying intensity between the Surveys. Decline in both age at OGS and APHV between 1966 and 2012 (1.5 and 1.4 years, respectively) indicated an acceleration in developmental tempo, on average, by 0.3 year/decade. Increases in the contribution to the trend in adult height gained during growth in particular developmental periods between 1966 and 2012 were as followed-childhood: 0.6%, pre-adolescent growth: -3.1%, adolescent growth: 3.1%. CONCLUSIONS: Secular trend in developmental tempo and growth among boys reflects changes in living conditions and socio-political aspirations in Poland during nearly 50 years. Acceleration in tempo is already visible at age at OGS, whereas the trend in adult height occurs largely during adolescence, pointing to different regulation of developmental tempo and growth in body height. This finding emphasizes the importance of extending public health intervention into children's growth up until adolescence.

Secular trend and social variation in height of Polish schoolchildren between 1966 and 2012.

AIM: To assess secular trend and changes in social inequalities of children's height across nearly 50 years, when vast socio-political changes took place in Poland. METHODS: Data on schoolchildren aged 7-18 years were collected in 1966, 1978, 1988 and 2012 in Poland. Height was standardised for age using the LMS method. Socio-economic status (SES) was based on 4 factors: urbanisation level, mother's and father's education, and family size (number of children). Statistics included 2-way ANOVA with post hoc Tukey's test and effect size calculations. RESULTS: Positive secular trend in height was observed across all years. All analysed SES factors had significant effect on height which differed depending on SES category and year of Survey. Differences in height between extreme categories of SES factors decreased gradually, starting from 1978. However, only general SES in girls and urbanisation level in both sexes became insignificant in 2012. CONCLUSION: Improvement of living conditions across nearly 50 years was reflected in the secular trend in children's height. Despite this improvement, however, the biological effects of social inequalities, visible in differences in height, to some extent, are still present in Poland.

Body structure, muscular strength and living conditions of primary school children in Warsaw.

The aim of this study was to determine whether the living conditions of school children affects their body structure and muscular strength. Data were taken from 400 girls and 341 boys aged 7-15 years attending nine primary schools in Warsaw in 1997. A questionnaire was completed, anthropological measurements made and two muscular strength tests conducted. The questionnaire asked questions on the children's level of education, their parents' professions and monthly incomes, the number of persons in the family and the number of rooms in the family's apartment/home. Body height, body weight, chest and arm circumferences, grip strength and vertical jump height were measured and used to calculate body mass index, Marty's Index and the Sargent Vertical Jump Index. Statistical tests included Student's t-test, Principal Component Analysis (PCA) and multiple regression analysis. Body height, chest circumference, Sargent Vertical Jump Index and grip strength were significantly greater in the boys than the girls. Two factors, namely 'socioeconomic status' (F1) and 'family size' (F2), describing living conditions, were isolated after PCA. Boys from bigger families (F2) were shorter, with lower weights and BMIs, smaller chest and arm circumferences and greater grip strengths than those from smaller families, whereas girls from families of lower socioeconomic status (F1) weighed less and had greater BMIs and arm circumferences than those from higher socioeconomic status families. The results suggest that boys seem to be more 'ecosensitive' than girls.

Low-Cost Unattended Design of Miniaturized 4 × 4 Butler Matrices with Nonstandard Phase Differences.

Design of Butler matrices dedicated to Internet of Things and 5th generation (5G) mobile systems-where small size and high performance are of primary concern-is a challenging task that often exceeds capabilities of conventional techniques. Lack of appropriate, unified design approaches is a serious bottleneck for the development of Butler structures for contemporary applications. In this work, a low-cost bottom-up procedure for rigorous and unattended design of miniaturized 4 × 4 Butler matrices is proposed. The presented approach exploits numerical algorithms (governed by a set of suitable objective functions) to control synthesis, implementation, optimization, and fine-tuning of the structure and its individual building blocks. The framework is demonstrated using two miniaturized matrices with nonstandard output-port phase differences. Numerical results indicate that the computational cost of the design process using the presented framework is over 80% lower compared to the conventional approach. The footprints of optimized matrices are only 696 and 767 mm2, respectively. Small size and operation frequency of around 2.6 GHz make the circuits of potential use for mobile devices dedicated to work within a sub-6 GHz 5G spectrum. Both structures have been benchmarked against the state-of-the-art designs from the literature in terms of performance and size. Measurements of the fabricated Butler matrix prototype are also provided.