Symmetry reductions and new functional separable solutions of nonlinear Klein–Gordon and telegraph type equations

The paper is concerned with different classes of nonlinear Klein–Gordon and telegraph type equations with variable coefficient

Multi-parameter reaction–diffusion systems with quadratic nonlinearity and delays: new exact solutions in elementary functions

The study considers a nonlinear multi-parameter reaction–diffusion system of two Lotka–Volterra-type equations with several delays. It treats both cases of different diffusion coefficients and identical diffusion coefficients. The study describes a few different techniques to solve the system of interest, including (i) reduction to a single second-order linear ODE without delay, (ii) reduction to a system of three second-order ODEs without delay, (iii) reduction to a system of three first-order ODEs with delay, (iv) reduction to a system of two second-order ODEs without delay and a linear Schrödinger-type PDE, and (v) reduction to a system of two first-order ODEs with delay and a linear heat-type PDE. The study presents many new exact solutions to a Lotka–Volterra-type reaction–diffusion system with several arbitrary delay times, including over 50 solutions in terms of elementary functions. All of these are generalized or incomplete separable solutions that involve several free parameters (constants of integration). A special case is studied where a solution contains infinitely many free parameters. Along with that, some new exact solutions are obtained for a simpler nonlinear reaction–diffusion system of PDEs without delays that represents a special case of the original multi-parameter delay system. Several generalizations to systems with variable coefficients, systems with more complex nonlinearities, and hyperbolic type systems with delay are discussed. The solutions obtained can be used to model delay processes in biology, ecology, biochemistry and medicine and test approximate analytical and numerical methods for reaction–diffusion and other nonlinear PDEs with delays

The heat and mass transfer modeling with time delay

Nonlinear hyperbolic reaction-diffusion equations with a delay in time are investigated. All equations considered here contain one arbitrary function. Exact solutions are also presented for more complex nonlinear equations in which delay arbitrarily depends on time. Exact solutions with a generalized separation of variables are found. For special cases, new exact solutions in the form of a traveling waves are obtained, some of which can be represented in terms of elementary functions. All of these solutions contain free (arbitrary) parameters, so that one can use them to solve modeling problems of heat and mass transfer with relaxation phenomena

Multilevel analysis of the influence of maternal smoking and alcohol consumption on the facial shape of English adolescents

This cross-sectional study aims to assess the influence of maternal smoking and alcohol consumption during pregnancy on the facial shape of non-syndromic English adolescents and demonstrate the potential benefits of using multilevel principal component analysis (mPCA). A cohort of 3755 non-syndromic 15-year-olds from the Avon Longitudinal Study of Parents and Children (ALSPAC), England, were included. Maternal smoking and alcohol consumption during the 1st and 2nd trimesters of pregnancy were determined via questionnaire at 18 weeks gestation. 21 facial landmarks, used as a proxy for the main facial features, were manually plotted onto 3D facial scans of the participants. The effect of maternal smoking and maternal alcohol consumption (average 1–2 glasses per week) was minimal, with 0.66% and 0.48% of the variation in the 21 landmarks of non-syndromic offspring explained, respectively. This study provides a further example of mPCA being used effectively as a descriptive analysis in facial shape research. This is the first example of mPCA being extended to four levels to assess the influence of environmental factors. Further work on the influence of high/low levels of smoking and alcohol and providing inferential evidence is required

What's in a smile? Initial analyses of dynamic changes in facial shape and appearance

Single-level principal component analysis (PCA) and multi-level PCA (mPCA) methods are applied here to a set of (2D frontal) facial images from a group of 80 Finnish subjects (34 male; 46 female) with two different facial expressions (smiling and neutral) per subject. Inspection of eigenvalues gives insight into the importance of different factors affecting shapes, including: biological sex, facial expression (neutral versus smiling), and all other variations. Biological sex and facial expression are shown to be reflected in those components at appropriate levels of the mPCA model. Dynamic 3D shape data for all phases of a smile made up a second dataset sampled from 60 adult British subjects (31 male; 29 female). Modes of variation reflected the act of smiling at the correct level of the mPCA model. Seven phases of the dynamic smiles are identified: rest pre-smile, onset 1 (acceleration), onset 2 (deceleration), apex, offset 1 (acceleration), offset 2 (deceleration), and rest post-smile. A clear cycle is observed in standardized scores at an appropriate level for mPCA and in single-level PCA. mPCA can be used to study static shapes and images, as well as dynamic changes in shape. It gave us much insight into the question “what’s in a smile?

The influence of snoring, mouth breathing and apnoea on facial morphology in late childhood: a three-dimensional study

Objective: To explore the relationship between the prevalence of sleep disordered breathing (SDB) and face shape morphology in a large cohort of 15-year-old children. Design: Observational longitudinal cohort study Setting: Avon Longitudinal Study of Parents and Children (ALSPAC), South West of England. Participants: Three-dimensional surface laser scans were taken for 4784 white British children from the ALSPAC during a follow-up clinic. A total of 1724 children with sleep disordered breathing (SDB) and 1862 healthy children were identified via parents’ report of sleep disordered symptoms for their children. We excluded from the original cohort all children identified as having congenital abnormalities, diagnoses associated with poor growth and children with adenoidectomy and/or tonsillectomy. Main outcome measures: Parents in the ALSPAC reported sleep disordered symptoms (snoring, mouth breathing and apnoea) for their children at 6, 18, 30, 42, 57, 69 and 81 months. Average facial shells were created for children with and without SDB in order to explore surface differences. Results: Differences in facial measurements were found between the children with and without SDB throughout early childhood. The mean differences included an increase in face height in SDB children of 0.3 mm (95% CI −0.52 to −0.05); a decrease in mandibular prominence of 0.9° (95% CI −1.30 to −0.42) in SDB children; and a decrease in nose prominence and width of 0.12 mm (95% CI 0.00 to 0.24) and 0.72 mm (95% CI −0.10 to −0.25), respectively, in SDB children. The odds of children exhibiting symptoms of SDB increased significantly with respect to increased face height and mandible angle, but reduced with increased nose width and prominence. Conclusions: The combination of a long face, reduced nose prominence and width, and a retrognathic mandible may be diagnostic facial features of SBD that may warrant a referral to specialists for the evaluation of other clinical symptoms of SDB

A three-dimensional analysis of the effect of atopy on face shape

Three-dimensional (3D) imaging technology has been widely used to analyse facial morphology and has revealed an influence of some medical conditions on craniofacial growth and morphology. The aim of the study is to investigate whether craniofacial morphology is different in atopic Caucasian children compared with controls. Study design included observational longitudinal cohort study. Atopy was diagnosed via skin-prick tests performed at 7.5 years of age. The cohort was followed to 15 years of age as part of the Avon Longitudinal Study of Parents and Children (ALSPAC). A total of 734 atopic and 2829 controls were identified. 3D laser surface facial scans were obtained at 15 years of age. Twenty-one reproducible facial landmarks (x, y, z co-ordinates) were identified on each facial scan. Inter-landmark distances and average facial shells for atopic and non-atopic children were compared with explore differences in face shape between the groups. Both total anterior face height (pg–g, pg–men) and mid-face height (Is–men, sn–men, n–sn) were longer (0.6 and 0.4mm respectively) in atopic children when compared with non-atopic children. No facial differences were detected in the transverse and antero-posterior relationships. Small but statistically significant differences were detected in the total and mid-face height between atopic and non-atopic children. No differences were detected in the transverse and antero-posterior relationships

Spatially Dense 3D Facial Heritability and Modules of Co-heritability in a Father-Offspring Design

Introduction: The human face is a complex trait displaying a strong genetic component as illustrated by various studies on facial heritability. Most of these start from sparse descriptions of facial shape using a limited set of landmarks. Subsequently, facial features are preselected as univariate measurements or principal components and the heritability is estimated for each of these features separately. However, none of these studies investigated multivariate facial features, nor the co-heritability between different facial features. Here we report a spatially dense multivariate analysis of facial heritability and co-heritability starting from data from fathers and their children available within ALSPAC. Additionally, we provide an elaborate overview of related craniofacial heritability studies.Methods: In total, 3D facial images of 762 father-offspring pairs were retained after quality control. An anthropometric mask was applied to these images to establish spatially dense quasi-landmark configurations. Partial least squares regression was performed and the (co-)heritability for all quasi-landmarks (∼7160) was computed as twice the regression coefficient. Subsequently, these were used as input to a hierarchical facial segmentation, resulting in the definition of facial modules that are internally integrated through the biological mechanisms of inheritance. Finally, multivariate heritability estimates were obtained for each of the resulting modules.Results: Nearly all modular estimates reached statistical significance under 1,000,000 permutations and after multiple testing correction (p ≤ 1.3889 × 10-3), displaying low to high heritability scores. Particular facial areas showing the greatest heritability were similar for both sons and daughters. However, higher estimates were obtained in the former. These areas included the global face, upper facial part (encompassing the nasion, zygomas and forehead) and nose, with values reaching 82% in boys and 72% in girls. The lower parts of the face only showed low to moderate levels of heritability.Conclusion: In this work, we refrain from reducing facial variation to a series of individual measurements and analyze the heritability and co-heritability from spatially dense landmark configurations at multiple levels of organization. Finally, a multivariate estimation of heritability for global-to-local facial segments is reported. Knowledge of the genetic determination of facial shape is useful in the identification of genetic variants that underlie normal-range facial variation

Genome-wide association study of primary tooth eruption identifies pleiotropic loci associated with height and craniofacial distances

Twin and family studies indicate that the timing of primary tooth eruption is highly heritable, with estimates typically exceeding 80%. To identify variants involved in primary tooth eruption we performed a population based genome-wide association study of ‘age at first tooth’ and ‘number of teeth’ using 5998 and 6609 individuals respectively from the Avon Longitudinal Study of Parents and Children (ALSPAC) and 5403 individuals from the 1966 Northern Finland Birth Cohort (NFBC1966). We tested 2,446,724 SNPs imputed in both studies. Analyses were controlled for the effect of gestational age, sex and age of measurement. Results from the two studies were combined using fixed effects inverse variance meta-analysis. We identified a total of fifteen independent loci, with ten loci reaching genome-wide significance (p<5x10−8) for ‘age at first tooth’ and eleven loci for ‘number of teeth’. Together these associations explain 6.06% of the variation in ‘age of first tooth’ and 4.76% of the variation in ‘number of teeth’. The identified loci included eight previously unidentified loci, some containing genes known to play a role in tooth and other developmental pathways, including a SNP in the protein-coding region of BMP4 (rs17563, P= 9.080x10−17). Three of these loci, containing the genes HMGA2, AJUBA and ADK, also showed evidence of association with craniofacial distances, particularly those indexing facial width. Our results suggest that the genome-wide association approach is a powerful strategy for detecting variants involved in tooth eruption, and potentially craniofacial growth and more generally organ development