A new mathematical model for pattern formation by cranial sutures

Fractality of Cranial Sutures

It has long been known that skull suture has a typical fractal structure. Although the fractal dimension has been utilized to assess morphology, the mechanism of the fractal structure formation remains to be elucidated. Recent advances in the mathematical modeling of biological pattern formation provided useful frameworks for understanding this mechanism. This chapter describes how various proposed mechanisms tried to explain the formation of fractal structures in cranial sutures.

Finite element analysis of the influence of interdigitation pattern and collagen fibers on the mechanical behavior of the midpalatal suture

The midpalatal suture (MPS) corresponds to the tissue that joins the two maxillary bones. Understanding the mechanical behavior of this tissue is of particular interest to those patients who require orthodontic treatments such as Rapid Maxillary Expansion (RME). The objective of this research was to observe the influence of interdigitation and collagen fibers on the mechanical response of MPS. To this end, a finite element analysis in two-dimensional models of the bone-suture-bone interface was performed considering the characteristics of the MPS. The geometry of the suture was modeled with 4 different levels of interdigitation: null, moderate, scalloped and fractal. The influence of collagen fibers, aligned transversely along the suture, was considered by incorporating linked structures of the bone fronts. According to the results, the factor that has the greatest impact on the magnitude and distribution of stresses is the interdigitation degree. A higher level of interdigitation produces an increase in tissue stiffness and a lower influence of collagen fibers on the mechanical response of the tissue. Therefore, this research contributes to the understanding of the MPS biomechanics by providing information that may be useful to health staff when evaluating the feasibility of procedures such as RME.

Mechanism of interdigitation formation at apical boundary of MDCK cell

It has been reported that the MDCK cell tight junction shows stochastic fluctuation and forms the interdigitation structure, but the mechanism of the pattern formation remains to be elucidated. In the present study, we first quantified the shape of the cell-cell boundary at the initial phase of pattern formation. We found that the Fourier transform of the boundary shape shows linearity in the log-log plot, indicating the existence of scaling. Next, we tested several working hypotheses and found that the Edwards-Wilkinson equation, which consists of stochastic movement and boundary shortening, can reproduce the scaling property. Next, we examined the molecular nature of stochastic movement and found that myosin light chain puncta may be responsible. Quantification of boundary shortening indicates that mechanical property change may also play some role. Physiological meaning and scaling properties of the cell-cell boundary are discussed.

Cranial suture morphometry and mechanical response to loading: 2D vs. 3D assumptions and characterization

Cranial sutures are complex soft tissue structures whose mechanics are often studied due to their link with bone growth in the skull. Researchers will often use a cross-sectional two-dimensional slice to define suture geometry when studying morphometry and/or mechanical response to loading. However, using a single cross section neglects the full suture complexity and may introduce significant errors when defining their form. This study aims to determine trends in suture path variability through skull thickness in a swine model and the implications of using a 'representative' cross section on mechanical modeling. To explore these questions, a mixture of quantitative analysis of computed tomography images and finite element models was used. The linear interdigitation and width of coronal and sagittal sutures were analyzed on offset transverse planes through the skull thickness. It was found that sagittal suture width and interdigitation were largely consistent through the skull thickness, whereas the coronal suture showed significant variation in both. The finite element study found that average values of displacement and strain were similar between the two-dimensionally variable and three-dimensionally variable models. Larger ranges and more complex distributions of strain were found in the three-dimensionally variable model. Outcomes of this study indicate that the appropriateness of using a representative cross section to describe suture morphometry and predict mechanical response should depend on specific research questions and goals. Two-dimensional approximations can be sufficient for less-interdigitated sutures and when bulk site mechanics are of interest, while taking the true three-dimensional geometry into account is necessary when considering spatial variability and local mechanical response.

Mathematical modeling of palatal suture pattern formation: morphological differences between sagittal and palatal sutures

The median palatal suture serves as a growth center for the maxilla; inadequate growth at this site causes malocclusion and dental crowding. However, the pattern formation mechanism of palatal sutures is poorly understood compared with that of calvarial sutures such as the sagittal suture. In the present study, therefore, we compared the morphological characteristics of sagittal and palatal sutures in human bone specimens. We found that palatal suture width was narrower than sagittal suture width, and the interdigitation amplitude of the palatal suture was lower than that of the sagittal suture. These tendencies were also observed in the neonatal stage. However, such differences were not observed in other animals such as chimpanzees and mice. We also used a mathematical model to reproduce the differences between palatal and sagittal sutures. After an extensive parameter search, we found two conditions that could generate the difference in interdigitation amplitude and suture width: bone differentiation threshold [Formula: see text] and growth speed c. We discuss possible biological interpretations of the observed pattern difference and its cause.

Noise-induced scaling in skull suture interdigitation

Sutures, the thin, soft tissue between skull bones, serve as the major craniofacial growth centers during postnatal development. In a newborn skull, the sutures are straight; however, as the skull develops, the sutures wind dynamically to form an interdigitation pattern. Moreover, the final winding pattern had been shown to have fractal characteristics. Although various molecules involved in suture development have been identified, the mechanism underlying the pattern formation remains unknown. In a previous study, we reproduced the formation of the interdigitation pattern in a mathematical model combining an interface equation and a convolution kernel. However, the generated pattern had a specific characteristic length, and the model was unable to produce a fractal structure with the model. In the present study, we focused on the anterior part of the sagittal suture and formulated a new mathematical model with time–space-dependent noise that was able to generate the fractal structure. We reduced our previous model to represent the linear dynamics of the centerline of the suture tissue and included a time–space-dependent noise term. We showed theoretically that the final pattern from the model follows a scaling law due to the scaling of the dispersion relation in the full model, which we confirmed numerically. Furthermore, we observed experimentally that stochastic fluctuation of the osteogenic signal exists in the developing skull, and found that actual suture patterns followed a scaling law similar to that of the theoretical prediction.

Sclerostin Antibody-Induced Changes in Bone Mass Are Site Specific in Developing Crania

Sclerostin antibody (Scl-Ab) is an anabolic bone agent that has been shown to increase bone mass in clinical trials of adult diseases of low bone mass, such as osteoporosis and osteogenesis imperfecta (OI). Its use to decrease bone fragility in pediatric OI has shown efficacy in several growing mouse models, suggesting translational potential to pediatric disorders of low bone mass. However, the effects of pharmacologic inhibition of sclerostin during periods of rapid growth and development have not yet been described with respect to the cranium, where lifelong deficiency of functioning sclerostin leads to patterns of excessive bone growth, cranial compression, and facial palsy. In the present study, we undertook dimensional and volumetric measurements in the skulls of growing Brtl/+ OI mice treated with Scl-Ab to examine whether therapy-induced phenotypic changes were similar to those observed clinically in patients with sclerosteosis or Van Buchem disorder. Mice treated between 3 and 14 weeks of age with high doses of Scl-Ab show significant calvarial thickening capable of rescuing OI-induced deficiencies in skull thickness. Other changes in cranial morphology, such as lengths and distances between anatomic landmarks, intracranial volume, and suture interdigitation, showed minimal effects of Scl-Ab when compared with growth-induced differences over the treatment duration. Treatment-induced narrowing of foramina was limited to sites of vascular but not neural passage, suggesting patterns of local regulation. Together, these findings reveal a site specificity of Scl-Ab action in the calvaria with no measurable cranial nerve impingement or brainstem compression. This differentiation from the observed outcomes of lifelong sclerostin deficiency complements reports of Scl-Ab treatment efficacy at other skeletal sites with the prospect of minimal cranial secondary complications. © 2019 American Society for Bone and Mineral Research. © 2019 American Society for Bone and Mineral Research.

Soluble VEGFR1 signaling guides vascular patterns into dense branching morphologies

Vascular patterning is a key process during development and disease. The diffusive decoy receptor sVEGFR1 (sFlt1) is a known regulator of endothelial cell behavior, yet the mechanism by which it controls vascular structure is little understood. We propose computational models to shed light on how vascular patterning is guided by self-organized gradients of the VEGF/sVEGFR1 factors. We demonstrate that a diffusive inhibitor can generate structures with a dense branching morphology in models where the activator elicits directed growth. Inadequate presence of the inhibitor leads to compact growth, while excessive production of the inhibitor blocks expansion and stabilizes existing structures. Model predictions were compared with time-resolved experimental data obtained from endothelial sprout kinetics in fibrin gels. In the presence of inhibitory antibodies against VEGFR1 vascular sprout density increases while the speed of sprout expansion remains unchanged. Thus, the rate of secretion and stability of extracellular sVEGFR1 can modulate vascular sprout density.

Cell Mechanics of Craniosynostosis

Craniosynostosis is the premature fusion of the calvarial sutures that is associated with a number of physical and intellectual disabilities spanning from pediatric to adult years. Over the past two decades, techniques in molecular genetics and more recently, advances in high-throughput DNA sequencing have been used to examine the underlying pathogenesis of this disease. To date, mutations in 57 genes have been identified as causing craniosynostosis and the number of newly discovered genes is growing rapidly as a result of the advances in genomic technologies. While contributions from both genetic and environmental factors in this disease are increasingly apparent, there remains a gap in knowledge that bridges the clinical characteristics and genetic markers of craniosynostosis with their signaling pathways and mechanotransduction processes. By linking genotype to phenotype, outlining the role of cell mechanics may further uncover the specific mechanotransduction pathways underlying craniosynostosis. Here, we present a brief overview of the recent findings in craniofacial genetics and cell mechanics, discussing how this information together with animal models is advancing our understanding of craniofacial development.