A fractal analysis of human 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.

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.

Fourier Analysis of Human Sagittal Sutures

Objective:

To evaluate the complexity of human sagittal suture patterns and to investigate whether the suture complexity correlates with age.

Design:

Geometric patterns of the sagittal sutures from 104 dry human skulls from the Terry Collection and 16 computed tomography images from the Bosma Collection, aged 2 months to 60 years, were digitized. The complexity of the patterns was presented by suture length, curved suture (or skull) length, and length ratio and the frequency and amplitude contents by the discrete Fourier transform (DFT) analysis.

Results:

The suture length along the skull showed a positive correlation with age from 2 months to 10 years, reflecting the growth of the skull. The suture length ratio, R, a measure of the complexity of the suture pattern, had a similar trend to suture length (i.e., increased with age to about 10 years and leveled off afterward, accompanied by a large scatter). The major frequency from the DFT analysis indicated an age-related development in suture complexity from infants to about 10 years and no further change for individuals older than 10 years.

Conclusions:

Quantitative analyses of human sagittal suture using length, length ratio, and DFT indicated that there is a progressive increase in the complexity of sagittal sutural waveform with age, especially in the early ages. These findings agree with the observations from animal experiments that sagittal sutural waveform is the result of intrinsic tissue response to extrinsic forces such as those generated by the temporalis.

Tissue Dynamics: Lessons Learned From Sutural Morphogenesis and Cancer Growth

Why are cranial sutures the way they are? How do cancers grow? Merging physics and mathematics with biology, we develop equations describing these complex adaptive systems, to which all biological entities belong, calling them laws of tissue dynamics:Where t is time, E is energy, M is body mass, X is the biological characteristic of interest, C is a constant, a is an exponent.(1) is based on conservation of matter: for any given tissue, materials in must equal to materials out +/- assimilated or degraded. (2) is based on energy conservation. All living systems require energy, without which life becomes impossible. Equation (2) is a power spectrum.

OBJECTIVES

This study aimed to introduce the laws of tissue dynamics and to illustrate them using observations from craniofacial and cancer growth.

METHODS

We use cranial sutures as a model system to test Equation (1), we also measure the in vitro growth rate of normal murine liver and spleen cells, comparing them to B16F10 melanoma cells. We show the increase in compound growth rate and energetic requirement of malignant versus normal cells as partial proof of Equation (2).

RESULTS

The constant width and wavy form of cranial sutures are the inevitable results of repeated iteration from coupling of growth and stress. The compound growth rate of B10F16 melanoma cells exceeds that of normal cells by 1.0 to 1.5%, whereas their glucose uptake is equal to 3.6 billion glucose molecules/cell per minute.

SUMMARY

Living things are complex adaptive systems, thus a different way of thinking and investigating, going beyond the current reductive approach, is required.

Novel methodologies and technologies to assess mid-palatal suture maturation: a systematic review

INTRODUCTION

A reliable method to assess midpalatal suture maturation to drive clinical decision-making, towards non-surgical or surgical expansion, in adolescent and young adult patients is needed. The objectives were to systematically review and evaluate what is known regarding contemporary methodologies capable of assessing midpalatal suture maturation in humans.

METHODS

A computerized database search was conducted using Medline, PubMed, Embase and Scopus to search the literature up until October 5, 2016. A supplemental hand search was completed of references from retrieved articles that met the final inclusion criteria.

RESULTS

Twenty-nine abstracts met the initial inclusion criteria. Following assessment of full articles, only five met the final inclusion criteria. The number of subjects involved and quality of studies varied, ranging from an in-vitro study using autopsy material to prospective studies with in vivo human patients. Three types of evaluations were identified: quantitative, semi-quantitative and qualitative evaluations. Four of the five studies utilized computed tomography (CT), while the remaining study utilized non-invasive ultrasonography (US). No methodology was validated against a histological-based reference standard.

CONCLUSIONS

Weak limited evidence exists to support the newest technologies and proposed methodologies to assess midpalatal suture maturation. Due to the lack of reference standard validation, it is advised that clinicians still use a multitude of diagnostic criteria to subjectively assess palatal suture maturation and drive clinical decision-making.

Stretch force guides finger-like pattern of bone formation in suture

Mechanical tension is widely applied on the suture to modulate the growth of craniofacial bones. Deeply understanding the features of bone formation in expanding sutures could help us to improve the outcomes of clinical treatment and avoid some side effects. Although there are reports that have uncovered some biological characteristics, the regular pattern of sutural bone formation in response to expansion forces is still unknown. Our study was to investigate the shape, arrangement and orientation of new bone formation in expanding sutures and explore related clinical implications. The premaxillary sutures of rat, which histologically resembles the sutures of human beings, became wider progressively under stretch force. Micro-CT detected new bones at day 3. Morphologically, these bones were forming in a finger-like pattern, projecting from the maxillae into the expanded sutures. There were about 4 finger-like bones appearing on the selected micro-CT sections at day 3 and this number increased to about 18 at day 7. The average length of these projections increased from 0.14 mm at day 3 to 0.81 mm at day 7. The volume of these bony protuberances increased to the highest level of 0.12 mm³ at day 7. HE staining demonstrated that these finger-like bones had thick bases connecting with the maxillae and thin fronts stretching into the expanded suture. Nasal sections had a higher frequency of finger-like bones occuring than the oral sections at day 3 and day 5. Masson-stained sections showed stretched fibers embedding into maxillary margins. Osteocalcin-positive osteoblasts changed their shapes from cuboidal to spindle and covered the surfaces of finger-like bones continuously. Alizarin red S and calcein deposited in the inner and outer layers of finger-like bones respectively, which showed that longer and larger bones formed on the nasal side of expanded sutures compared with the oral side. Interestingly, these finger-like bones were almost paralleling with the direction of stretch force. Inclined force led to inclined finger-like bones formation and deflection of bilateral maxillae. Additionally, heavily compressive force caused fracture of finger-like bones in the sutures. These data together proposed the special finger-like pattern of bone formation in sutures guided by stretch force, providing important implications for maxillary expansion.

Quantitative evaluation of midpalatal suture maturation via fractal analysis

Objective

The purpose of this study was to determine whether the results of fractal analysis can be used as criteria for midpalatal suture maturation evaluation.

Methods

The study included 131 subjects aged over 18 years of age (range 18.1–53.4 years) who underwent cone-beam computed tomography. Skeletonized images of the midpalatal suture were obtained via image processing software and used to calculate fractal dimensions. Correlations between maturation stage and fractal dimensions were calculated using Spearman's correlation coefficient. Optimal fractal dimension cut-off values were determined using a receiver operating characteristic curve.

Results

The distribution of maturation stages of the midpalatal suture according to the cervical vertebrae maturation index was highly variable, and there was a strong negative correlation between maturation stage and fractal dimension (−0.623, p < 0.001). Fractal dimension was a statistically significant indicator of dichotomous results with regard to maturation stage (area under curve = 0.794, p < 0.001). A test in which fractal dimension was used to predict the resulting variable that splits maturation stages into ABC and D or E yielded an optimal fractal dimension cut-off value of 1.0235.

Conclusions

There was a strong negative correlation between fractal dimension and midpalatal suture maturation. Fractal analysis is an objective quantitative method, and therefore we suggest that it may be useful for the evaluation of midpalatal suture maturation.

Cranial sutures: a multidisciplinary review

INTRODUCTION

Progress in cranial suture research is shaping our current understanding of the topic; however, emphasis has been placed on individual contributing components rather than the cranial sutural system as a whole. Improving our holistic view helps further guide clinicians who treat cranial sutural abnormalities as well as researchers who study them.

MATERIALS AND METHODS

Information from anatomy, anthropology, surgery, and computed modeling was integrated to provide a perspective to interpret suture formation and variability within the cranial functional and structural system.

RESULTS

Evidence from experimental settings, simulations, and evolution suggest a multifactorial morphogenetic process associated with functions and morphology of the sutures. Despite molecular influences, the biomechanical cranial environment has a main role in both the ontogenetic and phylogenetic suture dynamics.

CONCLUSIONS

Furthering our holistic understanding of the intricate cranial sutural system promises to expand our knowledge and enhance our ability to treat associated anomalies.

The mouse palate and its cellular responses to midpalatal suture expansion forces

OBJECTIVES

To investigate the anatomy of the mouse palate, the midpalatal suture, and the cellular characteristics in the sutures before and immediately after midpalatal suture expansion.

MATERIALS AND METHODS

Wild-type C57BL/6 male mice, aged between 6 weeks and 12 months, were chosen for all the experiments. The complete palate of the non-operated group and the midpalatal suture-expanded group at different ages was used for histological, micro-CT, immunohistochemistry, and sutural cell analyses.

RESULTS

This study documents precise morphological and histological characteristics of the mouse palatal sutures. In addition to the opening of the midpalatal suture caused by expansion, both transverse and interpalatine sutures were also seen to be affected. Cellular density was decreased in different types of sutures following the application of mechanical force.

CONCLUSIONS

The detailed morphology and histology of the mouse palate and the cellular changes that occur following midpalatal suture expansion, as described here, will be helpful as a basis for further investigations of palatal suture tissue responses to mechanical force.

Changes in biomechanical strain and morphology of rat calvarial sutures and bone after Tgf-β3 inhibition of posterior interfrontal suture fusion

Craniofacial sutures are bone growth fronts that respond and adapt to biomechanical environments. Little is known of the role sutures play in regulating the skull biomechanical environment during patency and fusion conditions, especially how delayed or premature suture fusion will impact skull biomechanics. Tgf-β3 has been shown to prevent or delay suture fusion over the short term in rat skulls, yet the long-term patency or its consequences in treated sutures is not known. It was therefore hypothesized that Tgf-β3 had a long-term impact to prevent suture fusion and thus alter the skull biomechanics. In this study, collagen gels containing 3 ng Tgf-β3 were surgically placed superficial to the posterior interfrontal suture (IFS) and deep to the periosteum in postnatal day 9 (P9) rats. At P9, P24, and P70, biting forces and strains over left parietal bone, posterior IFS, and sagittal suture were measured with masticatory muscles bilaterally stimulated, after which the rats were sacrificed and suture patency analyzed histologically. Results demonstrated that Tgf-β3 treated sutures showed less fusion over time than control groups, and strain patterns in the skulls of the Tgf-β3-treated group were different from that of the control group. Although bite force increased with age, no alterations in bite force were attributable to Tgf-β3 treatment. These findings suggest that the continued presence of patent sutures can affect strain patterns, perhaps when higher bite forces are present as in adult animals.

The role of the sutures in biomechanical dynamic simulation of a macaque cranial finite element model: implications for the evolution of craniofacial form

The global biomechanical impact of cranial sutures on the face and cranium during dynamic conditions is not well understood. It is hypothesized that sutures act as energy absorbers protecting skulls subjected to dynamic loads. This hypothesis predicts that sutures have a significant impact on global patterns of strain and cranial structural stiffness when analyzed using dynamic simulations; and that this global impact is influenced by suture material properties. In a finite element model developed from a juvenile Rhesus macaque cranium, five different sets of suture material properties for the zygomaticotemporal sutures were tested. The static and dynamic analyses produced similar results in terms of strain patterns and reaction forces, indicating that the zygomaticotemporal sutures have limited impact on global skull mechanics regardless of loading design. Contrary to the functional hypothesis tested in this study, the zygomaticotemporal sutures did not absorb significant amounts of energy during dynamic simulations regardless of loading speed. It is alternatively hypothesized that sutures are mechanically significant only insofar as they are weak points on the cranium that must be shielded from unduly high stresses so as not to disrupt vitally important growth processes. Thus, sutural and overall cranial form in some vertebrates may be optimized to minimize or otherwise modulate sutural stress and strain.

A bidirectional interface growth model for cranial interosseous suture morphogenesis

Interosseous sutures exhibit highly variable patterns of interdigitation and corrugation. Recent research has identified fundamental molecular mechanisms of suture formation, and computer models have been used to simulate suture morphogenesis. However, the role of bone strain in the development of complex sutures is largely unknown, and measuring suture morphologies beyond the evaluation of fractal dimensions remains a challenge. Here we propose a morphogenetic model of suture formation, which is based on the paradigm of Laplacian interface growth. Computer simulations of suture morphogenesis under various boundary conditions generate a wide variety of synthetic sutural forms. Their morphologies are quantified with a combination of Fourier analysis and principal components analysis, and compared with natural morphological variation in an ontogenetic sample of human interparietal suture lines. Morphometric analyses indicate that natural sutural shapes exhibit a complex distribution in morphospace. The distribution of synthetic sutures closely matches the natural distribution. In both natural and synthetic systems, sutural complexity increases during morphogenesis. Exploration of the parameter space of the simulation system indicates that variation in strain and/or morphogen sensitivity and viscosity of sutural tissue may be key factors in generating the large variability of natural suture complexity.

Mechanism of skull suture maintenance and interdigitation

Skull sutures serve as growth centers whose function involves multiple molecular pathways. During periods of brain growth the sutures remain thin and straight, later developing complex fractal interdigitations that provide interlocking strength. The nature of the relationship between the molecular interactions and suture pattern formation is not understood. Here we show that by classifying the molecules involved into two groups, stabilizing factors and substrate molecules, complex molecular networks can be modeled by a simple two-species reaction-diffusion model that recapitulates all the known behavior of suture pattern formation. This model reproduces the maintenance of thin sutural tissue at early stages, the later modification of the straight suture to form osseous interdigitations, and the formation of fractal structures. Predictions from the model are in good agreement with experimental observations, indicating that the model captures the essential nature of the interdigitation process.

Tensile strain-induced Ets-2 phosphorylation by CaMKII and the homeostasis of cranial sutures

BACKGROUND

Mechanotransduction underpins the homeostasis of musculoskeletal tissues, including cranial sutures. Intracellular calcium, [Ca 2+]ic, and protein phosphorylation are two intermediate variables in signal relay during mechanotransduction. This project establishes a chain of cause and effect, linking cellular strain to substrate phosphorylation, and identifies the agent and target sites of phosphorylation.

METHODS

Cyclic tensile force (0.5 N at 1 Hz) was applied to 1-day-old rat sagittal sutures. [Ca 2+]ic was measured by FURA-2. Ets-2 phosphorylation by CaMKII was tested using Western blot autoradiography. Peptide array was constructed to determine the precise sites of phosphorylation. The results were confirmed with mass spectroscopy and Western blots using phospho-specific antibodies.

RESULTS

[Ca 2+]ic increased rapidly in response to tensile stress. In the presence of Ca2+, CaMKII caused Ets-2 phosphorylation. Of the three possible sites for phosphorylation of Ets-2 by CaMKII, RVPS, FESF, RLSS, Serine 246, 310, and 313 were the targets. Furthermore, the contiguous sequence modified this effect. Mass spectroscopy showed 80 Da (molecular weight of phosphate group, -PO3) right shifts consistent with phosphorylation. There was cytosolic translocation of Ets-2 on tensile deformation of suture cells. CaMKII binding of Ets-2 occurred within 30 minutes after the onset of tensile strain.

CONCLUSIONS

Cranial suture cells can respond to tensile forces by increasing [Ca 2+]ic, which causes CaMKII to phosphorylate Ets-2, thus altering Ets-2 binding to its downstream promoters. Of note, Ets-2 is at the intersection of three key pathways important in craniosynostosis: fibroblast growth factor-2, transforming growth factor-beta, and mechanotransduction.

Error estimation of the fractal dimension measurements of cranial sutures

The fractal exponents used to quantify the complexity of cranial sutures were computed for 17 coronal and 17 sagittal sutures of adults from different populations, using the box-counting algorithm. This paper discusses the main sources of error for the fractal exponents, and gives the error estimates. We then compare our results with those obtained by other authors. We suggest that the usual error estimates implied by the standard deviation for the regression line are too low. We emphasize the crucial role played by the choice of regression line in the log-log plot. For the coronal and sagittal sutures we found mean fractal dimensions of 1.48 and 1.56, respectively. Our values are close to the value for Brownian random walk.