Geodesic Convexity of the Symmetric Eigenvalue Problem and Convergence of Steepest Descent.

We study the convergence of the Riemannian steepest descent algorithm on the Grassmann manifold for minimizing the block version of the Rayleigh quotient of a symmetric matrix. Even though this problem is non-convex in the Euclidean sense and only very locally convex in the Riemannian sense, we discover a structure for this problem that is similar to geodesic strong convexity, namely, weak-strong convexity. This allows us to apply similar arguments from convex optimization when studying the convergence of the steepest descent algorithm but with initialization conditions that do not depend on the eigengap δ . When δ > 0 , we prove exponential convergence rates, while otherwise the convergence is algebraic. Additionally, we prove that this problem is geodesically convex in a neighbourhood of the global minimizer of radius O ( δ ) .

Endoscopic Cystoventricular Stenting in Awake Patients with ESM for Convexity Cysts: Three Case Reports and a Systematic Review.

Surgical interventions for arachnoid cysts and glioependymal cysts primarily focus on cyst decompression and establishing communication with cerebrospinal fluid (CSF) spaces. However, a standardized surgical strategy for symptomatic convexity cysts lacking surrounding CSF space is still lacking due to the limited number of cases. This systematic review aims to evaluate surgical interventions for symptomatic convexity cysts and proposes a safe and effective treatment approach we have developed. A systematic review of the literature was conducted following PRISMA guidelines. Surgical techniques were categorized into cystoperitoneal (CP) shunt, cyst excision, cyst fenestration, cyst ventriculostomy, and several uniquely modified methods. Using a method different from these previous reports, we performed endoscopic cystoventricular stenting via burr hole in three cases of convexity cysts. While traditional methods have proven effective, CP shunt may lead to malfunction, infection, and subdural hematoma; wall excision is invasive and increases morbidity risks. Fenestration and cystoventriculostomy have also reported cases of recurrence. Other methods also pose issues, including off-label use of instruments and inadequate consideration for removal. The method we performed for placing the stent catheter with multiple side holes demonstrated long-term effectiveness. Furthermore, the surgery was conducted with the patient fully awake, and the use of electrical stimulation mapping enabled preservation of higher neurocognitive functions, including language, working memory, and spatial cognition, at the puncture site. This novel approach offers advantages such as minimally invasive procedure, preservation of safe brain function, and long-term patency. Further studies are warranted to validate these findings and improve surgical strategies for convexity cysts.

Treatment-Related Reversible Cerebral Vasoconstriction Syndrome.

Reversible cerebral vasoconstriction syndrome (RCVS) is a rare but significant cause of intracranial arteriopathy and stroke in young adults. The syndrome encompasses a spectrum of disorders radiologically characterized by reversible narrowing and dilation of intracranial arteries, often triggered by vasoactive drugs or the postpartum period. The hallmark clinical feature of RCVS is thunderclap headache with or without other neurological signs. Though endothelial dysfunction and sympathetic hyperactivation are hypothesized to be key mechanisms, the exact pathogenesis of RCVS is still unclear. RCVS's diagnosis could be challenging, since vasospasm proceeds centripetally, initially involving distal small pial and cortical arteries, and angiographic studies, especially brain magnetic resonance angiography (MRA) and computed tomography angiography (CTA), may miss it in the early phase of the disease, while early signs such as vascular hyperintensities may be visible on T2/FLAIR sequences before vasospasm onset. Catheter angiography is the gold standard and it could be used to assess vasospasm reversibility post-intra-arterial vasodilator administration. Treatment is mainly symptomatic, and nimodipine is the most commonly administered therapy, given orally or intra-arterially in severe cases. Since many aspects of RCVS remain partially known, further research is needed to better understand the complex pathophysiology of this unique clinical condition and to optimize specific management strategies.

A case of convexity non-aneurysmal subarachnoid hemorrhage caused by cerebral sinus thrombosis.

BACKGROUND: Convexity subarachnoid hemorrhage (cSAH) is an uncommon presentation of subarachnoid bleeding, referring to bleeding more localized to the convexities of the brain. The diagnosis of cerebral venous sinus thrombosis (CVST) can be difficult especially when patients initially present with cSAH. The authors present a case and then discuss the pathophysiology and management. CASE PRESENTATION: A 56-year-old woman with a previous history of hypertension and ischemic heart disease presented to the emergency department after experiencing it. Two seizures following a severe headache. The patient's history was negative for recent illnesses, head trauma, history of migraines, smoking, alcohol consumption, or intravenous drug use. The patient was diagnosed with CVST based on magnetic resonance venography (MRV). Genetic studies further identified homozygous mutations in the Prothrombin and MTHFR genes. Anticoagulant therapy was initiated with 60 mg of Enoxaparin twice daily and subsequently transitioned to Warfarin after 48 h continued for 3 months, and then replaced by rivaroxaban. CONCLUSIONS: This study highlights the importance of considering CVST as a cause of SAH, emphasizes the role of advanced imaging in diagnosis, and demonstrates a successful treatment approach using both traditional and direct oral anticoagulants. The insights provided in this article can contribute to improving the management of patients with CVST-related SAH.

Convexity subarachnoid hemorrhage revealed contralateral internal carotid artery dissection due to Eagle syndrome: a case report.

BACKGROUND: Atraumatic localized convexity subarachnoid hemorrhage (cSAH) is an uncommon form of nonaneurysmal subarachnoid hemorrhage characterized by bleeding limited to the cerebral convexities. Ipsilateral cSAH can result from a variety of causes, such as internal carotid artery stenosis, obstruction, and dissection, although concomitant contralateral cSAH is exceptionally rare. In this case, the initial findings of cSAH led us to discovering contralateral internal carotid artery dissection (ICAD) and an elongated styloid process (ESP). ESP is recognized as a risk factor for ICAD, which is a hallmark of Eagle syndrome. This sequence of findings led to the diagnosis of Eagle syndrome, illustrating a complex and intriguing interplay between cerebrovascular conditions and anatomical variations. CASE PRESENTATION: A 47-year-old Japanese woman experienced acute onset of headache radiating to her neck, reaching its zenith approximately two hours after onset. Given the intractable nature of the headache and its persistence for three days, she presented to the emergency department. Neurological examination revealed no abnormalities, and the coagulation screening parameters were within normal ranges. Brain computed tomography (CT) revealed right parietal cSAH, while CT angiography (CTA) revealed ICAD and an ESP measuring 30.1 mm on the left side, positioned only 1.4 mm from the dissected artery. The unusual occurrence of contralateral cSAH prompted extensive and repeated imaging reviews that excluded reversible cerebral vasoconstriction syndrome (RCVS), leading to a diagnosis of left ICAD secondary to Eagle syndrome. The patient underwent conservative management, and the dissected ICA spontaneously resolved. The patient has remained recurrence-free for two and a half years. CONCLUSIONS: Managing cSAH requires diligent investigation for ICAD, extending beyond its identification to explore underlying causes. Recognizing Eagle syndrome, though rare, as a potential etiology of ICAD necessitates the importance of evaluating ESPs. The method for preventing recurrent cervical artery dissection due to Eagle syndrome is controversial; however, conservative management is a viable option.

Spontaneous Non-Aneurysmal Convexity Subarachnoid Hemorrhage: A Scoping Review of Different Etiologies beyond Cerebral Amyloid Angiopathy.

Spontaneous convexity subarachnoid hemorrhage (cSAH) is a vascular disease different from aneurysmal SAH in neuroimaging pattern, causes, and prognosis. Several causes might be considered in individual patients, with a limited value of the patient's age for discriminating among these causes. Cerebral amyloid angiopathy (CAA) is the most prevalent cause in people > 60 years, but reversible cerebral vasoconstriction syndrome (RCVS) has to be considered in young people. CAA gained attention in the last years, but the most known manifestation of cSAH in this context is constituted by transient focal neurological episodes (TFNEs). CAA might have an inflammatory side (CAA-related inflammation), whose diagnosis is relevant due to the efficacy of immunosuppression in resolving essudation. Other causes are hemodynamic stenosis or occlusion in extracranial and intracranial arteries, infective endocarditis (with or without intracranial infectious aneurysms), primary central nervous system angiitis, cerebral venous thrombosis, and rarer diseases. The diagnostic work-up is fundamental for an etiological diagnosis and includes neuroimaging techniques, nuclear medicine techniques, and lumbar puncture. The correct diagnosis is the first step for choosing the most effective and appropriate treatment.

Secretory Meningioma of the Right Frontal Lobe: Clinical Presentation and Pathological Insights.

This article presents a case study of a rare convexity meningioma located in the frontal lobe of the right cerebellar hemisphere. Meningiomas comprise a substantial part of central nervous system neoplasms and are classified into benign, atypical, or anaplastic categories, each encompassing a variety of histological subtypes, among which the secretory meningioma is notably rare. A 77-year-old male presented with a clinical history of headache, impaired memory functions, an initial form of apathetic-abulic syndrome, and a single seizure, which were considered to be indicative of epileptic symptoms that had been present for several weeks. The imaging studies conducted showed a convexity tumor characterized by a rounded morphology and homogeneous contrast enhancement, positioned adjacent to the frontal lobe's cortical surface. This clinical report details the pathology of a secretory type of meningioma, which is distinguished by the atypical epithelial differentiation of meningothelial cells, resulting in hyaline fiber production. The neoplasm's anatomical accessibility permitted successful surgical resection. The tumor's position was appropriate for surgical removal, and the histological variant, along with the patient's favorable clinical course, is of particular scientific interest.

Supraorbital Bone Shaving (S.O.S Procedure) in Restoration of Upper Lateral Convexity in Sunken Eyes.

BACKGROUND: Sunken eyes have become a most important target of periorbital area aesthetics. Throughout history, the aesthetics of the periorbital region have been emphasized, and various surgical techniques related to this region have been described. Most of these techniques provide only soft tissue solutions; therefore, additional surgical interventions may be required. The aim of our study was to introduce an endoscopic supraorbital shaving (SOS) technique for the treatment of individuals with sunken eyes. METHODS: Between 2020 and 2021, 34 patients (30 females, 4 males; mean age 36.2 years) with sunken eyes were treated with our described technique. All patients underwent an endoscopic SOS procedure under general anesthesia. RESULTS: A total of 34 patients (30 women and four men), aged 23-59 years old (mean = 36.2 years), underwent the endoscopic SOS procedure. The mean follow-up period was 13 months (range: 12-16 months). Postoperatively, significant improvement in lateral convexity was achieved in all patients. Physical examinations performed at the control visits revealed no functional problems in any patients and no visible or palpable irregularities or contour deformities. No complications were encountered regarding the SOS procedure. CONCLUSIONS: The technique described here provides significant improvement in lateral convexity compared to other techniques used in patients with sunken eyes. No additional eyelid intervention is needed. Unlike the techniques previously described in the literature, intervention is made in the bone structure, thereby providing more accurate results. LEVEL OF EVIDENCE V: This journal requires that authors assign a level of evidence to each article. For a full description of these Evidence-Based Medicine ratings, please refer to the Table of Contents or the online Instructions to Authors www.springer.com/00266 .

A Convexity-Preserving Level-Set Method for the Segmentation of Tumor Organoids.

Tumor organoid cultures play a crucial role in clinical practice, particularly in guiding medication by accurately determining the morphology and size of the organoids. However, segmenting individual tumor organoids is challenging due to their inhomogeneous internal intensity and overlapping structures. This paper proposes a convexity-preserving level-set segmentation 4 model based on the characteristics of tumor organoid images to segment individual tumor organoids precisely. Considering the predominant spherical shape exhibited by organoid growth, we propose a level-set model that includes a data-driven term, a curvature term, and a regularization term. The data-driven term pulls the contour to the vicinity of the boundary; the curvature term ensures the maintenance of convexity in the targeted segmentation, and the regularization term controls the smoothness and propagation of the contour. The proposed model aids in overcoming interference from factors such as overlap and noise, enabling the evolving curve to converge to the actual boundary of the target accurately. Furthermore, we propose a selectable and targeted initialization method that guarantees precise segmentation of specific regions of interest. Experiments on 51 pancreatic ductal adenocarcinoma organoid images show that our model achieved excellent segmentation results. The average Dice value and computation time are 98.81±0.48% and 20.67 s. Compared with the C-V and CPLSE models, it is more accurate and takes less time.

In Vitro Study on the Influence of the Buccal Surface Convexity of the Tooth upon Enamel Loss after Bracket Removal.

Polishing after the removal of brackets is the final step in orthodontic treatment. It is simple to perform, though some studies have reported that polishing causes damage to the enamel surface. An in vitro study was made of the influence of the buccal surface convexity of the tooth upon possible enamel loss when the remaining resin and adhesive are removed after bracket decementing using two different polishing modes: a tungsten carbide bur at low and high speeds. The convexity of the buccal surface was quantified in 30 incisors and 30 premolars. A stereoscopic microscope was used to obtain photographs of the profile of the crown, and Image J software was used to calculate convexity by dividing the length of a line from the cementoenamel junction to the incisal margin by another line from the mentioned junction to the maximum convexity of the buccal surface. Brackets were cemented on all the teeth and were decemented 24 h later. In both groups, the residual composite was removed with a tungsten carbide bur at a low speed in one-half of the teeth and at a high speed in the other half. The buccal surface of each tooth was then photographed again, and the convexity was calculated and compared against the baseline value. The difference between the two values were taken to represent the enamel loss. The convexity of the premolars was significantly greater than that of the incisors, but this did not result in greater enamel loss when the same polishing mode was used. However, the tungsten carbide bur at a high speed proved more aggressive, causing significantly greater enamel loss than when used at a low speed.

Convexity Dural Arteriovenous Fistula without Cortical Venous Reflux Presenting with Pure Acute Subdural Hematoma.

Hemorrhagic changes in a dural arteriovenous fistula are typically associated with cortical venous reflux and occur as intracerebral or subarachnoid hemorrhages. A convexity dural arteriovenous fistula (DAVF) usually flows directly into the cortical veins and exhibits cortical venous reflux. Herein, we report a rare case of a convexity DAVF without cortical venous reflux presenting with a pure acute subdural hematoma. A 19-year-old man complaining of headache without any history of head injury was diagnosed with a left acute subdural hematoma on magnetic resonance imaging (MRI) and referred to our hospital. The patient was conscious and exhibited no neurological signs. The MRI did not reveal any possible abnormalities leading to hemorrhage. Cerebral angiography revealed a dural arteriovenous fistula in the left parietal cranium with a feeder from the middle meningeal artery and a drainer into the main transverse sinus via a diploic vein. Part of the shunt blood flowed into the superior sagittal sinus from the meningeal vein; however, there was no reflux into the cortical vein or stasis of the cerebral vein, suggesting venous hypertension. A convexity DAVF was diagnosed as the source of bleeding, and transarterial embolization was performed. The patient recovered without any neurological deficits. In the absence of trauma, an acute subdural hematoma requires an appropriate evaluation of the vascular lesions and a treatment plan.

Divergences Induced by the Cumulant and Partition Functions of Exponential Families and Their Deformations Induced by Comparative Convexity.

Exponential families are statistical models which are the workhorses in statistics, information theory, and machine learning, among others. An exponential family can either be normalized subtractively by its cumulant or free energy function, or equivalently normalized divisively by its partition function. Both the cumulant and partition functions are strictly convex and smooth functions inducing corresponding pairs of Bregman and Jensen divergences. It is well known that skewed Bhattacharyya distances between the probability densities of an exponential family amount to skewed Jensen divergences induced by the cumulant function between their corresponding natural parameters, and that in limit cases the sided Kullback-Leibler divergences amount to reverse-sided Bregman divergences. In this work, we first show that the α-divergences between non-normalized densities of an exponential family amount to scaled α-skewed Jensen divergences induced by the partition function. We then show how comparative convexity with respect to a pair of quasi-arithmetical means allows both convex functions and their arguments to be deformed, thereby defining dually flat spaces with corresponding divergences when ordinary convexity is preserved.

Formation Pattern Analysis of Spheroids Formed by a Droplet-Based Microfluidic System for Predicting the Aggressiveness of Tumor Cells.

Examining tumor heterogeneity is essential for selecting an appropriate anticancer treatment for an individual. This study aimed to distinguish low- and high-aggressive tumor cells by analyzing the formation patterns of spheroids. The droplet-based microfluidic system was employed for the formation of each spheroid from four different subtypes of breast tumor cells. Additionally, heterotypic spheroids with T lymphocytes and cancer-associated fibroblasts (CAFs) were produced, and distinctions between low- and high-aggressive tumor cells were explored through the analysis of formation patterns using circularity, convexity, and cell distributions. In both homotypic spheroids and heterotypic spheroids with T lymphocytes, spheroids formed from low-aggressive tumor cells exhibited high circularity and convexity. On the other hand, spheroids formed from high-aggressive tumor cells had relatively low circularity and convexity. In the case of heterotypic spheroids with CAFs, circularity and convexity did not exhibit clear differences between low- and high-aggressive tumor cells, but distinct variations were observed in cell distributions. CAFs and low-aggressive tumor cells were evenly distributed, whereas the CAFs were predominantly located in the inner layer, and high-aggressive tumor cells were primarily located in the outer layer. This finding can offer valuable insights into predicting the aggressiveness of unknown tumor cells.

Second-Order Effects of Chemotherapy Pharmacodynamics and Pharmacokinetics on Tumor Regression and Cachexia.

Drug dose response curves are ubiquitous in cancer biology, but these curves are often used to measure differential response in first-order effects: the effectiveness of increasing the cumulative dose delivered. In contrast, second-order effects (the variance of drug dose) are often ignored. Knowledge of second-order effects may improve the design of chemotherapy scheduling protocols, leading to improvements in tumor response without changing the total dose delivered. By considering treatment schedules with identical cumulative dose delivered, we characterize differential treatment outcomes resulting from high variance schedules (e.g. high dose, low dose) and low variance schedules (constant dose). We extend a previous framework used to quantify second-order effects, known as antifragility theory, to investigate the role of drug pharmacokinetics. Using a simple one-compartment model, we find that high variance schedules are effective for a wide range of cumulative dose values. Next, using a mouse-parameterized two-compartment model of 5-fluorouracil, we show that schedule viability depends on initial tumor volume. Finally, we illustrate the trade-off between tumor response and lean mass preservation. Mathematical modeling indicates that high variance dose schedules provide a potential path forward in mitigating the risk of chemotherapy-associated cachexia by preserving lean mass without sacrificing tumor response.

Photometric analysis of the relationship between craniovertebral angle and facial profile in children aged 10-12 years.

Background: Head posture deviation is seen in 52.5 % of children aged 6-15 years. Studies have shown that poor posture habits can impair muscle function during craniofacial growth and development. A muscle imbalance causes abnormal positioning of dental and skeletal structures, a condition that exerts negative impacts, such as changes in facial morphology. Objective: To determine through photometric analysis the relationship between craniovertebral angle as a function of head posture and glabella-subnasale-pogonion (G-Sn-Pg) angle as a function of facial profile in 10-12-year-old children, and the results will help to avoid facial development issues. Methods: Thirty-three subjects met the inclusion criteria. Their craniovertebral angles and facial profiles were measured using lateral photometry and ImageJ. The craniovertebral angle was determined by connecting the tragus and C7 with a horizontal line, whereas the facial profile angle was determined by connecting the glabella, subnasale, and pogonion. The relationship between the craniovertebral and G-Sn-Pg angle values was analyzed using the Pearson correlation test. Results: A significant relationship was observed between the craniovertebral angle and the G-Sn- Pg angle (p < 0.05), although such a relationship was weak (r = 0.373). Conclusion: A more forward head posture is associated with a more convex facial profile, and this relationship is useful for the early prevention and treatment.

Energetically stable curve fitting to hyperelastic models based on uniaxial and biaxial tensile tests.

Hyperelastic constitutive laws in biomechanics are used to model soft tissues, and material model parameters are often determined by performing curve fitting on data from uniaxial or biaxial tensile tests. The strain energy function of the applied constitutive law must to be energetically stable; however, this condition is not inherently provided by most currently available models. This study provides a procedure to determine stable strain energy functions in a biaxial strain space based on either uniaxial or biaxial tensile tests. Instead of conservative, strain-independent conditions, a stability region is defined in the strain space based on the sample's tensile tests, thus allowing optimisation within a wider parameter space, resulting in better approximations. An extension of the Levenberg-Marquardt algorithm incorporating user-defined stability constraints is proposed, and the constrained optimisation algorithm is applied to isotropic and anisotropic models. The uniqueness of solutions of the Fung model is also discussed. The material model parameters of stable solutions for soft tissue measurements from various literature sources are determined to demonstrate the proposed procedure. Applying appropriate constraints in the optimisation algorithm resulted in stable and physically permissible constrained solutions for the strain energy function, in contrast to the results of most unconstrained optimisation cases.

Impact of Stoma Baseplate Convexity on Tension and Compression Around the Stoma Site: A Finite Element Analysis.

For patients living with intestinal or urinary stomas, skin barriers play an essential role in protecting the peristomal skin and preventing peristomal complications. Convex baseplates press into the peristomal skin and are suitable for retracted stomas that do not protrude, peristomal skin with creases, folds, or dips, and stomas where frequent leaking can occur with flat pouching systems. However, there is a lack of data on the magnitude and location of tension applied to the abdomen by convex baseplates. We evaluated the impact of a range of convex baseplates applied to a simulated stoma site. A comparative finite element analysis investigation was conducted to evaluate the impact of eight different convex stoma system baseplates applied to an idealised flat abdomen, representing skin, subcutaneous tissue, and musculature layers. The baseplates considered had varying convexity with depths of 3.5 mm and 7 mm and internal structural diameters between ~30 mm and ~60 mm. The convex product range provided tension in the skin (maximum principal strain) and compression through the fat layer (minimum principal strain). Large differences in the locations and magnitudes of skin tension and fat layer compression were seen between the baseplates under analysis, with both the depth and diameter of convexity influencing the strain experienced across the abdominal topography. The results generated highlight the importance of having an appropriate range of convexity products available and selecting an appropriate option for use based on the stoma type and condition of the peristomal skin.

Intracranial Interdural and Extradural Cerebellar Convexity Dermoid Cyst: A Case Report of Rare Tumor at the Rarest Location.

Intracranial dermoid cysts are rare dysembryonic tumors of benign nature. These are uncommon in adults. If present, they are usually located in the midline or along the lines of embryonic fusion. The posterior fossa region is an infrequent site. Extradural or interdural locations are even more rare. In this case report, the authors report a laterally located large posterior fossa right cerebellar convexity interdural and extradural dermoid cyst over the sigmoid sinus. It was managed by totally extradural maximum possible safe decompression with microneurosurgical technique. The authors share their experience of addressing this rare pathology at the rarest location with unusual imaging findings.

Using a convex ostomy appliance to manage peristomal skin complications: introducing Aura Plus Soft Convex.

This article explores convex stoma appliances, introduces Aura Plus Soft Convex (CliniMed) and presents three case studies of its use. Convexity applies pressure to flatten uneven peristomal skin and form an effective adhesive seal, as well as increase protrusion of a poorly spouted stoma. This reduces the risk of leaks and peristomal skin damage, as well as minimising accessory use. Excess pressure can damage the skin, so convexity should be used with caution at the appropriate depth and firmness for the ostomate's body profile and stomal complications. Aura Plus Soft Convex has a soft and flexible baseplate for easy application and adherence, as well as a unique shape, comfort curves and a large adhesive area to reduce creases and leaks. The hydrocolloid contains Manuka honey to promote skin health, and integral belt loops offer additional security. The case studies show how this appliance can restore peristomal skin integrity and relieve stoma-related anxiety; provide gentle support for a flush stoma and a rounded abdomen; and prevent leaks and improve quality of life after years of stoma-related complications.

Using a convex ostomy appliance to manage peristomal skin complications: introducing Aura Plus Soft Convex.

This article explores convex stoma appliances, introduces Aura Plus Soft Convex (CliniMed) and presents three case studies of its use. Convexity applies pressure to flatten uneven peristomal skin and form an effective adhesive seal, as well as increase protrusion of a poorly spouted stoma. This reduces the risk of leaks and peristomal skin damage, as well as minimising accessory use. Excess pressure can damage the skin, so convexity should be used with caution at the appropriate depth and firmness for the ostomate's body profile and stomal complications. Aura Plus Soft Convex has a soft and flexible baseplate for easy application and adherence, as well as a unique shape, comfort curves and a large adhesive area to reduce creases and leaks. The hydrocolloid contains Manuka honey to promote skin health, and integral belt loops offer additional security. The case studies show how this appliance can restore peristomal skin integrity and relieve stoma-related anxiety; provide gentle support for a flush stoma and a rounded abdomen; and prevent leaks and improve quality of life after years of stoma-related complications.