Differential conditions for constrained nonlinear programming via Pareto optmization

We deal with differential conditions for local optimality. The conditions that we derive for inequality constrained problems do not require constraint qualifications and are the broadest conditions based on only first-order and second-order derivatives. A similar result is proved for equality constrained problems, although the necessary conditions require the regularity of the equality constraints

Mechanical Behavior of Elastic Self-Locking Nails for Intramedullary Fracture Fixation: A Numerical Analysis of Innovative Nail Designs

Intramedullary nails constitute a viable alternative to extramedullary fixation devices; their use is growing in recent years, especially with reference to self-locking nails. Different designs are available, and it is not trivial to foresee the respective in vivo performances and to provide clinical indications in relation to the type of bone and fracture. In this work a numerical methodology was set up and validated in order to compare the mechanical behavior of two new nailing device concepts with one already used in clinic. In detail, three different nails were studied: (1) the Marchetti-Vicenzi's nail (MV1), (2) a revised concept of this device (MV2), and (3) a new Terzini-Putame's nail (TP) concept. Firstly, the mechanical behavior of the MV1 device was assessed through experimental loading tests employing a 3D-printed component aimed at reproducing the bone geometry inside which the device is implanted. In the next step, the respective numerical model was created, based on a multibody approach including flexible parts, and this model was validated against the previously obtained experimental results. Finally, numerical models of the MV2 and TP concepts were implemented and compared with the MV1 nail, focusing the attention on the response of all devices to compression, tension, bending, and torsion. A stability index (SI) was defined to quantify the mechanical stability provided to the nail-bone assembly by the elastic self-locking mechanism for the various loading conditions. In addition, results in terms of nail-bone assembly stiffness, computed from force/moment vs. displacement/rotation curves, were presented and discussed. Findings revealed that numerical models were able to provide good estimates of load vs. displacement curves. The TP nail concept proved to be able to generate a significantly higher SI (27 N for MV1 vs. 380 N for TP) and a greater stiffening action (up to a stiffness difference for bending load that ranges from 370 Nmm/° for MV1 to 1,532 Nmm/° for TP) than the other two devices which showed similar performances. On the whole, a demonstration was given of information which can be obtained from numerical simulations of expandable fixation devices

Stochastic PCA-based bone models from inverse transform sampling: Proof of concept for mandibles and proximal femurs

Principal components analysis is a powerful technique which can be used to reduce data dimensionality. With reference to three-dimensional bone shape models, it can be used to generate an unlimited number of models, defined by thousands of nodes, from a limited (less than twenty) number of scalars. The full procedure has been here described in detail and tested. Two databases were used as input data: the first database comprised 40 mandibles, while the second one comprised 98 proximal femurs. The “average shape” and principal components that were required to cover at least 90% of the whole variance were identified for both bones, as well as the statistical distributions of the respective principal components weights. Fifteen principal components sufficed to describe the mandibular shape, while nine components sufficed to describe the proximal femur morphology. A routine has been set up to generate any number of mandible or proximal femur geometries, according to the actual statistical shape distributions. The set-up procedure can be generalized to any bone shape given a sufficiently large database of the respective 3D shapes

Positive Steady States of Evolution Equations with Finite Dimensional Nonlinearities

We study the question of existence of positive steady states of nonlinear evolution equations. We recast the steady state equation in the form of eigenvalue problems for a parametrised family of unbounded linear operators, which are generators of strongly continuous semigroups; and a xed point problem. In case of irreducible governing semigroups we consider evolution equations with non-monotone nonlinearities of dimension two, and we establish a new xed point theorem for set-valued maps. In case of reducible governing semigroups we establish results for monotone nonlinearities of any nite dimension n. In addition, we establish a non-quasinilpotency result for a class of strictly positive operators, which are neither irreducible nor compact, in general. We illustrate our theoretical results with examples of partial dierential equations arising in structured population dynamics. In particular, we establish existence of positive steady states of a size-structured juvenile- adult and a structured consumer-resource population model, as well as for a selection-mutation model with distributed recruitment process

Impact of lesion delineation and intensity quantisation on the stability of texture features from lung nodules on ct: A reproducible study

Computer-assisted analysis of three-dimensional imaging data (radiomics) has received a lot of research attention as a possible means to improve the management of patients with lung cancer. Building robust predictive models for clinical decision making requires the imaging features to be stable enough to changes in the acquisition and extraction settings. Experimenting on 517 lung lesions from a cohort of 207 patients, we assessed the stability of 88 texture features from the following classes: first-order (13 features), Grey-level Co-Occurrence Matrix (24), Grey-level Difference Matrix (14), Grey-level Run-length Matrix (16), Grey-level Size Zone Matrix (16) and Neighbouring Grey-tone Difference Matrix (five). The analysis was based on a public dataset of lung nodules and open-access routines for feature extraction, which makes the study fully reproducible. Our results identified 30 features that had good or excellent stability relative to lesion delineation, 28 to intensity quantisation and 18 to both. We conclude that selecting the right set of imaging features is critical for building clinical predictive models, particularly when changes in lesion delineation and/or intensity quantisation are involved

Stochastic PCA-based bone models from inverse transform sampling: proof of concept for mandibles and proximal femurs

Principal components analysis is a powerful technique which can be used to reduce data dimensionality. With reference to three-dimensional bone shape models, it can be used to generate an unlimited number of models, defined by thousands of nodes, from a limited (less than twenty) number of scalars. The full procedure has been here described in detail and tested. Two databases were used as input data: the first database comprised 40 mandibles, while the second one comprised 98 proximal femurs. The “average shape” and principal components that were required to cover at least 90% of the whole variance were identified for both bones, as well as the statistical distributions of the respective principal components weights. Fifteen principal components sufficed to describe the mandibular shape, while nine components sufficed to describe the proximal femur morphology. A routine has been set up to generate any number of mandible or proximal femur geometries, according to the actual statistical shape distributions. The set-up procedure can be generalized to any bone shape given a sufficiently large database of the respective 3D shapes

Chaotic dynamics in periodically forced asymmetric ordinary differential equations

AbstractUsing a topological approach, we prove the existence of infinitely many periodic solutions and the presence of chaotic dynamics for the periodically forced second order ODE u″+bu+−au−=p(t). The choice of the equation is motivated by the studies about the Dancer–Fučik spectrum and the Lazer–McKenna suspension bridge model