Métodos de un paso para ecuaciones diferenciales ordinarias: recursos didácticos

El objeto de este trabajo es mostrar la utilidad, desde el punto de vista didáctico, de las hojas de cálculo Excel en la aplicación de diferentes métodos numéricos a la resolución de problemas de valor inicial de ecuaciones diferenciales ordinarias. Con el fin de verificar la convergencia de cada uno de los métodos aplicados se consideran diferentes longitudes de paso h, comenzando por un cierto valor y dividiendo sucesivamente h entre dos hasta conseguir una aproximación adecuada. Aplicando los distintos métodos numéricos a un mismo problema de valor inicial es posible comparar la velocidad de convergencia de los diferentes métodos aplicados. Analizando los resultados obtenidos a través de las hojas de Excel es posible conseguir directamente cotas del error cometido al aplicar cada uno de los diferentes métodos, sin un conocimiento previo de la solución exacta del problema

Carbon Dioxide Emissions and the Macroeconomy: Evidence from the ASEAN Region

This paper examined the effects of income, trade, and foreign direct investments (FDI) on carbon dioxide (CO2) emissions in the Association of Southeast Asian Nations (ASEAN) member countries for the period of 1970–2011 using the panel corrected standard errors (PCSE) estimation method. The results of the study were consistent with the environmental Kuznet’s curve (EKC) hypothesis – CO2 emissions increase as per capita GDP increases up to a certain income threshold, beyond which further increase in income is accompanied by lower emissions. However, the threshold per capita GDP (estimated to be USD 20,017) is way above the income levels of the ASEAN countries (with the exception of Brunei and Singapore). This suggests that most of the ASEAN region will still be in the upward-sloping portion of the EKC for several more years, and this necessitates an economic growth strategy that includes a stringent program to curb CO2 emissions. Nonetheless, both trade and FDIs do not significantly contribute to CO2 emissions in the ASEAN region, auguring well for the trade- and FDI-oriented development strategies adopted by most ASEAN member countries. Since low-carbon technologies and production methods are owned by high-income investing countries, trade and FDI can also be encouraged to facilitate and hasten the transfer of low-carbon technologies to the fast-developing countries of the ASEAN region

Tourism and Crime: Evidence from the Philippines

Using panel data gathered from 16 regions of the Philippines for the period 2009-11, this paper investigates the relationship between tourism and crime. The findings of the study show that the relation between tourism and crime may largely depend on the characteristics of visitors and the types of crime. For all types of crime and their aggregate, no significant correlation between the crime rate (defined as the number of crime cases divided by population) and total tourist arrivals is found. However, a statistically significant positive relation is found between foreign tourism and robbery and theft cases as well as between overseas Filipino tourism and robbery. On the other hand, domestic tourism is not significantly correlated with any of the four types of crimes. These results, together with a strong evidence of the negative relationship between crime and the crime clearance efficiency, present much opportunity for policy intervention in order to minimize the crime externality of the country\u27s tourism-led development strategy

Uncertainties of synchrotron microCT-based digital volume correlation bone strain measurements under simulated deformation

Digital Volume Correlation (DVC) is used to measure internal displacements and strains in bone. Recent studies have shown that synchrotron radiation micro-computed tomography (SR-microCT) can improve the accuracy and precision of DVC. However, only zero-strain or virtually-moved test have been used to quantify the DVC uncertainties, leading to potential underestimation of the measurement errors. In this study, for the first time, the uncertainties of a global DVC approach have been evaluated on repeated SR-microCT scans of bovine cortical bone (voxel size: 1.6μm), which were virtually deformed for different magnitudes and along different directions. The results showed that systematic and random errors of the normal strain components along the deformation direction were higher than the errors along unstrained directions. The systematic percentage errors were smaller for larger virtual deformations. The random percentage error was in the order of 10% of the virtual deformation. However, higher errors were localized at the boundary of the volumes of interest, perpendicular to the deformation direction. When only the central region of the samples was considered (100 micrometers layers removed from the borders where the deformation was applied), the errors in the direction of virtual deformation were comparable to the errors in the unstrained directions. In conclusion, the method presented to estimate the uncertainties of DVC is suitable for testing anisotropic specimens as cortical bone. The good agreement between the uncertainties in measurements of strain components obtained with this approach and with the simpler zero-strain-test suggests that the latter is adequate in the tested deformation scenarios

Precision of Digital Volume Correlation Approaches for Strain Analysis in Bone Imaged with Micro-Computed Tomography at Different Dimensional Levels

Accurate measurement of local strain in heterogeneous and anisotropic bone tissue is fundamental to understand the pathophysiology of musculoskeletal diseases, to evaluate the effect of interventions from preclinical studies, and to optimize the design and delivery of biomaterials. Digital volume correlation (DVC) can be used to measure the three-dimensional displacement and strain fields from micro-computed tomography (μCT) images of loaded specimens. However, this approach is affected by the quality of the input images, by the morphology and density of the tissue under investigation, by the correlation scheme, and by the operational parameters used in the computation. Therefore, for each application, the precision of the method should be evaluated. In this paper, we present the results collected from datasets analyzed in previous studies as well as new data from a recent experimental campaign for characterizing the relationship between the precision of two different DVC approaches and the spatial resolution of the outputs. Different bone structures scanned with laboratory source μCT or synchrotron light μCT (SRμCT) were processed in zero-strain tests to evaluate the precision of the DVC methods as a function of the subvolume size that ranged from 8 to 2,500 µm. The results confirmed that for every microstructure the precision of DVC improves for larger subvolume size, following power laws. However, for the first time, large differences in the precision of both local and global DVC approaches have been highlighted when SRμCT or in vivo μCT images were used instead of conventional ex vivo μCT. These findings suggest that in situ mechanical testing protocols applied in SRμCT facilities should be optimized to allow DVC analyses of localized strain measurements. Moreover, for in vivo μCT applications, DVC analyses should be performed only with relatively course spatial resolution for achieving a reasonable precision of the method. In conclusion, we have extensively shown that the precision of both tested DVC approaches is affected by different bone structures, different input image resolution, and different subvolume sizes. Before each specific application, DVC users should always apply a similar approach to find the best compromise between precision and spatial resolution of the measurements

Experimental validation of a subject-specific finite element model of lumbar spine segment using digital image correlation

Pathologies such as cancer metastasis and osteoporosis strongly affect the mechanical properties of the vertebral bone and increase the risk of fragility fractures. The prediction of the fracture risk with a patient-specific model, directly generated from the diagnostic images of the patient, could help the clinician in the choice of the correct therapy to follow. But before such models can be used to support any clinical decision, their credibility must be demonstrated through verification, validation, and uncertainty quantification. In this study we describe a procedure for the generation of such patient-specific finite element models and present a first validation of the kinematics of the spine segment. Quantitative computed tomography images of a cadaveric lumbar spine segment presenting vertebral metastatic lesions were used to generate the model. The applied boundary conditions replicated a specific experimental test where the spine segment was loaded in compression-flexion. Model predictions in terms of vertebral surface displacements were compared against the full-field experimental displacements measured with Digital Image Correlation. A good agreement was obtained from the local comparison between experimental data and simulation results (R2 > 0.9 and RMSE% <8%). In conclusion, this work demonstrates the possibility to apply the developed modelling pipeline to predict the displacement field of human spine segment under physiological loading conditions, which is a first fundamental step in the credibility assessment of these clinical decision-support technology

Experimental study exploring the factors that promote rib fragility in the elderly

Rib fractures represent a common injury type due to blunt chest trauma, affecting hospital stay and mortality especially in elderly patients. Factors promoting rib fragility, however, are little investigated. The purpose of this in vitro study was to explore potential determinants of human rib fragility in the elderly. 89 ribs from 13 human donors (55\u201399&nbsp;years) were loaded in antero-posterior compression until fracture using a material testing machine, while surface strains were captured using a digital image correlation system. The effects of age, sex, bone mineral density, rib level and side, four global morphological factors (e.g. rib length), and seven rib cross-sectional morphological factors (e.g. cortical thickness, determined by \u3bcCT), on fracture load were statistically examined using Pearson correlation coefficients, Mann\u2013Whitney U test as well as Kruskal\u2013Wallis test with Dunn-Bonferroni post hoc correction. Fracture load showed significant dependencies (p &lt; 0.05) from bone mineral density, age, antero-posterior rib length, cortical thickness, bone volume/tissue volume ratio, trabecular number, trabecular separation, and both cross-sectional area moments of inertia and was significantly higher at rib levels 7 and 8 compared to level 4 (p = 0.001/0.013), whereas side had no significant effect (p = 0.989). Cortical thickness exhibited the highest correlation with fracture load (r = 0.722), followed by the high correlation of fracture load with the area moment of inertia around the longitudinal rib cross-sectional axis (r = 0.687). High correlations with maximum external rib surface strain were detected for bone volume/tissue volume ratio (r = 0.631) and trabecular number (r = 0.648), which both also showed high correlations with the minimum internal rib surface strain (r = 12 0.644/ 12 0.559). Together with rib level, the determinants cortical thickness, area moment of inertia around the longitudinal rib cross-sectional axis, as well as bone mineral density exhibited the largest effects on human rib fragility with regard to the fracture load. Sex, rib cage side, and global morphology, in contrast, did not affect rib fragility in this study. When checking elderly patients for rib fractures due to blunt chest trauma, patients with low bone mineral density and the mid-thoracic area should be carefully examined

Local displacement and strain uncertainties in different bone types by digital volume correlation of synchrotron microtomograms

Understanding bone mechanics at different hierarchical levels is fundamental to improve preclinical and clinical assessments of bone strength. Digital Volume Correlation (DVC) is the only experimental measurement technique used for measuring local displacements and calculating local strains within bones. To date, its combination with laboratory source micro-computed tomography (LS-microCT) data typically leads to high uncertainties, which limit its application. Here, the benefits of synchrotron radiation micro-computed tomography (SR-microCT) for DVC are reported. Specimens of cortical and trabecular bovine bone and murine tibiae, were each scanned under zero-strain conditions with an effective voxel size of 1.6 μm. In order to consider the effect of the voxel size, analyses were also performed on downsampled images with voxel size of 8 μm. To evaluate displacement and strain uncertainties, each pair of tomograms was correlated using a global DVC algorithm (ShIRT-FE). Displacement random errors for original SR-microCT ranged from 0.024 to 0.226 μm, depending on DVC nodal spacing. Standard deviation of strain errors was below 200 microstrain (ca. 1/10 of the strain associated with physiological loads) for correlations performed with a measurement spatial resolution better than 40 μm for cortical bovine bone (240 μm for downsampled images), 80 μm for trabecular bovine bone (320 μm for downsampled images) and murine tibiae (120 μm for downsampled images). This study shows that the uncertainties of SR-microCT-based DVC, estimated from repeated scans, are lower than those obtained from LS-microCT-based DVC on similar specimens and low enough to measure accurately the local deformation at the tissue level

Assessing the mechanical weakness of vertebrae affected by primary tumors : a feasibility study

Patients spend months between the primary spinal tumor diagnosis and the surgical treatment, due to the need for performing chemotherapy and/or radiotherapy. During this period, they are exposed to an unknown risk of fracture. The aim of this study was to assess if it is possible to measure the mechanical strain in vertebrae affected by primary tumors, so as to open the way to an evidence-based scoring or prediction tool. We performed biomechanical tests on three vertebrae with bone tumor removed from patients. The tests were designed so as not to compromise the standard surgical and diagnostic procedures. Non-destructive mechanical tests in combination with state-of-the-art digital image correlation allowed to measure the distribution of strain on the surface of the vertebra. Our study has shown that the strains in the tumor region is circa 3 times higher than in the healthy bones, with principal strain peaks of 40,000/−20,000 microstrain, indicating a stress concentration potentially triggering vertebral fracture. This study has proven it is possible to analyze the mechanical behavior of primary tumor vertebrae as part of the clinical treatment protocol. This will allow building a tool for quantifying the risk of fracture and improving decision making in spine tumors

MicroFE models of porcine vertebrae with induced bone focal lesions : validation of predicted displacements with digital volume correlation

The evaluation of the local mechanical behavior as a result of metastatic lesions is fundamental for the characterization of the mechanical competence of metastatic vertebrae. Micro finite element (microFE) models have the potential of addressing this challenge through laboratory studies but their predictions of local deformation due to the complexity of the bone structure compromized by the lesion must be validated against experiments. In this study, the displacements predicted by homogeneous, linear and isotropic microFE models of vertebrae were validated against experimental Digital Volume Correlation (DVC) measurements. Porcine spine segments, with and without mechanically induced focal lesions, were tested in compression within a micro computed tomography (microCT) scanner. The displacement within the bone were measured with an optimized global DVC approach (BoneDVC). MicroFE models of the intact and lesioned vertebrae, including or excluding the growth plates, were developed from the microCT images. The microFE and DVC boundary conditions were matched. The displacements measured by the DVC and predicted by the microFE along each Cartesian direction were compared. The results showed an excellent agreement between the measured and predicted displacements, both for intact and metastatic vertebrae, in the middle of the vertebra, in those cases where the structure was not loaded beyond yield (0.69 < R2 < 1.00). Models with growth plates showed the worst correlations (0.02 < R2 < 0.99), while a clear improvement was observed if the growth plates were excluded (0.56 < R2 < 1.00). In conclusion, these simplified models can predict complex displacement fields in the elastic regime with high reliability, more complex non-linear models should be implemented to predict regions with high deformation, when the bone is loaded beyond yield