Estimation of the mechanical properties of the eye through the study of its vibrational modes

Measuring the eye's mechanical properties in vivo and with minimally invasive techniques can be the key for individualized solutions to a number of eye pathologies. The development of such techniques largely relies on a computational modelling of the eyeball and, it optimally requires the synergic interplay between experimentation and numerical simulation. In Astrophysics and Geophysics the remote measurement of structural properties of the systems of their realm is performed on the basis of (helio-)seismic techniques. As a biomechanical system, the eyeball possesses normal vibrational modes encompassing rich information about its structure and mechanical properties. However, the integral analysis of the eyeball vibrational modes has not been performed yet. Here we develop a new finite difference method to compute both the spheroidal and, specially, the toroidal eigenfrequencies of the human eye. Using this numerical model, we show that the vibrational eigenfrequencies of the human eye fall in the interval 100 Hz - 10 MHz. We find that compressible vibrational modes may release a trace on high frequency changes of the intraocular pressure, while incompressible normal modes could be registered analyzing the scattering pattern that the motions of the vitreous humour leave on the retina. Existing contact lenses with embebed devices operating at high sampling frequency could be used to register the microfluctuations of the eyeball shape we obtain. We advance that an inverse problem to obtain the mechanical properties of a given eye (e.g., Young's modulus, Poisson ratio) measuring its normal frequencies is doable. These measurements can be done using non-invasive techniques, opening very interesting perspectives to estimate the mechanical properties of eyes in vivo. Future research might relate various ocular pathologies with anomalies in measured vibrational frequencies of the eye.Comment: Published in PLoS ONE as Open Access Research Article. 17 pages, 5 color figure

Probabilistic analysis of a general class of nonlinear random differential equations with state-dependent impulsive terms via probability density functions

[EN] In this contribution, we rigorously construct a pathwise solution to a general scalar random differential equation with state-dependent Dirac-delta impulse terms at a finite number of time instants. Furthermore, we obtain the first probability density function of the solution by combining two main results, firstly, the Liouville-Gibbs equation between the impulse instants, and secondly, the Random Variable Transformation technique at the impulse times. Finally, all theoretical findings are illustrated on two stochastic models, widely used in mathematical modeling, carrying on computational simulations.(c) 2023 The Author(s). Published by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).This work has been partially supported by the Spanish Ministerio de Economia, Industria y Competitividad (MINECO), the Agencia Estatal de Investigacion (AEI), Spain, and Fondo Europeo de Desarrollo Regional (FEDER UE) grant PID2020-115270GB-I00, the Generalitat Valenciana, Spain (grant AICO/2021/302) and by el Fondo Social Europeo y la Iniciativa de Empleo Juvenil EDGJID/2021/185. Vicente Bevia acknowledges the doctorate scholarship granted by Programa de Ayudas de Investigacion y Desarrollo (PAID), Universitat Politecnica de Valencia, Spain.Bevia, VJ.; Cortés, J.; Jornet-Sanz, M.; Villanueva Micó, RJ. (2023). Probabilistic analysis of a general class of nonlinear random differential equations with state-dependent impulsive terms via probability density functions. Communications in Nonlinear Science and Numerical Simulation. 119. https://doi.org/10.1016/j.cnsns.2023.10709711

Extending the deterministic Riemann-Liouville and Caputo operators to the random framework: A mean square approach with applications to solve random fractional differential equations

[EN] This paper extends both the deterministic fractional Riemann¿Liouville integral and the Caputo fractional derivative to the random framework using the mean square random calculus. Characterizations and sufficient conditions to guarantee the existence of both fractional random operators are given. Assuming mild conditions on the random input parameters (initial condition, forcing term and diffusion coefficient), the solution of the general random fractional linear differential equation, whose fractional order of the derivative is ¿ ¿ [0, 1], is constructed. The approach is based on a mean square chain rule, recently established, together with the random Fröbenius method. Closed formulae to construct reliable approximations for the mean and the covariance of the solution stochastic process are also given. Several examples illustrating the theoretical results are included.This work has been partially supported by the Ministerio de Economia y Competitividad grant MTM2013-41765-P. The co-author Prof. L. Villafuerte acknowledges the support by Mexican Conacyt.Burgos, C.; Cortés, J.; Villafuerte, L.; Villanueva Micó, RJ. (2017). Extending the deterministic Riemann-Liouville and Caputo operators to the random framework: A mean square approach with applications to solve random fractional differential equations. Chaos, Solitons and Fractals. 102:305-318. https://doi.org/10.1016/j.chaos.2017.02.008S30531810

A nonlinear dynamic age-structured model of e-commerce in Spain: Stability analysis of the equilibrium by delay and stochastic perturbations

[EN] First, we propose a deterministic age-structured epidemiological model to study the diffusion of e-commerce in Spain. Afterwards, we determine the parameters (death, birth and growth rates) of the underlying demographic model as well as the parameters (transmission of the use of e-commerce rates) of the proposed epidemiological model that best fit real data retrieved from the Spanish National Statistical Institute. Motivated by the two following facts: first the dynamics of acquiring the use of a new technology as e-commerce is mainly driven by the feedback after interacting with our peers (family, friends, mates, mass media, etc.), hence having a certain delay, and second the inherent uncertainty of sampled real data and the social complexity of the phenomena under analysis, we introduce aftereffect and stochastic perturbations in the initial deterministic model. This leads to a delayed stochastic model for e-commerce. We then investigate sufficient conditions in order to guarantee the stability in probability of the equilibrium point of the dynamic e-commerce delayed stochastic model. Our theoretical findings are numerically illustrated using real data. (C) 2018 Elsevier B.V. All rights reserved.This work has been partially supported by the Ministerio de Economia y Competitividad grant MTM2017-89664-P.Burgos-Simon, C.; Cortés, J.; Shaikhet, L.; Villanueva Micó, RJ. (2018). A nonlinear dynamic age-structured model of e-commerce in Spain: Stability analysis of the equilibrium by delay and stochastic perturbations. Communications in Nonlinear Science and Numerical Simulation. 64:149-158. https://doi.org/10.1016/j.cnsns.2018.04.022S1491586

Probabilistic Fitting of Glucose Models with Real-Coded Genetic Algorithms

[EN] Type 1 Diabetes patients have to control their blood glucose levels using insulin therapy. Numerous factors (such as carbohydrate intake, physical activity, time of day, etc.) greatly complicate this task. In this article we propose a modeling method that will allow us to make predictions of blood glucose level evolution with a time horizon of 24 hours. This may allow the adjustment of insulin doses in advance and could help to improve the living conditions of diabetes patients. Our approach starts from a system of finite difference equations that characterizes the interaction between insulin and glucose (in the field, this is known as a minimal model). This model has several parameters whose values vary widely depending on patient characteristics and time. Thus, in the first phase of our strategy, We will enrich the patient¿s historical data by adding white Gaussian noise, which will allow us to perform a probabilistic fitting with a 95% confidence interval. Then, the model¿s parameters are adjusted based on the history of each patient using a genetic algorithm and dividing the day into 12 time intervals. In the final stage, we will perform a whole-day forecast from an ensemble of the models fitted in the previous phase. Th e validity of our strategy will be tested using the Parkers¿ error grid analysis. Our experimental results based on data from real diabetic patients show that this technique is capable of robust predictions that take into account all the uncertainty associated with the interaction between insulin and glucose.We acknowledge support from Spanish Ministry of Economy and Competitiveness under project RTI2018-095180- B-I00 and Madrid Regional Goverment - FEDER grants B2017/BMD3773 (GenObIA-CM) and Y2018/NMT-4668 (Micro-Stress- MAP-CM). Devices for adquiring data from patients were adquired with the support of Fundacion Eugenio Rodriguez Pascual 2019 grant - Desarrollo de sistemas adaptativos y bioinspirados para el control glucemico con infusores subcutaneos continuos de insulina y monitores continuos de glucosa (Development of adaptive and bioinspired systems for glycaemic control with continuous subcutaneous insulin infusors and continuous glucose monitors).Cervigón, C.; Velasco, JM.; Burgos-Simon, C.; Villanueva Micó, RJ.; Hidalgo, JI. (2021). Probabilistic Fitting of Glucose Models with Real-Coded Genetic Algorithms. IEEE. 736-743. https://doi.org/10.1109/CEC45853.2021.9504836S73674

Probabilistic prediction of outbreaks of meningococcus W-135 infections over the next few years in Spain

[EN] The genogroups of meningococcal and other bacteria are in competition in the ecosystem they form with the human hosts. Changes in vaccination strategies, prophylactic measures or usual habits, may also change the distribution of the genogroups in the ecosystem but, usually, this competition is ignored in most epidemiological models, despite it can be highly influential in the evolution of infection diseases and outbreaks. Our goal is to propose a susceptible carrier susceptible (SCS) epidemiological model to determine the percentage of carriers in the population, and introduce a fractional Lotka Volterra competition model to describe the evolution of the meningococcal genogroups in Spain among the carriers. Using data from the distribution of the genogroups in Spain in 2011 and 2012, we find the model parameters and their uncertainties according to a probabilistic fitting approach. On this basis, we predict the evolution of the carriers of the different genogroups over the next few years and, in particular, the percentage of carriers of meningococcus W-135 with a 95% confidence interval. Then, we estimate the probability of having a possible outbreak of meningococcus W-135 in Spain over the next few years. According to our model and, under the present conditions, the risk of a serious outbreak of W-135 in Spain in the next 3 years is below 0.3%.This work has been partially supported by the Ministerio de Economia y Competitividad grant MTM2013-41765-P and the FIS grant PI13/01459. We also acknowledge Dr. Julio Vazquez from the Carlos III Institute of Health for providing the epidemiological data used in this work.Acedo Rodríguez, L.; Burgos-Simon, C.; Cortés, J.; Villanueva Micó, RJ. (2017). Probabilistic prediction of outbreaks of meningococcus W-135 infections over the next few years in Spain. Physica A Statistical Mechanics and its Applications. 486:106-117. https://doi.org/10.1016/j.physa.2017.05.043S10611748

Improving adaptive generalized polynomial chaos method to solve nonlinear random differential equations by the random variable transformation technique

[EN] Generalized polynomial chaos (gPC) is a spectral technique in random space to represent random variables and stochastic processes in terms of orthogonal polynomials of the Askey scheme. One of its most fruitful applications consists of solving random differential equations. With gPC, stochastic solutions are expressed as orthogonal polynomials of the input random parameters. Different types of orthogonal polynomials can be chosen to achieve better convergence. This choice is dictated by the key correspondence between the weight function associated to orthogonal polynomials in the Askey scheme and the probability density functions of standard random variables. Otherwise, adaptive gPC constitutes a complementary spectral method to deal with arbitrary random variables in random differential equations. In its original formulation, adaptive gPC requires that both the unknowns and input random parameters enter polynomially in random differential equations. Regarding the inputs, if they appear as non-polynomial mappings of themselves, polynomial approximations are required and, as a consequence, loss of accuracy will be carried out in computations. In this paper an extended version of adaptive gPC is developed to circumvent these limitations of adaptive gPC by taking advantage of the random variable transformation method. A number of illustrative examples show the superiority of the extended adaptive gPC for solving nonlinear random differential equations. In addition, for the sake of completeness, in all examples randomness is tackled by nonlinear expressions.This work has been partially supported by the Ministerio de Economia y Competitividad grants MTM2013-41765-P.Cortés, J.; Romero, J.; Roselló, M.; Villanueva Micó, RJ. (2017). Improving adaptive generalized polynomial chaos method to solve nonlinear random differential equations by the random variable transformation technique. Communications in Nonlinear Science and Numerical Simulation. 50:1-15. https://doi.org/10.1016/j.cnsns.2017.02.011S1155

Predicting mobile apps spread: An epidemiological random network modeling approach

[EN] The mobile applications business is a really big market, growing constantly. In app marketing, a key issue is to predict future app installations. The influence of the peers seems to be very relevant when downloading apps. Therefore, the study of the evolution of mobile apps spread may be approached using a proper network model that considers the influence of peers. Influence of peers and other social contagions have been successfully described using models of epidemiological type. Hence, in this paper we propose an epidemiological random network model with realistic parameters to predict the evolution of downloads of apps. With this model, we are able to predict the behavior of an app in the market in the short term looking at its evolution in the early days of its launch. The numerical results provided by the proposed network are compared with data from real apps. This comparison shows that predictions improve as the model is fed back. Marketing researchers and strategy business managers can benefit from the proposed model since it can be helpful to predict app behavior over the time anticipating the spread of an appAlegre-Sanahuja, J.; Cortés, J.; Villanueva Micó, RJ.; Santonja, F. (2017). Predicting mobile apps spread: An epidemiological random network modeling approach. Transactions of the Society for Computer Simulation. 94(2):123-130. https://doi.org/10.1177/0037549717712600S12313094

Accommodative Stimulus-Response Curve with Emoji Symbols

Purpose. To evaluate the static measurement of the accommodative stimulus-response curve with emoji symbols. Methods. The accommodative stimulus-response curve was measured in 18 subjects using a Hartmann-Shack sensor to obtain the objective accommodative response from the Zernike defocus term. Measurements were acquired at different accommodative demands, from 0 to 3 D with a step of 0.5 D. Detailed and nondetailed emoji targets were used with two different sizes, corresponding to the two most common visual angles used in smartphones. Results. A regression analysis was performed to fit the mean results obtained for each target. The determination coefficient was R2≥0.988 for all targets. For the detailed targets, the slopes for the averaged stimulus-response curve were 0.65 and 0.66 for the bigger and smaller sizes, respectively. For the nondetailed targets, the slopes were 0.60 and 0.58 for the bigger and smaller sizes, respectively. p values for these slopes were statistically significant for the two types of targets (p<0.01). Conclusions. Our results reveal that the replacement of a word or several words by detailed or nondetailed emoji symbols seems not to provoke a different accommodative response in normal subjects and under standard viewing conditions in the use of smartphones