Holotomography and atomic force microscopy: a powerful combination to enhance cancer, microbiology and nanotoxicology research.

Modern imaging strategies are paramount to studying living systems such as cells, bacteria, and fungi and their response to pathogens, toxicants, and nanomaterials (NMs) as modulated by exposure and environmental factors. The need to understand the processes and mechanisms of damage, healing, and cell survivability of living systems continues to motivate the development of alternative imaging strategies. Of particular interest is the use of label-free techniques (microscopy procedures that do not require sample staining) that minimize interference of biological processes by foreign marking substances and reduce intense light exposure and potential photo-toxicity effects. This review focuses on the synergic capabilities of atomic force microscopy (AFM) as a well-developed and robust imaging strategy with demonstrated applications to unravel intimate details in biomedical applications, with the label-free, fast, and enduring Holotomographic Microscopy (HTM) strategy. HTM is a technique that combines holography and tomography using a low intensity continuous illumination laser to investigate (quantitatively and non-invasively) cells, microorganisms, and thin tissue by generating three-dimensional (3D) images and monitoring in real-time inner morphological changes. We first review the operating principles that form the basis for the complementary details provided by these techniques regarding the surface and internal information provided by HTM and AFM, which are essential and complimentary for the development of several biomedical areas studying the interaction mechanisms of NMs with living organisms. First, AFM can provide superb resolution on surface morphology and biomechanical characterization. Second, the quantitative phase capabilities of HTM enable superb modeling and quantification of the volume, surface area, protein content, and mass density of the main components of cells and microorganisms, including the morphology of cells in microbiological systems. These capabilities result from directly quantifying refractive index changes without requiring fluorescent markers or chemicals. As such, HTM is ideal for long-term monitoring of living organisms in conditions close to their natural settings. We present a case-based review of the principal uses of both techniques and their essential contributions to nanomedicine and nanotoxicology (study of the harmful effects of NMs in living organisms), emphasizing cancer and infectious disease control. The synergic impact of the sequential use of these complementary strategies provides a clear drive for adopting these techniques as interdependent fundamental tools.

Computer simulation of the dynamics of a spatial susceptible-infected-recovered epidemic model with time delays in transmission and treatment.

BACKGROUND AND OBJECTIVE: In this work, we analyze the spatial-temporal dynamics of a susceptible-infected-recovered (SIR) epidemic model with time delays. To better describe the dynamical behavior of the model, we take into account the cumulative effects of diffusion in the population dynamics, and the time delays in both the Holling type II treatment and the disease transmission process, respectively. METHODS: We perform linear stability analyses on the disease-free and endemic equilibria. We provide the expression of the basic reproduction number and set conditions on the backward bifurcation using Castillo's theorem. The values of the critical time transmission, the treatment delays and the relationship between them are established. RESULTS: We show that the treatment rate decreases the basic reproduction number while the transmission rate significantly affects the bifurcation process in the system. The transmission and treatment time-delays are found to be inversely proportional to the susceptible and infected diffusion rates. The analytical results are numerically tested. The results show that the treatment rate significantly reduces the density of infected population and ensures the transition from the unstable to the stable domain. Moreover, the system is more sensible to the treatment in the stable domain. CONCLUSIONS: The density of infected population increases with respect to the infected and susceptible diffusion rates. Both effects of treatment and transmission delays significantly affect the behavior of the system. The transmission time-delay at the critical point ensures the transition from the stable (low density) to the unstable (high density) domain.

A SEIR model with memory effects for the propagation of Ebola-like infections and its dynamically consistent approximation.

BACKGROUND AND OBJECTIVE: We present and analyze a nonstandard numerical method to solve an epidemic model with memory that describes the propagation of Ebola-type diseases. The epidemiological system contemplates the presence of sub-populations of susceptible, exposed, infected and recovered individuals, along with nonlinear interactions between the members of those sub-populations. The system possesses disease-free and endemic equilibrium points, whose stability is studied rigorously. METHODS: To solve the epidemic model with memory, a nonstandard approach based on Grünwald-Letnikov differences is used to discretize the problem. The discretization is conveniently carried out in order to produce a fully explicit and non-singular scheme. The discrete problem is thus well defined for any set of non-negative initial conditions. RESULTS: The existence and uniqueness of the solutions of the discrete problem for non-negative initial data is thoroughly proved. Moreover, the positivity and the boundedness of the approximations is also theoretically elucidated. Some simulations confirm the validity of these theoretical results. Moreover, the simulations prove that the computational model is capable of preserving the equilibria of the system (both the disease-free and the endemic equilibria) as well as the stability of those points. CONCLUSIONS: Both theoretical and numerical results establish that the computational method proposed in this work is capable of preserving distinctive features of an epidemiological model with memory for the propagation of Ebola-type diseases. Among the main characteristics of the numerical integrator, the existence and the uniqueness of solutions, the preservation of both positivity and boundedness, the preservation of the equilibrium points and their stabilities as well as the easiness to implement it computationally are the most important features of the approach proposed in this manuscript.

Analysis of a nonstandard computer method to simulate a nonlinear stochastic epidemiological model of coronavirus-like diseases.

BACKGROUND AND OBJECTIVE: We propose a nonstandard computational model to approximate the solutions of a stochastic system describing the propagation of an infectious disease. The mathematical model considers the existence of various sub-populations, including humans who are susceptible to the disease, asymptomatic humans, infected humans and recovered or quarantined individuals. Various mechanisms of propagation are considered in order to describe the propagation phenomenon accurately. METHODS: We propose a stochastic extension of the deterministic model, considering a random component which follows a Brownian motion. In view of the difficulties to solve the system exactly, we propose a computational model to approximate its solutions following a nonstandard approach. RESULTS: The nonstandard discretization is fully analyzed for positivity, boundedness and stability. It is worth pointing out that these properties are realized in the discrete scenario and that they are thoroughly established herein using rigorous mathematical arguments. We provide some illustrative computational simulations to exhibit the main computational features of this approach. CONCLUSIONS: The results show that the nonstandard technique is capable of preserving the distinctive characteristics of the epidemiologically relevant solutions of the model, while other (classical) approaches are not able to do it. For the sake of convenience, a computational code of the nonstandard discrete model may be provided to the readers at their requests.

Design of a nonlinear model for the propagation of COVID-19 and its efficient nonstandard computational implementation.

In this manuscript, we develop a mathematical model to describe the spreading of an epidemic disease in a human population. The emphasis in this work will be on the study of the propagation of the coronavirus disease (COVID-19). Various epidemiologically relevant assumptions will be imposed upon the problem, and a coupled system of first-order ordinary differential equations will be obtained. The model adopts the form of a nonlinear susceptible-exposed-infected-quarantined-recovered system, and we investigate it both analytically and numerically. Analytically, we obtain the equilibrium points in the presence and absence of the coronavirus. We also calculate the reproduction number and provide conditions that guarantee the local and global asymptotic stability of the equilibria. To that end, various tools from analysis will be employed, including Volterra-type Lyapunov functions, LaSalle's invariance principle and the Routh-Hurwitz criterion. To simulate computationally the dynamics of propagation of the disease, we propose a nonstandard finite-difference scheme to approximate the solutions of the mathematical model. A thorough analysis of the discrete model is provided in this work, including the consistency and the stability analyses, along with the capability of the discrete model to preserve the equilibria of the continuous system. Among other interesting results, our numerical simulations confirm the stability properties of the equilibrium points.

An optimal Bayesian threshold method for onset detection in electric biosignals.

In this work, we consider the problem of identifying activity phases in electromyography signals, and various other potential types of electrical and non-electrical biological signals such as electroneurograms, electroencephalograms, voice and ultrasounds. The solution to this problem has been provided under relatively limited scenarios. The purpose of the present work is to propose an optimal Bayesian classifier to solve the problem of detecting bursts on biological signals. To that end, a parametrization of the distribution of samples in signals is presented. We propose a model based on a linear combination of normal distributions with mean equal to zero and different variances. The threshold criterion is expressed in a closed-form, and the use of morphology operators in the post-processing treatment leads to accurate results. Various comparisons are provided against other techniques available in the literature. In all of our experiments, we show that our present approach yields superior results.

A computational method for the detection of activation/deactivation patterns in biological signals with three levels of electric intensity.

In the present work, we develop a computational technique to approximate the changes of phase in temporal series associated to electric signals of muscles which perform activities at three different levels of intensity. We suppose that the temporal series are samples of independent, normally distributed random variables with mean equal to zero, and variance equal to one of three possible values, each of them associated to a certain degree of electric intensity. For example, these intensity levels may represent a leg muscle at rest, or active during a light activity (walking), or active during a highly demanding performance (jogging). The model is presented as a maximum likelihood problem involving discrete variables. In turn, this problem is transformed into a continuous one via the introduction of continuous variables with penalization parameters, and it is solved recursively through an iterative numerical method. An a posteriori treatment of the results is used in order to avoid the detection of relatively short periods of silence or activity. We perform simulations with synthetic data in order to assess the validity of our technique. Our computational results show that the method approximates well the occurrence of the change points in synthetic temporal series, even in the presence of autocorrelated sequences. In the way, we show that a generalization of a computational technique for the change-point detection of electric signals with two phases of activity (Esquivel-Frausto et al., 2010 [40]), may be inapplicable in cases of temporal series with three levels of intensity. In this sense, the method proposed in the present manuscript improves previous efforts of the authors.

An efficient nonlinear finite-difference approach in the computational modeling of the dynamics of a nonlinear diffusion-reaction equation in microbial ecology.

In this manuscript, we present a computational model to approximate the solutions of a partial differential equation which describes the growth dynamics of microbial films. The numerical technique reported in this work is an explicit, nonlinear finite-difference methodology which is computationally implemented using Newton's method. Our scheme is compared numerically against an implicit, linear finite-difference discretization of the same partial differential equation, whose computer coding requires an implementation of the stabilized bi-conjugate gradient method. Our numerical results evince that the nonlinear approach results in a more efficient approximation to the solutions of the biofilm model considered, and demands less computer memory. Moreover, the positivity of initial profiles is preserved in the practice by the nonlinear scheme proposed.

Activity pattern detection in electroneurographic and electromyogram signals through a heteroscedastic change-point method.

In this work, we propose a heteroscedastic method in the detection of activity patterns of electroneurographic and electromyogram signals involved in rhythmic activities of nerves and muscles, respectively. The electric behavior observed in such signals is characterized by phases of activity and silence. The beginning and the length of electrically active and electrically silent phases in a signal allow us to quantitatively analyze the changes and the effects on a rhythmic activity produced by experimental changes. In order to distinguish between these two phases, signals are assumed to be a sample of a time-dependent, normally distributed random variable with non-constant variance, and that the determination of the variance at each point allows us to determine in which phase is the signal. The parameters of the model are determined by means of an iterative process which maximizes the log-likelihood under the proposed model. Moreover, we apply our method to the determination of the activity phases and silence phases in sequences of experimental and synthetic electroneurographic and electromyogram signals. The results obtained with synthetic data show that the method performs well in the determination of these activity patterns. Finally, the study of particular signals simulated under a generalized autoregressive conditional heteroscedasticity model suggests the robustness of the method with respect to the assumption of independence.