Mathematical models describing the spatial spreading and invasion of populations of biological cells are often developed in a continuum modelling framework using reaction-diffusion equations. While continuum models based on linear diffusion are routinely employed and known to capture key experimental observations, linear diffusion fails to predict well-defined sharp fronts that are often observed experimentally. This observation has motivated the use of nonlinear degenerate diffusion; however, these nonlinear models and the associated parameters lack a clear biological motivation and interpretation. Here, we take a different approach by developing a stochastic discrete lattice-based model incorporating biologically inspired mechanisms and then deriving the reaction-diffusion continuum limit. Inspired by experimental observations, agents in the simulation deposit extracellular material, which we call a substrate, locally onto the lattice, and the motility of agents is taken to be proportional to the substrate density. Discrete simulations that mimic a two-dimensional circular barrier assay illustrate how the discrete model supports both smooth and sharp-fronted density profiles depending on the rate of substrate deposition. Coarse-graining the discrete model leads to a novel partial differential equation (PDE) model whose solution accurately approximates averaged data from the discrete model. The new discrete model and PDE approximation provide a simple, biologically motivated framework for modelling the spreading, growth and invasion of cell populations with well-defined sharp fronts. Open-source Julia code to replicate all results in this work is available on GitHub.

BACKGROUND: The global prevalence of multiple sclerosis has shown a marked rise in recent decades, with Germany reporting the highest prevalence among European countries. This study aims to project the future number of people with multiple sclerosis in Germany until 2040 which is necessary for effective resource allocation and health care planning.
METHODS: Based on data from the German statutory health insurance, the age- and sex-specific prevalence of multiple sclerosis was estimated applying mathematical relations between prevalence, incidence rate, and mortality rate. Subsequently, the projected prevalence was applied to the age structure of the German population between 2015 and 2040 to calculate the future number of people with multiple sclerosis. Several temporal trend scenarios pertaining to the incidence and mortality rate were compared.
RESULTS: Application of current age-specific prevalence estimates combined with the projected population structure in 2040, results in a decline of 8% in the number of people with multiple sclerosis. More realistic scenarios that reflect on trends in mortality and incidence rates, project between 453,000 (+75%) and 477,000 (+85%) multiple sclerosis cases in 2040. It is expected that females will be affected nearly 2.5 times more frequently than males in 2040.
CONCLUSIONS: The findings indicate a substantial rise in the prevalence of multiple sclerosis, ranging from 75% to 85% in 2040 compared to 2015. Assuming a constant age-specific prevalence between 2015 and 2040 without any temporal trends in mortality and incidence rates may underestimate the actual number of cases and consequently, future requirements for healthcare resources.

Cancer invasion and migration play a pivotal role in tumor malignancy, which is a major cause of most cancer deaths. Rotating magnetic field (RMF), one of the typical dynamic magnetic fields, can exert substantial mechanical influence on cells. However, studying the effects of RMF on cell is challenging due to its complex parameters, such as variation of magnetic field intensity and direction. Here, we developed a systematic simulation method to explore the influence of RMF on tumor invasion and migration, including a finite element method (FEM) model and a cell-based hybrid numerical model. Coupling with the data of magnetic field from FEM, the cell-based hybrid numerical model was established to simulate the tumor cell invasion and migration. This model employed partial differential equations (PDEs) and finite difference method to depict cellular activities and solve these equations in a discrete system. PDEs were used to depict cell activities, and finite difference method was used to solve the equations in discrete system. As a result, this study provides valuable insights into the potential applications of RMF in tumor treatment, and a series of in vitro experiments were performed to verify the simulation results, demonstrating the model\'s reliability and its capacity to predict experimental outcomes and identify pertinent factors. Furthermore, these findings shed new light on the mechanical and chemical interplay between cells and the ECM, offering new insights and providing a novel foundation for both experimental and theoretical advancements in tumor treatment by using RMF.

Aiming to address the vibration noise problems on ships, we constructed a piezoelectric phononic crystal (PC) plate structure model, solved the governing equations of the structure using the partial differential equations module (PDE) in the finite element softwareCOMSOL6.1, and obtained the corresponding energy band structure, transmission curves, and vibration modal diagrams. The application of this method to probe the structural properties of two-dimensional piezoelectric PCs is described in detail. The calculation results obtained using this method were compared with the structures obtained using the traditional plane wave expansion method (PWE) and the finite element method (FE). The results were found to be in perfect agreement, which verified the feasibility of this method. To safely and effectively adjust the bandgap within a reasonable voltage range, this paper explored the order of magnitude of the plate thickness, the influence of the voltage on the bandgap, and the dependence between them. It was found that the smaller the order of magnitude of the plate thickness, the smaller the order of magnitude of the band in which the bandgap was located. The magnitude of the driving voltage that made the bandgap change became smaller accordingly. The new idea of attaching the PC plate to the conventional plate structure to achieve a vibration damping effect is also briefly introduced. Finally, the effects of lattice constant, plate width, and thickness on the bandgap were investigated.

Mounting evidence shows that Alzheimer\'s disease (AD) is characterized by the propagation of tau aggregates throughout the brain in a prion-like manner. Since current pathology imaging technologies only provide a spatial mapping of tau accumulation, computational modeling becomes indispensable in analyzing the spatiotemporal propagation patterns of widespread tau aggregates from the longitudinal data. However, current state-of-the-art works focus on the longitudinal change of focal patterns, lacking a system-level understanding of the tau propagation mechanism that can explain and forecast the cascade of tau accumulation. To address this limitation, we conceptualize that the intercellular spreading of tau pathology forms a dynamic system where each node (brain region) is ubiquitously wired with other nodes while interacting with the build-up of pathological burdens. In this context, we formulate the biological process of tau spreading in a principled potential energy transport model (constrained by brain network topology), which allows us to develop an explainable neural network for uncovering the spatiotemporal dynamics of tau propagation from the longitudinal tau-PET scans. Specifically, we first translate the transport equation into a GNN (graph neural network) backbone, where the spreading flows are essentially driven by the potential energy of tau accumulation at each node. Conventional GNNs employ a l2-norm graph smoothness prior, resulting in nearly equal potential energies across nodes, leading to vanishing flows. Following this clue, we introduce the total variation (TV) into the graph transport model, where the nature of system\'s Euler-Lagrange equations is to maximize the spreading flow while minimizing the overall potential energy. On top of this min-max optimization scenario, we design a generative adversarial network (GAN-like) to characterize the TV-based spreading flow of tau aggregates, coined TauFlowNet. We evaluate our TauFlowNet on ADNI and OASIS datasets in terms of the prediction accuracy of future tau accumulation and explore the propagation mechanism of tau aggregates as the disease progresses. Compared to the current counterpart methods, our physics-informed deep model yields more accurate and interpretable results, demonstrating great potential in discovering novel neurobiological mechanisms through the lens of machine learning.

Neural operators, as a powerful approximation to the non-linear operators between infinite-dimensional function spaces, have proved to be promising in accelerating the solution of partial differential equations (PDE). However, it requires a large amount of simulated data, which can be costly to collect. This can be avoided by learning physics from the physics-constrained loss, which we refer to it as mean squared residual (MSR) loss constructed by the discretized PDE. We investigate the physical information in the MSR loss, which we called long-range entanglements, and identify the challenge that the neural network requires the capacity to model the long-range entanglements in the spatial domain of the PDE, whose patterns vary in different PDEs. To tackle the challenge, we propose LordNet, a tunable and efficient neural network for modeling various entanglements. Inspired by the traditional solvers, LordNet models the long-range entanglements with a series of matrix multiplications, which can be seen as the low-rank approximation to the general fully-connected layers and extracts the dominant pattern with reduced computational cost. The experiments on solving Poisson\'s equation and (2D and 3D) Navier-Stokes equation demonstrate that the long-range entanglements from the MSR loss can be well modeled by the LordNet, yielding better accuracy and generalization ability than other neural networks. The results show that the Lordnet can be 40× faster than traditional PDE solvers. In addition, LordNet outperforms other modern neural network architectures in accuracy and efficiency with the smallest parameter size.

Retirees meet a number of problems as they are growing older which needs persistent attention. Hence, without a doubt, the outcomes of the financial markets influence the choices that people make when nearing retirement. In our model, the stock price dynamics follow Geometric Brownian motion (GBM) and our goal was to optimize the expected discounted utility of consumption and terminal wealth whilst considering health expenses. The investment return process comprises risk free asset and risky assets, and the health expenses. We choose power utility functions where comprehensive solutions for Hyperbolic Absolute Risk Aversion (HARA) utility functions are obtained and optimal investment, consumption and health expenditure strategies are derived by applying dynamic programming and variable change technique on the Hamilton-Jacobi-Bellman (HJB) equations. In our numerical results it showed various effects of some economic and market parameters on the optimal investment, consumption and health expense strategies. The inflation price market risk governs the amount invested in stock, bond and also how much to be put in health to sustain a given period of the retiree\'s lifetime. As the health welfare rate R increases, the proportion of wealth invested in the stock increases. We also investigated the effects of the high correlation coefficients and low correlation coefficients on consumption and income rate respectively. As the constant variance discounting coefficient increases, seasoned enterprise annuity retirees decrease their allocation to the risky assets. Finally, a numerical example is presented to depict the effects of financial parameters on the optimal investment strategy with health expenditure.

Susceptibility of airborne ultrasonic power to augment heat and mass transfer during hot air dehydration of peppermint leaves was investigated in the present study. To predict the moisture removal curves, a unique non-equilibrium mathematical model was developed. For the samples dried at temperatures of 40‒70 °C and the power intensities of 0‒104 kW m-3, the diffusion of moisture inside the leaves and coefficients for of mass and heat transfer varied from 0.601 × 10-4 to 5.937 × 10-4 s-1, 4.693 × 10-4 to 7.975 × 10-4 m s-1 and 49.2 to 78.1 W m-2 K-1, respectively. In general, at the process temperatures up to 60 °C, all the studied transfer parameters were augmented in the presence of ultrasonic power.

UNASSIGNED: The German Federal Statistical Office routinely collects and reports aggregated numbers of people in need of long-term care (NLTC) stratified by age and sex. Age- and sex-specific prevalence of NLTC from 2011 to 2021 is reported as well. One estimation of the incidence rate of NLTC based on the age- and sex-specific prevalence exists that did not explore possible trends in incidence [based on MRR (mortality rate ratio)], which is important for an adequate projection of the future number of people with NLTC.
UNASSIGNED: We aim to explore possible trends in age-specific incidence of NLTC in German men and women from 2011 to 2021 based on different scenarios about excess mortality (in terms of MRR).
UNASSIGNED: The incidence of NLTC was calculated based on an illness-death model and a related partial differential equation based on data from the Federal Statistical Office. Estimation of annual percent change (APC) of the incidence rate was conducted in eight scenarios.
UNASSIGNED: There are consistent indications for trends in incidence for men and women aged 50-79 years with APC in incidence rate of more than +9% per year (up to nearly 19%). For ages 80+ the APC is between +0.4% and +12.5%. In all scenarios, women had higher age-specific APCs than men.
UNASSIGNED: We performed the first analysis of APC in the age- and sex-specific incidence rate of NLTC in Germany and revealed an increasing trend in the incidences. With these findings, a future prevalence of NLTC can be estimated which may exceed current prognoses.

BACKGROUND: In image processing, image segmentation is a more challenging task due to different shapes, locations, image intensities, etc. Brain tumors are one of the most common diseases in the world. So, the detection and segmentation of brain tumors are important in the medical field.
OBJECTIVE: The primary goal of this work is to use the proposed methodology to segment brain MRI images into tumor and non-tumor segments or pixels.
METHODS: In this work, we first selected the MRI medical images from the BraTS2020 database and transferred them to the contrast enhancement phase. Then, we applied thresholding for contrast enhancement to enhance the visibility of structures like blood arteries, tumors, or abnormalities. After the contrast enhancement process, the images were transformed into the image denoising phase. In this phase, a fourth-order partial differential equation was used for image denoising. After the image denoising process, these images were passed on to the segmentation phase. In this segmentation phase, we used an elephant herding algorithm for centroid optimization and then applied the multi-kernel fuzzy c-means clustering for image segmentation.
RESULTS: Peak signal-to-noise ratio, mean square error, sensitivity, specificity, and accuracy were used to assess the performance of the proposed methods. According to the findings, the proposed strategy produced better outcomes than the conventional methods.
CONCLUSIONS: Our proposed methodology was reported to be a more effective technique than existing techniques.