Numerical simulation and study on heat and mass transfer in a hybrid ultrasound/convective dryer

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.

TauFlowNet: Revealing latent propagation mechanism of tau aggregates using deep neural transport equations

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.

An optimal investment consumption model for retirees with no health insurance

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.

Mathematical modeling of adipocyte size distributions: Identifiability and parameter estimation from rat data

Fat cells, called adipocytes, are designed to regulate energy homeostasis by storing energy in the form of lipids. Adipocyte size distribution is assumed to play a role in the development of obesity-related diseases. These cells that do not have a characteristic size, indeed a bimodal size distribution is observed in adipose tissue. We propose a model based on a partial differential equation to describe adipocyte size distribution. The model includes a description of the lipid fluxes and the cell size fluctuations and using a formulation of a stationary solution fast computation of bimodal distribution is achieved. We investigate the parameter identifiability and estimate parameter values with CMA-ES algorithm. We first validate the procedure on synthetic data, then we estimate parameter values with experimental data of 32 rats. We discuss the estimated parameter values and their variability within the population, as well as the relation between estimated values and their biological significance. Finally, a sensitivity analysis is performed to specify the influence of parameters on cell size distribution and explain the differences between the model and the measurements. The proposed framework enables the characterization of adipocyte size distribution with four parameters and can be easily adapted to measurements of cell size distribution in different health conditions.

Partial mean-field model for neurotransmission dynamics

This article addresses reaction networks in which spatial and stochastic effects are of crucial importance. For such systems, particle-based models allow us to describe all microscopic details with high accuracy. However, they suffer from computational inefficiency if particle numbers and density get too large. Alternative coarse-grained-resolution models reduce computational effort tremendously, e.g., by replacing the particle distribution by a continuous concentration field governed by reaction-diffusion PDEs. We demonstrate how models on the different resolution levels can be combined into hybrid models that seamlessly combine the best of both worlds, describing molecular species with large copy numbers by macroscopic equations with spatial resolution while keeping the spatial-stochastic particle-based resolution level for the species with low copy numbers. To this end, we introduce a simple particle-based model for the binding dynamics of ions and vesicles at the heart of the neurotransmission process. Within this framework, we derive a novel hybrid model and present results from numerical experiments which demonstrate that the hybrid model allows for an accurate approximation of the full particle-based model in realistic scenarios.

Rayleigh step-selection functions and connections to continuous-time mechanistic movement models

BACKGROUND

The process known as ecological diffusion emerges from a first principles view of animal movement, but ecological diffusion and other partial differential equation models can be difficult to fit to data. Step-selection functions (SSFs), on the other hand, have emerged as powerful practical tools for ecologists studying the movement and habitat selection of animals.

METHODS

SSFs typically involve comparing resources between a set of used and available points at each step in a sequence of observed positions. We use change of variables to show that ecological diffusion implies certain distributions for available steps that are more flexible than others commonly used. We then demonstrate advantages of these distributions with SSF models fit to data collected for a mountain lion in Colorado, USA.

RESULTS

We show that connections between ecological diffusion and SSFs imply a Rayleigh step-length distribution and uniform turning angle distribution, which can accommodate data collected at irregular time intervals. The results of fitting an SSF model with these distributions compared to a set of commonly used distributions revealed how precision and inference can vary between the two approaches.

CONCLUSIONS

Our new continuous-time step-length distribution can be integrated into various forms of SSFs, making them applicable to data sets with irregular time intervals between successive animal locations.

Segmentation of Brain MRI Images using Multi-Kernel FCM EHO Method

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.

CONCLUSION

Our proposed methodology was reported to be a more effective technique than existing techniques.

Influence of telomerase activity and initial distribution on human follicular aging: Moving from a discrete to a continuum model

A discrete model is proposed for the temporal evolution of a population of cells sorted according to their telomeric length. This model assumes that, during cell division, the distribution of the genetic material to daughter cells is asymmetric, i.e. chromosomes of one daughter cell have the same telomere length as the mother, while in the other daughter cell telomeres are shorter. Telomerase activity and cell death are also taken into account. The continuous model is derived from the discrete model by introducing the generational age as a continuous variable in [0,h], being h the Hayflick limit, i.e. the number of times that a cell can divide before reaching the senescent state. A partial differential equation with boundary conditions is obtained. The solution to this equation depends on the initial telomere length distribution. The initial and boundary value problem is solved exactly when the initial distribution is of exponential type. For other types of initial distributions, a numerical solution is proposed. The model is applied to the human follicular growth from preantral to preovulatory follicle as a case study and the aging rate is studied as a function of telomerase activity, the initial distribution and the Hayflick limit. Young, middle and old cell-aged initial normal distributions are considered. In all cases, when telomerase activity decreases, the population ages and the smaller the h value, the higher the aging rate becomes. However, the influence of these two parameters is different depending on the initial distribution. In conclusion, the worst-case scenario corresponds to an aged initial telomere distribution.

Passivity-based boundary control with the backstepping observer for the vibration suppression of the flexible beam

In engineering applications, flexible beam vibration control is an important issue. Although several researchers have discussed controlling beam vibration, there are few strategies for implementing it in actual applications. The passivity-based boundary control for suppressing flexible beam vibration was investigated in this paper. The controller was implemented using a moving base, and the beam model was an undamped shear beam. The control law was established using the storage function in the design technique. The finite-gain L₂ - stability of the feedback control system was then proven. This method dealt directly with the PDE of the beam model with no model reduction. Because of the non-collocated measurement and actuation in many applications, the backstepping observer was required for state estimation. Since the controller was implemented at the end of the beam via a moving base, the beam domain remained intact. Therefore, the method is simple to apply in applications. With the use of the finite-difference approach, the PDEs were numerically solved. The controller's performance of the proposed control scheme was demonstrated using computer simulation.

Detecting minimum energy states and multi-stability in nonlocal advection-diffusion models for interacting species

Deriving emergent patterns from models of biological processes is a core concern of mathematical biology. In the context of partial differential equations, these emergent patterns sometimes appear as local minimisers of a corresponding energy functional. Here we give methods for determining the qualitative structure of local minimum energy states of a broad class of multi-species nonlocal advection–diffusion models, recently proposed for modelling the spatial structure of ecosystems. We show that when each pair of species respond to one another in a symmetric fashion (i.e. via mutual avoidance or mutual attraction, with equal strength), the system admits an energy functional that decreases in time and is bounded below. This suggests that the system will eventually reach a local minimum energy steady state, rather than fluctuating in perpetuity. We leverage this energy functional to develop tools, including a novel application of computational algebraic geometry, for making conjectures about the number and qualitative structure of local minimum energy solutions. These conjectures give a guide as to where to look for numerical steady state solutions, which we verify through numerical analysis. Our technique shows that even with two species, multi-stability with up to four classes of local minimum energy states can emerge. The associated dynamics include spatial sorting via aggregation and repulsion both within and between species. The emerging spatial patterns include a mixture of territory-like segregation as well as narrow spike-type solutions. Overall, our study reveals a general picture of rich multi-stability in systems of moving and interacting species.

A theory-guided graph networks based PM2.5 forecasting method

The theory-guided air quality model solves the mathematical equations of chemical and physical processes in pollution transportation numerically. While the data-driven model, as another scientific research paradigm with powerful extraction of complex high-level abstractions, has shown unique advantages in the PM_2.5 prediction applications. In this paper, to combine the two advantages of strong interpretability and feature extraction capability, we integrated the partial differential equation of PM_2.5 dispersion with deep learning methods based on the newly proposed DPGN model. We extended its ability to perform long-term multi-step prediction and used advection and diffusion effects as additional constraints for graph neural network training. We used hourly PM_2.5 monitoring data to verify the validity of the proposed model, and the experimental results showed that our model achieved higher prediction accuracy than the baseline models. Besides, our model significantly improved the correct prediction rate of pollution exceedance days. Finally, we used the GNNExplainer model to explore the subgraph structure that is most relevant to the prediction to interpret the results. We found that the hybrid model is more biased in selecting stations with Granger causality when predicting.

Physics-incorporated convolutional recurrent neural networks for source identification and forecasting of dynamical systems

Spatio-temporal dynamics of physical processes are generally modeled using partial differential equations (PDEs). Though the core dynamics follows some principles of physics, real-world physical processes are often driven by unknown external sources. In such cases, developing a purely analytical model becomes very difficult and data-driven modeling can be of assistance. In this paper, we present a hybrid framework combining physics-based numerical models with deep learning for source identification and forecasting of spatio-temporal dynamical systems with unobservable time-varying external sources. We formulate our model PhICNet as a convolutional recurrent neural network (RNN) which is end-to-end trainable for spatio-temporal evolution prediction of dynamical systems and learns the source behavior as an internal state of the RNN. Experimental results show that the proposed model can forecast the dynamics for a relatively long time and identify the sources as well.

Diffusion modeling reveals effects of multiple release sites and human activity on a recolonizing apex predator

BACKGROUND

Reintroducing predators is a promising conservation tool to help remedy human-caused ecosystem changes. However, the growth and spread of a reintroduced population is a spatiotemporal process that is driven by a suite of factors, such as habitat change, human activity, and prey availability. Sea otters (Enhydra lutris) are apex predators of nearshore marine ecosystems that had declined nearly to extinction across much of their range by the early 20th century. In Southeast Alaska, which is comprised of a diverse matrix of nearshore habitat and managed areas, reintroduction of 413 individuals in the late 1960s initiated the growth and spread of a population that now exceeds 25,000.

METHODS

Periodic aerial surveys in the region provide a time series of spatially-explicit data to investigate factors influencing this successful and ongoing recovery. We integrated an ecological diffusion model that accounted for spatially-variable motility and density-dependent population growth, as well as multiple population epicenters, into a Bayesian hierarchical framework to help understand the factors influencing the success of this recovery.

RESULTS

Our results indicated that sea otters exhibited higher residence time as well as greater equilibrium abundance in Glacier Bay, a protected area, and in areas where there is limited or no commercial fishing. Asymptotic spread rates suggested sea otters colonized Southeast Alaska at rates of 1-8 km/yr with lower rates occurring in areas correlated with higher residence time, which primarily included areas near shore and closed to commercial fishing. Further, we found that the intrinsic growth rate of sea otters may be higher than previous estimates suggested.

CONCLUSIONS

This study shows how predator recolonization can occur from multiple population epicenters. Additionally, our results suggest spatial heterogeneity in the physical environment as well as human activity and management can influence recolonization processes, both in terms of movement (or motility) and density dependence.

Quantifying compliance with COVID-19 mitigation policies in the US: A mathematical modeling study

The outbreak of COVID-19 disrupts the life of many people in the world. In response to this global pandemic, various institutions across the globe had soon issued their prevention guidelines. Governments in the US had also implemented social distancing policies. However, those policies, which were designed to slow the spread of COVID-19, and its compliance, have varied across the states, which led to spatial and temporal heterogeneity in COVID-19 spread. This paper aims to propose a spatio-temporal model for quantifying compliance with the US COVID-19 mitigation policies at a regional level. To achieve this goal, a specific partial differential equation (PDE) is developed and validated with short-term predictions. The proposed model describes the combined effects of transboundary spread among state clusters in the US and human mobilities on the transmission of COVID-19. The model can help inform policymakers as they decide how to react to future outbreaks.

Invading and Receding Sharp-Fronted Travelling Waves

Biological invasion, whereby populations of motile and proliferative individuals lead to moving fronts that invade vacant regions, is routinely studied using partial differential equation models based upon the classical Fisher-KPP equation. While the Fisher-KPP model and extensions have been successfully used to model a range of invasive phenomena, including ecological and cellular invasion, an often-overlooked limitation of the Fisher-KPP model is that it cannot be used to model biological recession where the spatial extent of the population decreases with time. In this work, we study the Fisher-Stefan model, which is a generalisation of the Fisher-KPP model obtained by reformulating the Fisher-KPP model as a moving boundary problem. The nondimensional Fisher-Stefan model involves just one parameter, [Formula: see text], which relates the shape of the density front at the moving boundary to the speed of the associated travelling wave, c. Using numerical simulation, phase plane and perturbation analysis, we construct approximate solutions of the Fisher-Stefan model for both slowly invading and receding travelling waves, as well as for rapidly receding travelling waves. These approximations allow us to determine the relationship between c and [Formula: see text] so that commonly reported experimental estimates of c can be used to provide estimates of the unknown parameter [Formula: see text]. Interestingly, when we reinterpret the Fisher-KPP model as a moving boundary problem, many overlooked features of the classical Fisher-KPP phase plane take on a new interpretation since travelling waves solutions with [Formula: see text] are normally disregarded. This means that our analysis of the Fisher-Stefan model has both practical value and an inherent mathematical value.

A Dual-Dimer method for training physics-constrained neural networks with minimax architecture

Data sparsity is a common issue to train machine learning tools such as neural networks for engineering and scientific applications, where experiments and simulations are expensive. Recently physics-constrained neural networks (PCNNs) were developed to reduce the required amount of training data. However, the weights of different losses from data and physical constraints are adjusted empirically in PCNNs. In this paper, a new physics-constrained neural network with the minimax architecture (PCNN-MM) is proposed so that the weights of different losses can be adjusted systematically. The training of the PCNN-MM is searching the high-order saddle points of the objective function. A novel saddle point search algorithm called Dual-Dimer method is developed. It is demonstrated that the Dual-Dimer method is computationally more efficient than the gradient descent ascent method for nonconvex-nonconcave functions and provides additional eigenvalue information to verify search results. A heat transfer example also shows that the convergence of PCNN-MMs is faster than that of traditional PCNNs.

Adaptive boundary control of a vibrating cantilever nanobeam considering small scale effects

This paper presents vibration control analysis for a cantilever nanobeam system. The dynamics of the system is obtained by the non-local elastic relationship which characterizes the small scale effects. The boundary conditions and governing equation are respectively expressed by several ordinary differential equations (ODE) and a partial differential equation (PDE) with the help of the Hamilton's principle. Model-based control and adaptive control are both designed at the free end to regulate the vibration in the control section. By employing the Lyapunov stability approach, the system state can be proven to be substantiated to converge to zero's small neighbourhood with appropriate parameters. Simulation results illustrate that the designed control is feasible for the nanobeam system.

Spread mechanism and control strategy of social network rumors under the influence of COVID-19

Since the outbreak of coronavirus disease in 2019 (COVID-19), the disease has rapidly spread to the world, and the cumulative number of cases is now more than 2.3 million. We aim to study the spread mechanism of rumors on social network platform during the spread of COVID-19 and consider education as a control measure of the spread of rumors. Firstly, a novel epidemic-like model is established to characterize the spread of rumor, which depends on the nonautonomous partial differential equation. Furthermore, the registration time of network users is abstracted as 'age,' and the spreading principle of rumors is described from two dimensions of age and time. Specifically, the susceptible users are divided into higher-educators class and lower-educators class, in which the higher-educators class will be immune to rumors with a higher probability and the lower-educators class is more likely to accept and spread the rumors. Secondly, the existence and uniqueness of the solution is discussed and the stability of steady-state solution of the model is obtained. Additionally, an interesting conclusion is that the education level of the crowd is an essential factor affecting the final scale of the spread of rumors. Finally, some control strategies are presented to effectively restrain the rumor propagation, and numerical simulations are carried out to verify the main theoretical results.

Assessing two methods for estimating excess mortality of chronic diseases from aggregated data

Objective

To assess the numerical properties of two recently published estimation techniques for excess mortality based on aggregated data about diabetes in Germany.

Results

Application of the new methods to the claims data yields implausible findings for the excess mortality of type 2 diabetes in ages below 50 years of age.

Bayesian inversion for a biofilm model including quorum sensing

We propose a mathematical model based on a system of partial differential equations (PDEs) for biofilms. This model describes the time evolution of growth and degradation of biofilms which depend on environmental factors. The proposed model also includes quorum sensing (QS) and describes the cooperation among bacteria when they need to resist against external factors such as antibiotics. The applications include biofilms on teeth and medical implants, in drinking water, cooling water towers, food processing, oil recovery, paper manufacturing, and on ship hulls. We state existence and uniqueness of solutions of the proposed model and implement the mathematical model to discuss numerical simulations of biofilm growth and cooperation. We also determine the unknown parameters of the presented biofilm model by solving the corresponding inverse problem. To this end, we propose Bayesian inversion techniques and the delayed-rejection adaptive-Metropolis (DRAM) algorithm for the simultaneous extraction of multiple parameters from the measurements. These quantities cannot be determined directly from the experiments or from the computational model. Furthermore, we evaluate the presented model by comparing the simulations using the estimated parameter values with the measurement data. The results illustrate a very good agreement between the simulations and the measurements.

Assessment of steady-state seepage through dams with nonsymmetric boundary conditions: analytical approach

In this study, new analytical solutions were developed for 2D and 3D steady-state water seepage through dams with nonsymmetric boundary conditions. The nonsymmetric boundary conditions for the 2D cases were created with different unit step functions on a part and/or parts of the right boundary of dam plane. Six cases were investigated in 2D, where a constant hydraulic head is applied at the left boundary of the dam plane and rectangular, ramp, triangular, trapezoidal, tunnel, and piecewise rectangular distributions of hydraulic head are applied at the right boundary of the dam plane. Then, a 3D case with a constant hydraulic head at the upstream and a linearly distributed hydraulic head at the downstream of the dam was investigated. Subsequently, the performance of proposed analytical solutions was examined by comparison with numerical finite difference modeling. The results demonstrate reasonable accuracy of the developed equations. The developed analytical solutions can be utilized as a benchmark to verify numerical models with similar boundary conditions.

Dating and localizing an invasion from post-introduction data and a coupled reaction-diffusion-absorption model

Invasion of new territories by alien organisms is of primary concern for environmental and health agencies and has been a core topic in mathematical modeling, in particular in the intents of reconstructing the past dynamics of the alien organisms and predicting their future spatial extents. Partial differential equations offer a rich and flexible modeling framework that has been applied to a large number of invasions. In this article, we are specifically interested in dating and localizing the introduction that led to an invasion using mathematical modeling, post-introduction data and an adequate statistical inference procedure. We adopt a mechanistic-statistical approach grounded on a coupled reaction-diffusion-absorption model representing the dynamics of an organism in an heterogeneous domain with respect to growth. Initial conditions (including the date and site of the introduction) and model parameters related to diffusion, reproduction and mortality are jointly estimated in the Bayesian framework by using an adaptive importance sampling algorithm. This framework is applied to the invasion of Xylella fastidiosa, a phytopathogenic bacterium detected in South Corsica in 2015, France.

New ways of estimating excess mortality of chronic diseases from aggregated data: insights from the illness-death model

Background

Recently, we have shown that the age-specific prevalence of a disease can be related to the transition rates in the illness-death model via a partial differential equation (PDE). The transition rates are the incidence rate, the remission rate and mortality rates from the ‘Healthy’ and ‘Ill’ states. In case of a chronic disease, we now demonstrate that the PDE can be used to estimate the excess mortality from age-specific prevalence and incidence data. For the prevalence and incidence, aggregated data are sufficient - no individual subject data are needed, which allows application of the methods in contexts of strong data protection or where data from individual subjects is not accessible.

Methods

After developing novel estimators for the excess mortality derived from the PDE, we apply them to simulated data and compare the findings with the input values of the simulation aiming to evaluate the new approach. In a practical application to claims data from 35 million men insured by the German public health insurance funds, we estimate the population-wide excess mortality of men with diagnosed type 2 diabetes.

Results

In the simulation study, we find that the estimation of the excess mortality is feasible from prevalence and incidence data if the prevalence is given at two points in time. The accuracy of the method decreases as the temporal difference between these two points in time increases. In our setting, the relative error was 5% and below if the temporal difference was three years or less. Application of the new method to the claims data yields plausible findings for the excess mortality of type 2 diabetes in German men.

Conclusions

The described approach is useful to estimate the excess mortality of a chronic condition from aggregated age-specific incidence and prevalence data.

Trial registration

The article does not report the results of any health care intervention.

Electronic supplementary material

The online version of this article (10.1186/s12889-019-7201-7) contains supplementary material, which is available to authorized users.

Series: Utilization of Differential Equations and Methods for Solving Them in Medical Physics (4)

Partial differential equations are often used in the field of medical physics. In this (final) issue, the methods for solving the partial differential equations were introduced, which include separation of variables, integral transform (Fourier and Fourier-sine transforms), Green's function, and series expansion methods. Some examples were also introduced, in which the integral transform and Green's function methods were applied to solving Pennes' bioheat transfer equation and the Fourier series expansion method was applied to Navier-Stokes equation for analyzing the wall shear stress in blood vessels.Finally, the author hopes that this series will be helpful for people who engage in medical physics.

The Role of Exosomes in Pancreatic Cancer Microenvironment

Exosomes are nanovesicles shed by cells as a means of communication with other cells. Exosomes contain mRNAs, microRNAs (miRs) and functional proteins. In the present paper, we develop a mathematical model of tumor-immune interaction by means of exosomes shed by pancreatic cancer cells and dendritic cells. Cancer cells' exosomes contain miRs that promote their proliferation and that inhibit immune response by dendritic cells, and by CD4+ and CD8+ T cells. Dendritic cells release exosomes with proteins that induce apoptosis of cancer cells and that block regulatory T cells. Simulations of the model show how the size of the pancreatic cancer can be determined by measurement of specific miRs (miR-21 and miR-203 in the case of pancreatic cancer), suggesting these miRs as biomarkers for cancer.

Quantifying the roles of random motility and directed motility using advection-diffusion theory for a 3T3 fibroblast cell migration assay stimulated with an electric field

Background

Directed cell migration can be driven by a range of external stimuli, such as spatial gradients of: chemical signals (chemotaxis); adhesion sites (haptotaxis); or temperature (thermotaxis). Continuum models of cell migration typically include a diffusion term to capture the undirected component of cell motility and an advection term to capture the directed component of cell motility. However, there is no consensus in the literature about the form that the advection term takes. Some theoretical studies suggest that the advection term ought to include receptor saturation effects. However, others adopt a much simpler constant coefficient. One of the limitations of including receptor saturation effects is that it introduces several additional unknown parameters into the model. Therefore, a relevant research question is to investigate whether directed cell migration is best described by a simple constant tactic coefficient or a more complicated model incorporating saturation effects.

Results

We study directed cell migration using an experimental device in which the directed component of the cell motility is driven by a spatial gradient of electric potential, which is known as electrotaxis. The electric field (EF) is proportional to the spatial gradient of the electric potential. The spatial variation of electric potential across the experimental device varies in such a way that there are several subregions on the device in which the EF takes on different values that are approximately constant within those subregions. We use cell trajectory data to quantify the motion of 3T3 fibroblast cells at different locations on the device to examine how different values of the EF influences cell motility. The undirected (random) motility of the cells is quantified in terms of the cell diffusivity, D, and the directed motility is quantified in terms of a cell drift velocity, v. Estimates D and v are obtained under a range of four different EF conditions, which correspond to normal physiological conditions. Our results suggest that there is no anisotropy in D, and that D appears to be approximately independent of the EF and the electric potential. The drift velocity increases approximately linearly with the EF, suggesting that the simplest linear advection term, with no additional saturation parameters, provides a good explanation of these physiologically relevant data.

Conclusions

We find that the simplest linear advection term in a continuum model of directed cell motility is sufficient to describe a range of different electrotaxis experiments for 3T3 fibroblast cells subject to normal physiological values of the electric field. This is useful information because alternative models that include saturation effects involve additional parameters that need to be estimated before a partial differential equation model can be applied to interpret or predict a cell migration experiment.

Electronic supplementary material

The online version of this article (doi:10.1186/s12918-017-0413-5) contains supplementary material, which is available to authorized users.

Numerical rate function determination in partial differential equations modeling cell population dynamics

This paper introduces a method to solve the inverse problem of determining an unknown rate function in a partial differential equation (PDE) based on discrete measurements of the modeled quantity. The focus is put on a size-structured population balance equation (PBE) predicting the evolution of the number distribution of a single cell population as a function of the size variable. Since the inverse problem at hand is ill-posed, an adequate regularization scheme is required to avoid amplification of measurement errors in the solution method. The technique developed in this work to determine a rate function in a PBE is based on the approximate inverse method, a pointwise regularization scheme, which employs two key ideas. Firstly, the mollification in the directions of time and size variables are separated. Secondly, instable numerical data derivatives are circumvented by shifting the differentiation to an analytically given function. To examine the performance of the introduced scheme, adapted test scenarios have been designed with different levels of data disturbance simulating the model and measurement errors in practice. The success of the method is substantiated by visualizing the results of these numerical experiments.

A model of space-fractional-order diffusion in the glial scar

Implantation of neuroprosthetic electrodes induces a stereotypical state of neuroinflammation, which is thought to be detrimental for the neurons surrounding the electrode. Mechanisms of this type of neuroinflammation are still poorly understood. Recent experimental and theoretical results point to a possible role of the diffusing species in this process. The paper considers a model of anomalous diffusion occurring in the glial scar around a chronic implant in two simple geometries - a separable rectilinear electrode and a cylindrical electrode, which are solvable exactly. We describe a hypothetical extended source of diffusing species and study its concentration profile in steady-state conditions. Diffusion transport is assumed to obey a fractional-order Fick law, derivable from physically realistic assumptions using a fractional calculus approach. Presented fractional-order distribution morphs into integer-order diffusion in the case of integral fractional exponents. The model demonstrates that accumulation of diffusing species can occur and the scar properties (i.e. tortuosity, fractional order, scar thickness) and boundary conditions can influence such accumulation. The observed shape of the concentration profile corresponds qualitatively with GFAP profiles reported in the literature. The main difference with respect to the previous studies is the explicit incorporation of the apparatus of fractional calculus without assumption of an ad hoc tortuosity parameter. The approach can be adapted to other studies of diffusion in biological tissues, for example of biomolecules or small drug molecules.

Modeling biological gradient formation: combining partial differential equations and Petri nets

Both Petri nets and differential equations are important modeling tools for biological processes. In this paper we demonstrate how these two modeling techniques can be combined to describe biological gradient formation. Parameters derived from partial differential equation describing the process of gradient formation are incorporated in an abstract Petri net model. The quantitative aspects of the resulting model are validated through a case study of gradient formation in the fruit fly.

A one-dimensional moving-boundary model for tubulin-driven axonal growth

A one-dimensional continuum-mechanical model of axonal elongation due to assembly of tubulin dimers in the growth cone is presented. The conservation of mass leads to a coupled system of three differential equations. A partial differential equation models the dynamic and the spatial behaviour of the concentration of tubulin that is transported along the axon from the soma to the growth cone. Two ordinary differential equations describe the time-variation of the concentration of free tubulin in the growth cone and the speed of elongation. All steady-state solutions of the model are categorized. Given a set of the biological parameter values, it is shown how one easily can infer whether there exist zero, one or two steady-state solutions and directly determine the possible steady-state lengths of the axon. Explicit expressions are given for each stationary concentration distribution. It is thereby easy to examine the influence of each biological parameter on a steady state. Numerical simulations indicate that when there exist two steady states, the one with shorter axon length is unstable and the longer is stable. Another result is that, for nominal parameter values extracted from the literature, in a large portion of a fully grown axon the concentration of free tubulin is lower than both concentrations in the soma and in the growth cone.