Coupled Reaction-diffusion Model Research Articles

Higher-order galerkin finite element method for nonlinear coupled reaction-diffusion models

In this article, generalized nonlinear coupled reaction-diffusion (NCRD) models are analyzed using the higher-order Galerkin finite element method (FEM). The authors present finite element analysis for one- and two-dimensional NCRD models by applying quadratic Lagrange basis functions. Optimal order convergence in space is proved by applying quadratic FEM in spatial discretization and utilizing the Crank-Nicolson (CN) scheme for time discretization. To address the nonlinearity of NCRD models, CN extrapolation and predictorcorrector schemes are employed. The stability analysis for generalized NCRD models and the applied CN scheme are discussed by utilizing the eigenvalues method. Finally, to assess the accuracy and efficiency of the proposed scheme and to capture different patterns of NCRD models, various one- and two-dimensional models are considered. The obtained results are numerically and graphically compared with existing literature. It is observed that quadratic basis functions in FEM improve accuracy and efficiency of the results.

Dynamical Behavior of Nonlinear Coupled Reaction-Diffusion Model: A Numerical Study Utilizing ADI and Staggered Grid Finite Volume Method in Matlab

Dynamical Behavior of Nonlinear Coupled Reaction-Diffusion Model: A Numerical Study Utilizing ADI and Staggered Grid Finite Volume Method in Matlab

Wavelets computational modeling of nonlinear coupled reaction–diffusion models arising in chemical processes

This work submits the computational modeling of nonlinear coupled reaction–diffusion (RD) models via uniform scale‐2 Haar wavelets (S2HWs) based numerical algorithm. The nonlinear RD models have many applications in mathematical chemistry, chemical and biological processes, etc. In the design of the algorithm, first of all, semi‐discretization is done in time, and after that, the quasi‐linearization process is used for linearization. Finally, the full discretization is done with the help of S2HWs, and the enlisted linear system is solved by LU decomposition method. Also, the convergence analysis of the algorithm has discussed. In the simulation part, different types of patterns such as Turing, pulse‐splitting, self‐replicating pulse patterns, etc. of different models are captured.

RETRACTED ARTICLE: A cubic B-spline quasi-interpolation method for solving hyperbolic partial differential equations

We, the Editors and Publisher of International Journal of Computer Mathematics have retracted the following article: Sudhir Kumar, R.C. Mittal & Ram Jiwari (2023) A cubic B-spline quasi-interpolation method for solving hyperbolic partial differential equations, International Journal of Computer Mathematics, DOI: 10.1080/00207160.2023.2205963 Since publication, significant concerns have been raised about the fact that this article has substantial overlap with the following article which was not acknowledged, particularly in Sections 1-4 where there were verbatim sentences and sentence fragments: Mittal, R. C., Kumar, S., & Jiwari, R. (2021). A cubic B-spline quasi-interpolation algorithm to capture the pattern formation of coupled reaction-diffusion models. Engineering with Computers, 1-17 https://doi.org/10.1007/s00366-020-01278-3. Upon query, the authors were not able to address the concerns with their paper or provide a satisfactory explanation for this significant level of overlap. As this is a serious breach of our Editorial Policies, we are retracting the article from the journal. The authors do not agree with the retraction. We have been informed in our decision-making by our policy on publishing ethics and integrity and the COPE guidelines on retractions. The retracted article will remain online to maintain the scholarly record, but it will be digitally watermarked on each page as ‘Retracted’.

Tunable Sulfur Incorporation into Atomic Layer Deposition Films Using Solution Anion Exchange

Metal sulfide and oxysulfide thin films deposited by atomic layer deposition (ALD) are important functional materials for a range of applications including solar cells, catalysts, and electronic devices. However, ALD of sulfur-containing films typically requires H2S, a toxic, corrosive, and flammable gas. To circumvent these challenges, we propose a method of tuning sulfur incorporation into ALD films by first growing a metal oxide, using CuOx as a model system, followed by a solution-phase sulfur anion-exchange process. By controlling the reaction time, solution molarity, and temperature, we demonstrate tunable sulfur incorporation into the films, which is described using a coupled reaction-diffusion model. The evolution of the film crystallinity, composition, and morphology was quantified as a function of the anion-exchange process parameters. Conformal anion exchange on a ZnO@TiO2@CuO core–shell–shell nanowire is shown, demonstrating the ability to maintain the conformality of the initial ALD process on high-aspect-ratio structures. Additionally, area-selective anion exchange was performed on micro-patterned substrates, illustrating a pathway toward device fabrication. Finally, the electrical properties of the converted films were evaluated, indicating a tunable reduction in sheet resistance of up to 4 orders of magnitude. In the future, the combination of ALD and anion-exchange chemistry can be used to incorporate sulfur without the need for H2S gas, while maintaining the atomically precise and conformal properties of the original ALD process.

A cubic B-spline quasi-interpolation algorithm to capture the pattern formation of coupled reaction-diffusion models

Diffusion plays a significant role in complex pattern formulations occurred in biological and chemical reactions. In this work, the authors study the effect of diffusion in coupled reaction-diffusion systems named the Gray-Scott model for complex pattern formation with the help of cubic B-spline quasi-interpolation (CBSQI) method and capture various formates of these patterns. The idea of Kronecker product is used first time with CBSQI method for 2D problems. Linear stability analysis of the reaction-diffusion system as well as stability of the proposed method is studied. Four test problems are considered to check the accuracy and efficiency of the method and found the stable patterns.

Deciphering role of inter and intracity human dispersal on epidemic spread via coupled reaction-diffusion models

Human mobility has been significantly influencing public health since time immemorial. A susceptible-infected-deceased epidemic reaction diffusion network model using asymptotic transmission rate is proposed to portray the spatial spread of the epidemic among two cities due to population dispersion. Qualitative behaviour including global attractor and persistence property are obtained. We also study asymptotic behaviour of the whole network with the help of asymptotic behaviour at individual cities. The epidemic model shows up two equilibria, (i) the disease-free, and (ii) unique endemic equilibria. An expression that can be used to calculate the basic reproduction number for heterogeneous environment, for the entire network is obtained. We use graph theory to analyze the global stability of our diffusive two-city model. We also performed bifurcation analysis and discovered that endemic equilibrium changes stability via Hopf bifurcations. A significant reduction in the number of infectives were observed when proper migration rate is maintained between the cities. Numerical results are provided to illuminate and clarify theoretical findings. Simulation experiments for two-dimensional spatial models show that infectious populations will increase if contact heterogeneity is increased, but it will decline if infective populations perform more local random movement. We observe that infection risk may be understated if the parameters used to estimate the basic reproduction number remains unchanged through space or time.

Pattern formation in superdiffusion predator–prey‐like problems with integer‐ and noninteger‐order derivatives

This paper focuses on the modeling and application of fractional derivative to model the interactions between two different species in which the one named predator depends on the other called prey solely for survival. The interaction between predator and prey has been one of the most intriguing and interesting subjects in applied mathematical biology and ecology. In the models, the classical reaction–diffusion equations subject to the Neumann boundary conditions are formulated on a finite but large domainx ∈ [0, L]by replacing the second‐order spatial derivatives with the fractional Laplacian operator of order1 < α ≤ 2, which is classified as superdiffusion process. We examine the resulting coupled reaction–diffusion models for linear stability analysis and derive conditions under which the spatial patterns is evolved. In a view to understand our theoretical findings, the species spatial interactions is described in one and two dimensions. Through numerical experiments, we observe that a number of patterns can arise, including Turing spots, spiral‐like structures, and seemingly complex spatiotemporal distributions.

Global Behavior of Solutions in a Predator-Prey Cross-Diffusion Model with Cannibalism

The global asymptotic behavior of solutions in a cross-diffusive predator-prey model with cannibalism is studied in this paper. Firstly, the local stability of nonnegative equilibria for the weakly coupled reaction-diffusion model and strongly coupled cross-diffusion model is discussed. It is shown that the equilibria have the same stability properties for the corresponding ODE model and semilinear reaction-diffusion model, but under suitable conditions on reaction coefficients, cross-diffusion-driven Turing instability occurs. Secondly, the uniform boundedness and the global existence of solutions for the model with SKT-type cross-diffusion are investigated when the space dimension is one. Finally, the global stability of the positive equilibrium is established by constructing a Lyapunov function. The result indicates that, under certain conditions on reaction coefficients, the model has no nonconstant positive steady state if the diffusion matrix is positive definite and the self-diffusion coefficients are large enough.

Taking paclitaxel coated balloons to a higher level: Predicting coating dissolution kinetics, tissue retention and dosing dynamics

Paclitaxel coated balloons (PCBs) are a promising non-implantable alternative to drug-eluting stents, whereby drug is delivered to the arterial wall in solid form as a semi-continuous solid coating or as micro drug depots. To date, it has been impossible to predict or even infer local tissue dosing levels and persistence, making it difficult to compare in vivo performance of different devices in healthy animals or to extrapolate such data to diseased human arteries. Here we derive and analyze a coupled reaction diffusion model that accounts for coating dissolution and tissue distribution, and predicts the concentration of dissolved drug in the tissue during and post dissolution.Time scale analysis and numerical simulations based on estimated diffusion coefficients in healthy animal and diseased human arteries both imply that dissolution of crystalline paclitaxel coating is mass transfer coefficient-limited, and can therefore be solved for independently of the tissue transport equations. Specifically, coating retention is predicted to follow piecewise linear kinetics, reflecting the differential and faster dissolution of lumenal versus tissue-embedded coating owing to a disparity in convective forces. This prediction is consistent with published data on a range of PCBs and allowed for the estimation of the associated dissolution rate-constants and the maximal soluble drug concentration in the tissue during coating dissolution. Maximal soluble drug concentration in the tissue scales as the product of the solubility and ratio of the dissolution and diffusion rate-constants. Thus, coatings characterized by micromolar solubilities give rise to nanomolar soluble concentrations in healthy animal arteries and ~0.1 micromolar in calcified atherosclerotic arteries owing to slower tissue diffusion. During dissolution, retention in porcine iliofemoral arteries is predicted to be dominated by solid coating, whereas post dissolution it is dominated by receptor-bound drug (3.7 ng receptors/g tissue).Paclitaxel coating dissolution and dosing kinetics can now be modeled based upon accepted principles of surface dissolution and tissue transport to provide insights into the dependence of clinical efficacy on device properties and the interplay of lesion complexity and procedural parameters.

Self-Assembled Monolayer of Metal Oxide Nanosheet and Structure and Gas-Sensing Property Relationship.

Semiconducting 2D metal oxides have attracted great research interests for gas-sensing applications because of their considerable specific surface area and highly homogeneous surface. Developing a method for fabricating thin films of 2D metal oxides is crucial for minimizing the negative effects on sensing performance caused by slow diffusion. In this work, a simple, versatile, and highly reproducible self-assembly method is developed for fabricating monolayer film sensors made from metal oxide nanosheets with much superior sensing performance compared with their thick film counterparts. To prepare the monolayer film sensors, a monolayer film of metal oxide nanosheets, self-assembled at the air-water interface, is transferred onto a sensor substrate. The CuO monolayer sensors prepared with this self-assembly method show much improved gas sensitivity (sensor signal: 350% vs 100% at 5 ppm of H2S) and faster response and recovery rate (τres: 20 s vs 110 s; τrec: 120 s vs 320 s) than the thick film sensors prepared from the same sensing material. The enhanced sensing performance demonstrated by the monolayer film of CuO nanosheets is explained quantitively with a modified coupled reaction-diffusion model. Similar enhancement on gas-sensing performance is also observed for the ZnO-nanosheet-based monolayer sensors prepared by the same self-assembly method.

Coupled lattice Boltzmann simulation method for bidomain type models in cardiac electrophysiology with multiple time-delays

In this work, we propose a mathematical model of the cardiac electrophysiology which take into account time delays in signal transmission, in order to capture the whole activities of macro- to micro-scale transport processes, and use this model to analyze the propagation of electrophysiological waves in the heart by using a developed coupling Lattice Boltzmann Method (LBM). The propagation of electrical activity in the heart is mathematically modeled by a modified bidomain system. As transmembrane potential evolves, the domain has anisotropical properties which are transposed into intracellular and extracellular conductivity. The new bidomain system is a multi-scale, stiff and strongly nonlinear coupled reaction-diffusion model in the shape of a set of ordinary differential equations coupled with a set of partial differential equations with multiple time delays. Due to delays, dynamic and geometry complexity, numerical simulation and implementation of this type of coupled systems are very ambitious mathematical and computational problems but are crucial in several biomedical applications. We introduce a modified LBM scheme, reliable, efficient, stable and easy to implement in the context of such bidomain systems with multiple time delays. Numerical tests to confirm effectiveness and accuracy of our approach are provided and, the influence and impact of delays to restore normal heart rhythm are analyzed.

A finite element approach for analysis and computational modelling of coupled reaction diffusion models

In this article, we establish the existence and uniqueness of solutions to the coupled reaction–diffusion models using Banach fixed point theorem. The Galerkin finite element method is used for the approximation of solutions, and an a priori error estimate is derived for such approximations. A scheme is proposed by combining the Crank–Nicolson and the predictor–corrector methods for the time discretization. Some numerical examples are considered to illustrate the accuracy and efficiency of the proposed scheme. It is found that the scheme is second‐order convergent. In addition, nonuniform grids are used in some cases to enhance the accuracy of the scheme.

Mathematical modeling and analysis of dynamic effects of multiple time-varying delays on electrophysiological wave propagation in the heart

In this work, we propose a mathematical model of the cardiac electrophysiology which takes into account time delays in signal transmission, in order to capture the whole activities of macro- to micro-scale transport processes, and use this model to analyze the propagation of electrophysiological waves in the heart. The propagation of electrical activity in the heart is mathematically modeled by a modified bidomain system. As transmembrane potential evolves, the domain has anisotropical properties which are transposed into intracellular and extracellular conductivity. The new bidomain system is a multi-scale, stiff and strongly nonlinear coupled reaction–diffusion model in the shape of a set of ordinary differential equations coupled with a set of partial differential equations with multiple time delays. First, the mathematical models are introduced. Afterward the existence and uniqueness of solution of the state equation are proved as well as stability result. A numerical calculation is performed by using a coupling Lattice Boltzmann Method.

Conditional Gaussian Systems for Multiscale Nonlinear Stochastic Systems: Prediction, State Estimation and Uncertainty Quantification.

A conditional Gaussian framework for understanding and predicting complex multiscale nonlinear stochastic systems is developed. Despite the conditional Gaussianity, such systems are nevertheless highly nonlinear and are able to capture the non-Gaussian features of nature. The special structure of the system allows closed analytical formulae for solving the conditional statistics and is thus computationally efficient. A rich gallery of examples of conditional Gaussian systems are illustrated here, which includes data-driven physics-constrained nonlinear stochastic models, stochastically coupled reaction–diffusion models in neuroscience and ecology, and large-scale dynamical models in turbulence, fluids and geophysical flows. Making use of the conditional Gaussian structure, efficient statistically accurate algorithms involving a novel hybrid strategy for different subspaces, a judicious block decomposition and statistical symmetry are developed for solving the Fokker–Planck equation in large dimensions. The conditional Gaussian framework is also applied to develop extremely cheap multiscale data assimilation schemes, such as the stochastic superparameterization, which use particle filters to capture the non-Gaussian statistics on the large-scale part whose dimension is small whereas the statistics of the small-scale part are conditional Gaussian given the large-scale part. Other topics of the conditional Gaussian systems studied here include designing new parameter estimation schemes and understanding model errors.

Mechanically Coupled Reaction-Diffusion Model to Predict Glioma Growth: Methodological Details.

Biophysical models designed to predict the growth and response of tumors to treatment have the potential to become a valuable tool for clinicians in care of cancer patients. Specifically, individualized tumor forecasts could be used to predict response or resistance early in the course of treatment, thereby providing an opportunity for treatment selection or adaption. This chapter discusses an experimental and modeling framework in which noninvasive imaging data is used to initialize and parameterize a subject-specific model of tumor growth. This modeling approach is applied to an analysis of murine models of glioma growth.

Biophysical Modeling of In Vivo Glioma Response After Whole-Brain Radiation Therapy in a Murine Model of Brain Cancer

Todevelop and investigate a set of biophysical models based on a mechanically coupled reaction-diffusion model of the spatiotemporal evolution of tumor growth after radiation therapy. Post-radiation therapy response is modeled using a cell death model (Md), a reduced proliferation rate model (Mp), and cell death and reduced proliferation model (Mdp). To evaluate each model, rats (n = 12) with C6 gliomas were imaged with diffusion-weighted magnetic resonance imaging (MRI) and contrast-enhanced MRI at 7 time points over 2weeks. Rats received either 20 or 40Gy between the third and fourth imaging time point. Diffusion-weighted MRI was used to estimate tumor cell number within enhancing regions in contrast-enhanced MRI data. Each model was fit to the spatiotemporal evolution of tumor cell number from time point 1 to time point 5 to estimate model parameters. The estimated model parameters were then used to predict tumor growth at the final 2 imaging time points. The model prediction was evaluated by calculating the error in tumor volume estimates, average surface distance, and voxel-based cell number. For both the rats treated with either 20 or 40Gy, significantly lower error in tumor volume, average surface distance, and voxel-based cell number was observed for the Mdp and Mp models compared with the Md model. The Mdp model fit, however, had significantly lower sum squared error compared with the Mp and Md models. The results of this study indicate that for both doses, the Mp and Mdp models result in accurate predictions of tumor growth, whereas the Md model poorly describes response to radiation therapy.

Numerical simulation to capture the pattern formation of coupled reaction-diffusion models

This work deals to capture the different types of patterns of nonlinear time dependent coupled reaction-diffusion models. To accomplish this work, a new differential quadrature (DQ) algorithm is developed with the help of modified trigonometric cubic B-spline functions. The stability part of the developed algorithm is studied by matrix stability analysis method. In the experimental part, different types of patterns of Gray–Scott, Schnakenberg, Isothermal Chemical and Brusselator Models are captured which are similar to the existing patterns of the models.

A mechanically coupled reaction–diffusion model that incorporates intra-tumoural heterogeneity to predict in vivo glioma growth

While gliomas have been extensively modelled with a reaction-diffusion (RD) type equation it is most likely an oversimplification. In this study, three mathematical models of glioma growth are developed and systematically investigated to establish a framework for accurate prediction of changes in tumour volume as well as intra-tumoural heterogeneity. Tumour cell movement was described by coupling movement to tissue stress, leading to a mechanically coupled (MC) RD model. Intra-tumour heterogeneity was described by including a voxel-specific carrying capacity (CC) to the RD model. The MC and CC models were also combined in a third model. To evaluate these models, rats (n = 14) with C6 gliomas were imaged with diffusion-weighted magnetic resonance imaging over 10 days to estimate tumour cellularity. Model parameters were estimated from the first three imaging time points and then used to predict tumour growth at the remaining time points which were then directly compared to experimental data. The results in this work demonstrate that mechanical-biological effects are a necessary component of brain tissue tumour modelling efforts. The results are suggestive that a variable tissue carrying capacity is a needed model component to capture tumour heterogeneity. Lastly, the results advocate the need for additional effort towards capturing tumour-to-tissue infiltration.

Abstract A11: Calibration, selection and validation of tumor growth models

Abstract As a result of the highly heterogeneous tumor microenvironment and a series of complex phenomena that occur at different scales, several modeling approaches have emerged. The tumor growth models can be categorized by the behaviors they aim to capture, the hallmarks of cancer that are being considered, and the temporal and biological scales represented. Moreover, these models can also be differentiated according to the mathematical framework that is being employed, being either discrete models, continuum models, or hybrid models. Therefore, with this large set of possible models, and with each model choice having many unknown parameters, it is important to develop an adaptive model selection and validation framework in order to choose the best model for the desired application. In this work, we employ the Occam Plausibility Algorithm (OPAL) framework [1] for model selection and validation of a large body of tumor growth models. We select several continuum tumor growth models, half of them using the phase-field modeling approach, derived from the models developed in [2], and the other half being reaction-diffusion type models derived from [3]. The OPAL framework brings together model calibration, determination of sensitivities of outputs to parameter variances, and calculation of model plausibilities for model selection. Firstly, we identify the model classes and compute the sensitivity of the quantity of interest (QoI), here the tumor volume as a function of time. The models with parameters that do not influence the QoI are eliminated. The next step consists in dividing the models into Occam categories according to their number of parameters, where the models with fewer parameters are assembled to category 1. Following the principle of Occam's Razor, one of main goals of OPAL is to select the simplest model among the several models that leads to the same prediction. The models from the first category are calibrated using Bayes' rule for a specific calibration scenario. The next step is to compute the plausibilities for the models in this category and choose the most plausible model. Finally, with the most plausible model known, we solve the problem in the validation scenario. The model is valid depending on the accuracy with which the model predictions of the QoI agree with experimental data. If the model is not valid, we move to the next category; otherwise, we select this model as being the simplest and the most plausible valid model. We demonstrate that the OPAL is able to assist in the selection of the best tumor growth model from a defined set of models. The model selected is valid for particular scenarios. In this work, we consider avascular tumor growth models to reproduce the data obtained in a murine model of glioma growth. The brain data is generated with diffusion weighted magnetic resonance imaging over a period of ten days. The methodologies and results confirm that the OPAL strategy provides a powerful framework for model selection, parameters estimation, and model validation in the presence of uncertainties.