Density-dependent Diffusion Research Articles

Cooperative transport with selective kinetic constraints.

We introduce and study a family of cooperative exclusion processes whose microscopic dynamics is governed by selective kinetic constraints. They display, in sharp contrast to the simple symmetric exclusion process, density profiles that can be concave, convex, or both, depending on the density of boundary particle reservoirs. A mean-field analysis based on a diffusion equation with a density-dependent diffusion coefficient qualitatively reproduces this behavior, and suggests its occurrence in liquids with a diffusivity anomaly.

Dynamical Stability Analysis Of Tumor Growth And Invasion: A Reaction- Diffusion Model

The acid-mediated tumor invasion hypothesis proposes that altered glucose metabolism exhibited by the vast majority of tumors leads to increased acid (H+ ion) production which subsequently facilitates tumor invasion [1-3]. The reaction-diffusion model [2] that captures the key elements of the hypothesis shows how the densities of normal cells, tumor cells, and excess H+ ions change with time due to both chemical reactions between these three populations and density-dependent diffusion by which they spread out in three-dimensional space.

Bose-Einstein-like condensation in scalar active matter with diffusivity edge.

Due to their remarkable properties, systems that exhibit self-organization of their components resulting from intrinsic microscopic activity have been extensively studied in the last two decades. In a generic class of active matter, the interactions between the active components are represented via an effective density-dependent diffusivity in a mean-field single-particle description. Here, a class of scalar active matter is proposed by incorporating a diffusivity edge into the dynamics: when the local density of the system surpasses a critical threshold, the diffusivity vanishes. The effect of the diffusivity edge is studied under the influence of an external potential, which introduces the ability to control the behavior of the system by changing an effective temperature, which is defined in terms of the single-particle diffusivity and mobility. At a critical effective temperature, a system that is trapped by a harmonic potential is found to undergo a condensation transition, which manifests formal similarities to Bose-Einstein condensation.

Abstract LB-219: Derivation of mathematical modeling for the diffusion and penetration of monoclonal antibody delivery into tumor

Abstract Background: Commonly malignant solid tumor have an abundant tumor-associated cellular matrix (ECM). Although radioimmunothrepy (RIT) have proven to be effective and precise treatment in cancer, yet RIT has a limited performance in solid tumor, due to highly distributed ECM inside solid tumor tissue. Previously, we designed the combination RIT to enhance the penetration of monoclonal antibody (mAb), where we investigated that, using Relaxin which contain oncolytic adenovirus (OA) as combination therapy decreases tumor associated ECM. In this study, we developed mathematical modeling to explain and describe the diffusion and penetration of mAb inside the tumor with ECM. To obtain the model parameters in our developed mathematical model, we performed related experiments such as immunofluorescence and qRT-PCR. Methods: We developed the reaction-diffusion partial differential equations (PDE) for describing the dependency of cancer cell, ECM density, diffusion and penetration of mAb and ECM modulated therapy that include our experimental data, also to overcome the classical limitation of PDE method. Model included the rapid decrease of ECM by OA, and density dependent diffusion coefficient. Initial condition for solving the PDE was derived using our experimental result. Diffusion coefficient for mAb and OA was modelled for simulating actual behavior of mAb and OA. Various condition of ECM density, and structure were modelled that derives the actual result of distribution of collagen. To confirm the mathematical prediction, we designed in vitro experiment with same condition as used as “in model”. Results: Using our developed model, we successfully described relationship between ECM and mAb transport by OA, and investigated the change of ECM. Diffusion and penetration of mAb and OA were well described from tumor vessel and tumor surface. Regardless of ECM structure, all ECM structure were degraded after spreading of OA. In addition, our model simulation results predicted that mAb uptake dramatically increased in combination approach and was more than mAb alone treatment. Our developed mathematical model predicted the optimal schedule of OA pretreatment and it shows great results more than mAb pretreatment. Pretreatment results with OA shows a great anti-tumor effect more than mAb pretreatment in experimental data. Our model could well predicted and matched the experimental results. Collectively, in our study, we suggested new theoretical mathematical model framework to explain various ECM condition with drug transport Citation Format: Hyeongi Kim, Yangjin Kim, Yong-Jin Lee, Jin-su Kim. Derivation of mathematical modeling for the diffusion and penetration of monoclonal antibody delivery into tumor [abstract]. In: Proceedings of the American Association for Cancer Research Annual Meeting 2019; 2019 Mar 29-Apr 3; Atlanta, GA. Philadelphia (PA): AACR; Cancer Res 2019;79(13 Suppl):Abstract nr LB-219.

Global attractors of some predator–prey reaction–diffusion systems with density-dependent diffusion and time-delays

This paper deals with a two-species and a three-species predator–prey reaction diffusion systems where the diffusion coefficients are density dependent and time-delays are involved in the reaction functions. The diffusion terms are of porous medium type which are degenerate and the boundary conditions are of Neumann type. The aim of the paper is to investigate the asymptotic behavior of the time-dependent solution in relation to various steady-state solutions of the system. This includes the existence of a unique positive classical solution, global attraction of a steady-state solution, and stability or instability of various semitrivial solutions and the positive steady-state solution.

On the hydrostatic stratification of the solar tachocline

We present an attempt to reconcile the solar tachocline glitch, a thin layer immediately beneath the convection zone in which the seismically inferred sound speed in the Sun exceeds corresponding values in standard solar models, with a degree of partial material mixing which we presume to have resulted from a combination of convective overshoot, wave transport and tachocline circulation. We first summarize the effects of either modifying in the models the opacity in the radiative interior or of incorporating either slow or fast tachocline circulation. Neither alone is successful. We then consider, without physical justification, incomplete material redistribution immediately beneath the convection zone which is slow enough not to disturb radiative equilibrium. It is modelled simply as a diffusion process. We find that, in combination with an appropriate opacity modification, it is possible to find a density-dependent diffusion coefficient that removes the glitch almost entirely, with a radiative envelope that is consistent with seismology.

Bulk diffusion in a kinetically constrained lattice gas

In the hydrodynamic regime, the evolution of a stochastic lattice gas with symmetric hopping rules is described by a diffusion equation with density-dependent diffusion coefficient encapsulating all microscopic details of the dynamics. This diffusion coefficient is, in principle, determined by a Green–Kubo formula. In practice, even when the equilibrium properties of a lattice gas are analytically known, the diffusion coefficient cannot be computed except when a lattice gas additionally satisfies the gradient condition. We develop a procedure to systematically obtain analytical approximations for the diffusion coefficient for non-gradient lattice gases with known equilibrium. The method relies on a variational formula found by Varadhan and Spohn which is a version of the Green–Kubo formula particularly suitable for diffusive lattice gases. Restricting the variational formula to finite-dimensional sub-spaces allows one to perform the minimization and gives upper bounds for the diffusion coefficient. We apply this approach to a kinetically constrained non-gradient lattice gas in two dimensions, viz. to the Kob–Andersen model on the square lattice.

Traveling wave solutions for density-dependent conformable fractional diffusion–reaction equation by the first integral method and the improved $$\textbf{tan}\left( {{\mathbf{\frac{1}{2}}}{\boldsymbol{\varphi }}\left({\boldsymbol{\upxi}} \right)} \right)$$ tan 1 2 φ ξ -expansion method

In this paper, we propose the first integral method (FIM) and the improved $$\text{tan}\left( {\frac{1}{2}{\varphi }\left(\upxi \right)} \right)$$ -expansion method (ITEM) for solving the density-dependent conformable fractional diffusion–reaction equation (CFDRE) which is commonly applied in mathematical biology. We received many new exact soliton solutions for the density-dependent CFDRE which are expressed by exponential function, rational function and hyperbolic function forms. The results show that FIM and ITEM are powerful mathematical tools and efficient techniques for solving the fractional nonlinear partial differential equations.

A kinetic view of acid-mediated tumor invasion.

According to the acid-mediated tumor invasion hypothesis, tumor-induced alteration of microenvironmental pH may provide a simple, yet complete mechanism for tumor invasion. The acid-mediation hypothesis analyzes the tumor growth and invasion process from a reaction-diffusion system perspective, where it incorporates the H+ ion concentration as a reaction factor and adds density-dependent diffusion parameters to the reaction terms, yielding independent reaction-diffusion equations for the normal, tumor, and acid populations. In this article, we apply the dynamical stability theory to the acid-mediation hypothesis. For reasonable biological parameters, we study the fixed points central to the model and their stability by calculating the eigenvalues of the Jacobian matrix of the partial differential equations that represent how these three populations evolve with time. For the case where a malignant behavior has not already taken place yet (the time rates of change of the densities of the three populations are equal to zero), our numerical results convey two different, yet possible configurations in three-dimensional space: stable and unstable dynamical equilibriums, and we discuss possible prospective trajectories for the normal and tumor populations starting from each configuration. Moreover, we discuss potential applications of our approach.

New method for investigating the density-dependent diffusion Nagumo equation

We apply reproducing kernel method to the density-dependent diffusion Nagumo equation. Powerful method has been applied by reproducing kernel functions. The approximations to the exact solution are obtained. In particular, series solutions are obtained. These solutions demonstrate the certainty of the method. The results acquired in this work conceive many attracted behaviors that assure further work on the Nagumo equation.

Gate-Driven Pure Spin Current in Graphene

Spintronics, which utilizes spin as information carrier, is a promising solution for nonvolatile memory and low-power computing in the post-Moore era. An important challenge is to realize long distance spin transport, together with efficient manipulation of spin current for novel logic-processing applications. Here, we describe a gate-variable spin current demultiplexer (GSDM) based on graphene, serving as a fundamental building block of reconfigurable spin current logic circuits. The concept relies on electrical gating of carrier density dependent conductivity and spin diffusion length in graphene. As a demo, GSDM is realized for both single-layer and bilayer graphene. The distribution and propagation of spin current in the two branches of GSDM depend on spin relaxation characteristics of graphene. Compared with Elliot-Yafet spin relaxation mechanism, D'yakonov-Perel mechanism results in more appreciable gate-tuning performance. These unique features of GSDM would give rise to abundant spin logic applications, such as on-chip spin current modulators and reconfigurable spin logic circuits.

Dynamics of degenerate quasilinear reaction diffusion systems with nonnegative initial functions

This paper is concerned with a system of quasilinear reaction–diffusion equations with density dependent diffusion coefficients and mixed quasimonotone reaction functions. The equations are allowed to be degenerate and the boundary conditions are of the nonlinear type. The main goals are to prove the existence and uniqueness of the weak solution between a pair of coupled upper and lower solutions; show that the weak solution evolves into the classical solution, and analyze the asymptotic behavior of the solution using quasi-solutions of the steady-state system. The general results are applied to a degenerate Lotka–Volterra competition model. Conditions are given for the solution to exist globally, to evolve into the classical solution, and to be attracted into a sector formed by quasi-solutions of the elliptic system. Especially for the Neumann problem we give a simple condition for the solution to converge to a unique constant steady-state solution which is a global attractor.

Dynamics of virus infection models with density-dependent diffusion

Density-dependent diffusion plays an important role in the process of viral infection. In this paper, we construct mathematical models to investigate the dynamics of the viruses and their control. Single strain and multi-strain viral infections are both considered in this work. Using the method proposed by Pao and Ruan (2013), we prove the well-posedness of the models. By constructing appropriate Lyapunov functions, we proved the global asymptotical stabilities of the models. For the multi-strain model, we show that when the basic reproduction number for each strain is greater than one, all viral strains coexist. Since the effect of different treatments may result in competitive exclusion, it is essential to employ the treatment with combined therapy. We find with surprise that the density-dependent diffusion of the virus does not influence the global stabilities of the model with homogeneous Neumann boundary conditions.

Hydrodynamics in kinetically constrained lattice-gas models.

Kinetically constrained models are lattice-gas models that are used for describing glassy systems. By construction, their equilibrium state is trivial and there are no equal-time correlations between the occupancy of different sites. We drive such models out of equilibrium by connecting them to two reservoirs of different densities, and we measure the response of the system to this perturbation. We find that under the proper coarse-graining, the behavior of these models may be expressed by a nonlinear diffusion equation, with a model- and density-dependent diffusion coefficient. We find a simple approximation for the diffusion coefficient, and we show that the relatively mild discrepancy between the approximation and our numerical results arises due to non-negligible correlations that appear as the system is driven out of equilibrium, even when the density gradient is infinitesimally small. Similar correlations appear when such kinetically constrained models are driven out of equilibrium by applying a uniform external force. We suggest that these correlations are the reason for the same discrepancy between the approximate diffusion coefficient and the numerical results for a broader group of models-nongradient lattice-gas models-for which kinetically constrained models are arguably the simplest example thereof.

A Mathematical Model for the Propagation of an Animal Species on a Plain

A mathematical model for the dynamics of an animal species propagating on a plain is constructed. Travelling wave solutions are then sought for two cases, the case with constant diffusion coefficient and that with density-dependent diffusion coefficient. The results show the existence of travelling wave solutions in both cases. The existence of travelling wave solutions for the two-dimensional model is important as it captures more realistically the physical interactions of species in a habitat. The minimum wave speeds as well as the basins of attraction were determined. The results also indicate the occurrence of a saddle-node bifurcation in the case with density-dependent diffusion coefficient. The basins of attraction in both cases are functions of the wave speedВ and is still a subject for further investigation.

A model of cell surface receptor aggregation

In this paper we construct and analyze a model of cell receptor aggregation. Experiments have shown that receptors in an aggregated state have greatly reduced mobility. We model the effects of this reduced mobility with a density dependent diffusion and study the impact of density dependent diffusion on aggregate formation in a one-dimensional domain. Critical values of receptor diffusivity and receptor activation are found and compared with numerical simulations. We find that the role of density dependant diffusion is quite limited in the formation of aggregate structures. In the case of receptor activation, the analytical results agree very well with the numerical calculations. Finally, we consider our model in higher dimensional domains. In this case our analysis is primarily numerical.

Close Form Solutions of the Fractional Generalized Reaction Duffing Model and the Density Dependent Fractional Diffusion Reaction Equation

The two variable (G'⁄G, 1⁄G)-expansion method is significant for finding the exact traveling wave solution to nonlinear evolution equations (NLEEs) in mathematical physics, applied mathematics and engineering. In this article, we exert the two variable (G'⁄G, 1⁄G)-expansion method for investigating the fractional generalized reaction Duffing model and density dependent fractional diffusion reaction equation and obtain exact solutions containing parameters. When the parameters are taken particular values, traveling wave solutions are transferred into the solitary wave solutions. The two variable (G'⁄G, 1⁄G)-expansion method is the generalization of the original (G'⁄G)-expansion method established by Wang et al [21].

A biophysical model of tumor invasion

Three-dimensional finite element computations of a cancer invasion model with nonlinear density-dependent diffusion and haptotactic sensitivity function are presented. The nonlinear model includes three key variables, namely the cancer cell density, the extra cellular matrix (ECM) density and the matrix degrading enzymes (MDE) concentration. In order to investigate the effects of tumor growth and invasion on a realistic geometry, the interactions between the cancer cells and the host tissue are incorporated into the model. The convergence study and the validation are first performed for the proposed numerical scheme. Then the effects of nonlinear diffusion and ECM-dependent haptotaxis on tumor growth and invasion in three-dimensional geometries are presented. Finally, several numerical simulations are performed with different combinations of nonlinear diffusion and haptotaxis functions to get an insight into the tumor invasion on a realistic (breast) geometry. The proposed computational model can be used to predict the location and shape of the tumor in realistic geometries at a particular instance.

Global existence and uniqueness of classical solutions for a generalized quasilinear parabolic equation with application to a glioblastoma growth model.

This paper studies the global existence and uniqueness of classical solutions for a generalized quasilinear parabolic equation with appropriate initial and mixed boundary conditions. Under some practicable regularity criteria on diffusion item and nonlinearity, we establish the local existence and uniqueness of classical solutions based on a contraction mapping. This local solution can be continued for all positive time by employing the methods of energy estimates, Lp-theory, and Schauder estimate of linear parabolic equations. A straightforward application of global existence result of classical solutions to a density-dependent diffusion model of in vitro glioblastoma growth is also presented.

A geometric approach for solving the density-dependent diffusion Nagumo equation

In this paper, some solutions of the density-dependent diffusion Nagumo equation are obtained by using a new approach, the Lie symmetry group-preserving scheme (LSGPS). The effects of various model parameters on the solution are investigated graphically using LSGPS. Finally, a different reduction method of PDEs is applied to construct two new analytical solutions and a first integral of the Nagumo equation.