Asymptotic-Preserving methods and multiscale models for plasma physics

Background 
Understanding cancer development crossing several spatial-temporal scales is of great practical significance to better understand and treat cancers. It is difficult to tackle this challenge with pure biological means. Moreover, hybrid modeling techniques have been proposed that combine the advantages of the continuum and the discrete methods to model multiscale problems.


Methods 
In light of these problems, we have proposed a new hybrid vascular model to facilitate the multiscale modeling and simulation of cancer development with respect to the agent-based, cellular automata and machine learning methods. The purpose of this simulation is to create a dataset that can be used for prediction of cell phenotypes. By using a proposed Q-learning based on SVR-NSGA-II method, the cells have the capability to predict their phenotypes autonomously that is, to act on its own without external direction in response to situations it encounters.


Results 
Computational simulations of the model were performed in order to analyze its performance. The most striking feature of our results is that each cell can select its phenotype at each time step according to its condition. We provide evidence that the prediction of cell phenotypes is reliable.


Conclusion 
Our proposed model, which we term a hybrid multiscale modeling of cancer cell behavior, has the potential to combine the best features of both continuum and discrete models. The in silico results indicate that the 3D model can represent key features of cancer growth, angiogenesis, and its related micro-environment and show that the findings are in good agreement with biological tumor behavior. To the best of our knowledge, this paper is the first hybrid vascular multiscale modeling of cancer cell behavior that has the capability to predict cell phenotypes individually by a self-generated dataset.

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Hybrid multiscale modeling and prediction of cancer cell behavior

We develop a new Monte Carlo method that solves hyperbolic transport equations with stiff terms, characterized by a (small) scaling parameter. In particular, we focus on systems which lead to a reduced problem of parabolic type in the limit when the scaling parameter tends to zero. Classical Monte Carlo methods suffer severe time step limitations in these situations, due to the fact that the characteristic speeds go to infinity in the diffusion limit. This makes the problem a real challenge, since the scaling parameter may differ by several orders of magnitude in the domain. To circumvent these time step limitations, we construct a new, asymptotic-preserving Monte Carlo method that is stable independently of the scaling parameter and degenerates to a standard probabilistic approach for solving the limiting equation in the diffusion limit. The method uses an implicit time discretization to formulate a modified equation in which the characteristic speeds do not grow indefinitely when the scaling factor tend...

/pdf/asymptotic-preserving-monte-carlo-methods-for-transport-37q8g5hjvq.pdf

Asymptotic-Preserving Monte Carlo Methods for Transport Equations in the Diffusive Limit

In this article, we are interested in the asymptotic analysis of a finite volume scheme for one dimensional linear kinetic equations, with either Fokker-Planck or linearized BGK collision operator. Thanks to appropriate uniform estimates, we establish that the proposed scheme is Asymptotic-Preserving in the diffusive limit. Moreover, we adapt to the discrete framework the hypocoercivity method proposed by [J. Dolbeault, C. Mouhot and C. Schmeiser, Trans. Amer. Math. Soc., 367, 6 (2015)] to prove the exponential return to equilibrium of the approximate solution. We obtain decay rates that are bounded uniformly in the diffusive limit.
Finally, we present an efficient implementation of the proposed numerical schemes, and perform numerous numerical simulations assessing their accuracy and efficiency in capturing the correct asymptotic behaviors of the models.

https://hal.archives-ouvertes.fr/hal-01957832/document

Hypocoercivity and diffusion limit of a finite volume scheme for linear kinetic equations

A high-order relaxation method with projective integration for solving nonlinear systems of hyperbolic conservation laws

We investigate a high-order, fully explicit, asymptotic-preserving scheme for a kinetic equation with linear relaxation, both in the hydrodynamic and diffusive scalings in which a hyperbolic, resp., parabolic, limiting equation exists. The scheme first takes a few small (inner) steps with a simple, explicit method (such as direct forward Euler) to damp out the stiff components of the solution and estimate the time derivative of the slow components. These estimated time derivatives are then used in an (outer) Runge--Kutta method of arbitrary order. We show that, with an appropriate choice of inner step size, the time-step restriction on the outer time step is similar to the stability condition for the limiting macroscopic equation. Moreover, the number of inner time steps is also independent of the scaling parameter. We analyze stability and consistency, and illustrate with numerical results.

/pdf/a-high-order-asymptotic-preserving-scheme-for-kinetic-182j6hzui4.pdf

A High-Order Asymptotic-Preserving Scheme for Kinetic Equations Using Projective Integration

We investigate a high-order, fully explicit, asymptotic-preserving scheme for a kinetic equation with linear relaxation, both in the hydrodynamic and diffusive scalings in which a hyperbolic, resp. parabolic, limiting equation exists. The scheme first takes a few small (inner) steps with a simple, explicit method (such as direct forward Euler) to damp out the stiff components of the solution and estimate the time derivative of the slow components. These estimated time derivatives are then used in an (outer) Runge-Kutta method of arbitrary order. We show that, with an appropriate choice of inner step size, the time-step restriction on the outer time step is similar to the stability condition for the limiting macroscopic equation. Moreover, the number of inner time steps is also independent of the scaling parameter. We analyse stability and consistency, and illustrate with numerical results.

A high-order asymptotic-preserving scheme for kinetic equations using projective integration

The use of stochastic modeling and simulation techniques is widespread in computational biology when fluctuations become important. In this chapter, we give a high-level overview of stochastic modeling techniques for biological problems, focussing on some common individual-based modeling and simulation methods. We pay particular attention to the equivalence between the stochastic process that governs the evolution of individual agents and the deterministic behaviour of the involved probability distributions, and we discuss numerical methods that exploit this relation for variance reduction purposes. The discussion will be illustrated using examples involving intracellular chemical reactions, bacterial chemotaxis and tumor growth, showing the effects of stochasticity at different scales and different levels of description.

Stochastic Modeling and Simulation Methods for Biological Processes: Overview

We investigate a hybrid PDE/Monte Carlo technique for the variance reduced simulation of an agent-based multiscale model for tumor growth. The variance reduction is achieved by combining a simulation of the stochastic agent-based model on the microscopic scale with a deterministic solution of a simplified (coarse) partial differential equation (PDE) on the macroscopic scale as a control variable. We show that this technique is able to significantly reduce the variance with only the (limited) additional computational cost associated with the deterministic solution of the coarse PDE. We illustrate the performance with numerical experiments in different regimes, both in the avascular and vascular stage of tumor growth.

Variance-reduced simulation of stochastic agent-based models for tumor growth

We are interested in the mean-field evolution of a growing tumor as it emerges from a stochastic agent-based multiscale model. To this end, we introduce a hybrid PDE/Monte Carlo variance reduction technique. The variance reduction on the cell densities is achieved by combining a simulation of the stochastic agent-based model on the microscopic scale with a deterministic solution of a simplified (coarse) PDE on the macroscopic scale as a control variable. We show that this technique is able to significantly reduce the variance with only the (limited) additional computational cost associated with the deterministic solution of the coarse PDE. We illustrate the performance with numerical experiments in different biological scenarios.

Annelies Lejon

Papers

A High-Order Asymptotic-Preserving Scheme for Kinetic Equations Using Projective Integration

A high-order asymptotic-preserving scheme for kinetic equations using projective integration

Stochastic Modeling and Simulation Methods for Biological Processes: Overview

Variance-reduced simulation of stochastic agent-based models for tumor growth

Variance-Reduced Simulation of Multiscale Tumor Growth Modeling