Higher-order models for glioma invasion: From a two-scale description to effective equations for mass density and momentum

Starting from a two-scale description involving receptor binding dynamics and a kinetic transport equation for the evolution of the cell density function under velocity reorientations, we deduce macroscopic models for glioma invasion featuring partial differential equations for the mass density and momentum of a population of glioma cells migrating through the anisotropic brain tissue. The proposed first and higher-order moment closure methods enable numerical simulations of the kinetic equation. Their performance is then compared to that of the diffusion limit. The approach allows for diffusion tensor imaging (DTI)-based, patient-specific predictions of the tumor extent and its dynamic behavior.

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Comparison of Markov Chain and Stochastic Differential Equation Population Models Under Higher-Order Moment Closure Approximations

Stochastic Analysis and Applications ◽

10.1080/07362990903415882 ◽

2010 ◽

Vol 28 (6) ◽

pp. 907-927 ◽

Cited By ~ 11

Author(s):

Amy J. Ekanayake ◽

Linda J. S. Allen

Keyword(s):

Differential Equation ◽

Markov Chain ◽

Stochastic Differential Equation ◽

Higher Order ◽

Population Models ◽

Moment Closure ◽

Order Moment ◽

Closure Approximations ◽

Moment Closure Approximations

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Quasi-Entropy Closure: A Fast and Reliable Approach to Close the Moment Equations of the Chemical Master Equation

10.1101/2021.12.01.470753 ◽

2021 ◽

Author(s):

Vincent Wagner ◽

Benjamin Castellaz ◽

Marco Oesting ◽

Nicole Radde

Keyword(s):

Master Equation ◽

Method Of Moments ◽

Lower Order ◽

Reaction System ◽

Higher Order ◽

Chemical Master Equation ◽

Moment Closure ◽

Order Moment ◽

True Solution ◽

Higher Order Moments

MotivationThe Chemical Master Equation is the most comprehensive stochastic approach to describe the evolution of a (bio-)chemical reaction system. Its solution is a time-dependent probability distribution on all possible configurations of the system. As the number of possible configurations is typically very large, the Master Equation is often practically unsolvable. The Method of Moments reduces the system to the evolution of a few moments of this distribution, which are described by a system of ordinary differential equations. Those equations are not closed, since lower order moments generally depend on higher order moments. Various closure schemes have been suggested to solve this problem, with different advantages and limitations. Two major problems with these approaches are first that they are open loop systems, which can diverge from the true solution, and second, some of them are computationally expensive.ResultsHere we introduce Quasi-Entropy Closure, a moment closure scheme for the Method of Moments which estimates higher order moments by reconstructing the distribution that minimizes the distance to a uniform distribution subject to lower order moment constraints. Quasi-Entropy closure is similar to Zero-Information closure, which maximizes the information entropy. Results show that both approaches outperform truncation schemes. Moreover, Quasi-Entropy Closure is computationally much faster than Zero-Information Closure. Finally, our scheme includes a plausibility check for the existence of a distribution satisfying a given set of moments on the feasible set of configurations. Results are evaluated on different benchmark problems.Abstract Figure

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More on the Performance of Higher Order Moment Estimators in Investment Equations

SSRN Electronic Journal ◽

10.2139/ssrn.1688804 ◽

2010 ◽

Cited By ~ 3

Author(s):

Heitor Almeida ◽

Murillo Campello ◽

Antonio F. Galvao

Keyword(s):

Higher Order ◽

Order Moment ◽

Moment Estimators

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ESTIMATION OF COEFFICIENTS OF AN AUTOREGRESSIVE PROCESS BY USING A HIGHER ORDER MOMENT

Journal of Time Series Analysis ◽

10.1111/j.1467-9892.1981.tb00314.x ◽

1981 ◽

Vol 2 (2) ◽

pp. 87-93 ◽

Cited By ~ 30

Author(s):

Mituaki Huzii

Keyword(s):

Autoregressive Process ◽

Higher Order ◽

Order Moment

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Relationship between the Corticospinal and Corticocerebellar Tracts and Their Role in Upper Extremity Motor Recovery in Stroke Patients

Journal of Personalized Medicine ◽

10.3390/jpm11111162 ◽

2021 ◽

Vol 11 (11) ◽

pp. 1162

Author(s):

Jungsoo Lee ◽

Won Hyuk Chang ◽

Yun-Hee Kim

Keyword(s):

Upper Extremity ◽

Predictive Accuracy ◽

Diffusion Tensor ◽

Stroke Onset ◽

Patient Specific ◽

Stroke Patients ◽

Predictive Values ◽

The Relationship ◽

Diffusion Tensor Imaging Dti

The corticospinal tract (CST) and corticocerebellar tract (CCT) are both involved in the upper extremity (UE) function after stroke. Understanding the relationship between the tracts and their functions can contribute to developing patient-specific rehabilitative strategies. Seventy ischemic stroke patients who underwent diffusion tensor imaging (DTI) two weeks after the stroke onset and motor function assessments two weeks and three months after the stroke onset were included in this study. To obtain the CST and CCT integrity, the functional anisotropy (FA) values of both tracts were extracted from the DTI data. Linear regression was used to identify the relationship and predictive accuracy. The CST FA data had predictive values, but CCT FA did not. There were interaction effects between the CST and CCT FA values (p = 0.011). The CCT was significantly associated with high CST FA but not low CST FA. When the CST or CCT FA were applied to patients depending on the CST status, the stratified model showed higher predictive accuracy (R2 = 0.380) than that of the CST-only model (R2 = 0.320). In this study, the conditional role of CCT depending on CST status was identified in terms of UE recovery in stroke patients. This result could provide useful information about individualized rehabilitative strategies in stroke patients.

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