MBE
Modeling and simulation for toxicity assessment
Cristina Anton Jian Deng Yau Shu Wong Yile Zhang Weiping Zhang Stephan Gabos Dorothy Yu Huang Can Jin
Mathematical Biosciences & Engineering 2017, 14(3): 581-606 doi: 10.3934/mbe.2017034

The effect of various toxicants on growth/death and morphology of human cells is investigated using the xCELLigence Real-Time Cell Analysis High Troughput in vitro assay. The cell index is measured as a proxy for the number of cells, and for each test substance in each cell line, time-dependent concentration response curves (TCRCs) are generated. In this paper we propose a mathematical model to study the effect of toxicants with various initial concentrations on the cell index. This model is based on the logistic equation and linear kinetics. We consider a three dimensional system of differential equations with variables corresponding to the cell index, the intracellular concentration of toxicant, and the extracellular concentration of toxicant. To efficiently estimate the model's parameters, we design an Expectation Maximization algorithm. The model is validated by showing that it accurately represents the information provided by the TCRCs recorded after the experiments. Using stability analysis and numerical simulations, we determine the lowest concentration of toxin that can kill the cells. This information can be used to better design experimental studies for cytotoxicity profiling assessment.

keywords: Mathematical model cytotoxicity parameter estimation persistence
MBE
Novel stability results for traveling wavefronts in an age-structured reaction-diffusion equation
Ming Mei Yau Shu Wong
Mathematical Biosciences & Engineering 2009, 6(4): 743-752 doi: 10.3934/mbe.2009.6.743
For a time-delayed reaction-diffusion equation of age-structured single species population, the linear and nonlinear stability of the traveling wavefronts were proved by Gourley [4] and Li-Mei-Wong [8] respectively. The stability results, however, assume the delay-time is sufficiently small. We now prove that the wavefronts remain stable even when the delay-time is arbitrarily large. This essentially improves the previous stability results obtained in [4, 8]. Finally, we point out that this novel stability can be applied to the age-structured reaction-diffusion equation with a more general maturation rate.
keywords: traveling wavefronts time-delayed reaction-diffusion equation exponential decay rate. nonlinear stability
DCDS-B
Phase transitions in a coupled viscoelastic system with periodic initial-boundary condition: (I) Existence and uniform boundedness
Ming Mei Yau Shu Wong Liping Liu
Discrete & Continuous Dynamical Systems - B 2007, 7(4): 825-837 doi: 10.3934/dcdsb.2007.7.825
This paper focuses on the phase transitions of a 2$\times$2 system of mixed type for viscosity-capillarity with periodic initial-boundary condition in a viscoelastic material. By the Liapunov functional method, we prove the existence, uniqueness, regularity and uniform boundedness of the solution. The results are correct even for large initial data.
keywords: global existence viscoelasticity uniform boundedness Phase transitions periodic initial-boundary value problem Liapunov functional.
DCDS-B
Phase transitions in a coupled viscoelastic system with periodic initial-boundary condition: (II) Convergence
Ming Mei Yau Shu Wong Liping Liu
Discrete & Continuous Dynamical Systems - B 2007, 7(4): 839-857 doi: 10.3934/dcdsb.2007.7.839
We present some new results on the asymptotic behavior of the periodic solution to a 2$\times$2 mixed-type system of viscosity-capillarity in a viscoelastic material. We prove that the solution converges to a certain stationary solution as time approaches to infinity, in particular, when the viscosity is large enough or the mean of the initial datum is in the hyperbolic regions, the solution converges exponentially to the trivial stationary solution with it any large initial datum. The location of the initial datum and the amplitude of viscosity play a key role for the phase transitions. Furthermore, we obtain the convergence rate to the stationary solutions. Finally, we carry out numerical simulations to confirm the theoretical predictions.
keywords: asymptotic convergence. periodic initial-boundary value problem stationary solutions Phase transitions viscoelasticity
MBE
Nonlinear stability of traveling wavefronts in an age-structured reaction-diffusion population model
Guangrui Li Ming Mei Yau Shu Wong
Mathematical Biosciences & Engineering 2008, 5(1): 85-100 doi: 10.3934/mbe.2008.5.85
The paper is devoted to the study of a time-delayed reaction- diffusion equation of age-structured single species population. Linear stability for this model was first presented by Gourley [4], when the time delay is small. Here, we extend the previous result to the nonlinear stability by using the technical weighted-energy method, when the initial perturbation around the wavefront decays to zero exponentially as x→-∞, but the initial perturbation can be arbitrarily large on other locations. The exponential convergent rate (in time) of the solution is obtained. Numerical simulations are carried out to confirm the theoretical results, and the traveling wavefronts with a large delay term in the model are reported.
keywords: time-delayed reaction-diffusion equation exponential decay rate. traveling wavefronts non-linear stability

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