Shocks in nonlinear diffusion

Witelski, Thomas P.

doi:10.1016/0893-9659(95)00062-u

Cited by 38 publications

(62 citation statements)

References 7 publications

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“…The ability to define relevant individual mechanisms in a random walk process, such as motility, proliferation, death and agent-agent adhesion, makes random walks a flexible modelling tool [72]. The stochastic nature of the random walk processes means that it can be computationally intensive to determine the average behaviour of the random walk, particularly for processes that are spatially heterogeneous.…”

Section: Overviewmentioning

confidence: 99%

“…More generally, throughout the physical and life sciences, lattice-based random walks are commonly employed to model collective behaviour [13,16,17,31,33,37,51,72]. For example, both Deroulers et al [16] and Khain et al [37] use a lattice-based random walk to describe the migration of glioma cells.…”

Section: Overviewmentioning

confidence: 99%

“…Due to these shortfalls, deterministic continuum approximations of the average random walk behaviour have been proposed [6,16,21,37,51,70,72,73]. The most common continuum description is a mean-field description, which involves an assumption that the spatial correlations between lattice sites are negligible [16,37,72].…”

Section: Overviewmentioning

confidence: 99%

“…Obtaining reliable predictions of the direction and speed of the moving front is critical for informing biological and ecological control measures. Lattice-based random walk processes provide a flexible framework for modelling moving fronts [16,21,33,37,72], and allow for the implementation of birth, death and movement parameters that depend on the size of the group of contiguous occupied sites that an agent belongs to.…”

Section: Research Questionsmentioning

confidence: 99%

“…The most common continuum description is a mean-field description, which involves an assumption that the spatial correlations between lattice sites are negligible [16,37,72]. Mean-field descriptions of random walk processes have been used to model glioma cell migration [16,37] and the migration of neural crest cells [72]. However, the assumption that the spatial correlations are negligible is only valid in an extremely limited set of parameter regimes [6,35,70].…”

Section: Overviewmentioning

confidence: 99%

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Mathematical Models for Quantifying Collective Cell Behaviour

Johnston¹

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Collective behaviour is critical to a variety of biological and ecological processes, including tumour invasion, wound healing and the spread of invasive species. Mathematical modelling techniques provide an opportunity for obtaining insight into the underlying mechanisms governing collective behaviour. Furthermore, mathematical models can be interfaced with experimental data to enhance quantitative information obtained from experiments. The aim of this thesis is two-fold. First, to investigate the application of mathematical models to experimental data to obtain robust estimates of the parameters governing collective behaviour. Second, to develop novel continuum mathematical descriptions of individual-based models of birth, death and movement.For the first part of this thesis, I begin by examining the application of a well-known mathematical model of cell motility and cell proliferation, the Fisher-Kolmogorov model, to IncuCyte ZOOM TM assay data for a prostate cancer (PC-3) cell population. Standard techniques used to interpret IncuCyte ZOOM TM assay data do not provide insight about the individual contributions of cell motility and cell proliferation to the overall migration of the cell population. I find that by combining experimental measurements of the evolution of the position of the leading edge of the cell population and the evolution of the cell density away from the leading edge, I am able to obtain unique estimates of the cell di↵u-sivity, cell proliferation rate and cell carrying capacity density. Furthermore, I am able to quantify how these parameters are influenced by the presence of varying concentrations of epidermal growth factor.Next, I investigate whether the position of the leading edge of a cell population in a scratch assay can be combined with an appropriate mathematical model to provide unique estimates of the cell motility rate and the cell proliferation rate. Leading edge data is a commonly-reported experimental measurement from scratch assays, which are typically interpreted in a qualitative fashion or with a quantitative technique that does not isolate the individual roles of motility and proliferation. I implement a lattice-based random walk model of motility and proliferation, mimic the geometry of a scratch assay, and combine this mathematical model with experimental leading edge data through an automated edge detection algorithm. I find that, provided the di↵erence in the time scales of cell proliferation and cell motility is accounted for, this technique produces unique estimates of the cell di↵usivity and cell proliferation rate.I then consider an approximate Bayesian computation (ABC) parameter recovery approach and examine which experimental measurements from a scratch assay provide the most information about the cell di↵usivity and cell proliferation rate parameters. ABC i techniques provide parameter distributions rather than point-estimates and hence contain quantitative information about the uncertainty associated with the parameter estimates. I find that an ABC a...

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Section: Overviewmentioning

confidence: 99%

Section: Overviewmentioning

confidence: 99%

Section: Overviewmentioning

confidence: 99%

Section: Research Questionsmentioning

confidence: 99%

Section: Overviewmentioning

confidence: 99%

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Mathematical Models for Quantifying Collective Cell Behaviour

Johnston¹

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The Structure of Internal Layers for Unstable Nonlinear Diffusion Equations

Witelski

1996

Stud Appl Math

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We study the structure of diffusive layers in solutions of unstable nonlinear diffusion equations. These equations are regularizations of the forward‐backward heat equation and have diffusion coefficients that become negative. Such models include the Cahn‐Hilliard equation and the pseudoparabolic viscous diffusion equation. Using singular perturbation methods we show that the balance between diffusion and higher‐order regularization terms uniquely determines the interface structure in these equations. It is shown that the well‐known “equal area” rule for the Cahn‐Hilliard equation is a special case of a more general rule for shock construction in the viscous Cahn‐Hilliard equation.

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Mechanics in Tumor Growth

Graziano

Preziosi

Modeling of Biological Materials

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Shocks in nonlinear diffusion

Cited by 38 publications

References 7 publications

Mathematical Models for Quantifying Collective Cell Behaviour

Mathematical Models for Quantifying Collective Cell Behaviour

The Structure of Internal Layers for Unstable Nonlinear Diffusion Equations

Mechanics in Tumor Growth

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