Mathematical models of the cell cycle with a view to tumor studies

Bertuzzi, Alessandro; Gandolfi, Alberto; Giovenco, Maria Adelaide

doi:10.1016/0025-5564(81)90017-1

Cited by 49 publications

(10 citation statements)

References 44 publications

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“…Our goal is to show the asynchronous exponential growth (AEG) property of the solutions of (2)- (4). To this end, the above equations can be rewritten as…”

Section: Formulation Of the Modelmentioning

confidence: 99%

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The asynchronous exponential growth property in a model for the kinetic heterogeneity of tumour cell populations

Arino¹,

Bertuzzi

Gandolfi

et al. 2005

Journal of Mathematical Analysis and Applications

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A continuous cell population model, which represents both the cell cycle phase structure and the kinetic heterogeneity of the population following Shackney's ideas [J. Theor. Biol. 38 (1973) 305-333], is studied. The asynchronous exponential growth property is proved in the framework of the theory of strongly continuous semigroups of bounded linear operators.  2004 Elsevier Inc. All rights reserved.

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“…Our goal is to show the asynchronous exponential growth (AEG) property of the solutions of (2)- (4). To this end, the above equations can be rewritten as…”

Section: Formulation Of the Modelmentioning

confidence: 99%

“…In the framework of deterministic models, the kinetic heterogeneity has been mainly represented by means of age-structured population models [1,2,4,15]. The age formalism, indeed, allows a simple representation of cell populations with variable (but uncorrelated) cell cycle times.…”

Section: Introductionmentioning

confidence: 99%

The asynchronous exponential growth property in a model for the kinetic heterogeneity of tumour cell populations

Arino¹,

Bertuzzi

Gandolfi

et al. 2005

Journal of Mathematical Analysis and Applications

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“…If not, but if α(x) = α, β(x) = β and γ (x) = γ for any age x, we consider the second-order PDE 19a) where γ represents the maturation acceleration (Bertuzzi et al 1981;Demongeot et al 1995). The main interest of the continuous formulation (2.19a) is the possibility to add a diffusion term, when u(x, t, s) depends on age x, time t and space s…”

Section: (C) Discrete Approachmentioning

confidence: 99%

“…In the class of models proposed by Takahashi (1966Takahashi ( , 1968, Hahn (1970) and Bertuzzi et al (1981), the physiological state or age of a cell is a discrete variable i and the cellular dynamics is parametrized by discrete entities depending on the discrete time t. The state density of the cell population at time t is described by…”

Section: (C) Discrete Approachmentioning

confidence: 99%

“…The five phases of the cell cycle can be further decomposed into different states, by taking into account critical control points sensitive to the regulation in the frame of genetic networks or to external or internal metabolite, morphogen or antibody fluctuations (e.g. Daszynkiewiez et al 1980;Bertuzzi et al 1981;Cohen et al 2001;Cinquin & Demongeot 2002a,b, 2005Aracena et al 2003Aracena et al , 2004aBasse et al 2003;Demongeot et al 2003a-c;Aracena & Demongeot 2004;Baum et al 2004Baum et al , 2006Laforge et al 2005;Guardavaccaro & Pagano 2006) accelerating the progress to the mitosis or provoking either cell death (apoptosis) or cell differentiation (a dramatic change of cell lineage).…”

Section: Cell Cycle Dynamics: Proliferation Growth and Apoptosis (A)mentioning

confidence: 99%

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Synchrony in reaction–diffusion models of morphogenesis: applications to curvature-dependent proliferation and zero-diffusion front waves

Abbas

Demongeot

Glade

2009

Phil. Trans. R. Soc. A.

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The paper presents the classical age-dependent approach of the morphogenesis in the framework of the von Foerster equation, in which we introduce a new constraint and study a new feature: (i) the new constraint concerns cell proliferation along the contour lines of the cell density, depending on the local curvature such as it favours the amplification of the concavities (like in the gastrulation process) and (ii) the new feature consists of considering, on the cell density surface, a remarkable line (the null mean Gaussian curvature line), on which the normal diffusion vanishes, favouring local coexistence of diffusing morphogens, metabolites or cells, and hence the auto-assemblages of these entities. Two applications to biological multi-agents systems are studied, gastrulation and feather morphogenesis.

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The S‐Distribution A Tool for Approximation and Classification of Univariate, Unimodal Probability Distributions

Voit¹

1992

Biometrical J

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SummaryIn many statistical applications a data set needs to be evaluated but there is no solid information about which probability distribution might be most appropriate. Typical solutions to this problems are: to make assumptions that facilitate mathematical treatment; to use a family of distribution functions that contains all relevant distributions as special cases; or, to employ nonparametric methods. All three solutions have disadvantages since assumptions are usually difficult to justify, families of distributions contain too many parameters to be of practical use, and nonparametric methods make it difficult to characterize data in a succinct quantitative form. The S-distribution introduced here is a compromise between the conflicting goals of simplicity in analysis and generality in scope. I t is characterized by four parameters, one of which reflects its location, the second one its spread, and the remaining two its shape; transformation to a standard form reduces the number of free parameters to two. Cumulatives and densities are computed numerically in fractions of seconds, key features like quantiles and moments are easily obtained, and results can be presented in terms of parameter values. The S-distribution rather accurately models different distribution functions, including central and noncentral distributions, and thus competes in flexibility with some distribution families. As an approximation, the S-distribution provides a graphical method for demonstrating relationships between distributions, such as the relationships between central F, xz and x-'. or central and noncentral t, I-', and normal.

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Mathematical models of the cell cycle with a view to tumor studies

Cited by 49 publications

References 44 publications

The asynchronous exponential growth property in a model for the kinetic heterogeneity of tumour cell populations

The asynchronous exponential growth property in a model for the kinetic heterogeneity of tumour cell populations

Synchrony in reaction–diffusion models of morphogenesis: applications to curvature-dependent proliferation and zero-diffusion front waves

The S‐Distribution A Tool for Approximation and Classification of Univariate, Unimodal Probability Distributions

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