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HOPF POINT ANALYSIS FOR ANGIOGENESIS MODELS

HOPF POINT ANALYSIS FOR ANGIOGENESIS MODELS,Z. AGUR,L. ARAKELYAN,P. DAUGULIS,Y. GINOSAR

HOPF POINT ANALYSIS FOR ANGIOGENESIS MODELS   (Citations: 29)
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In this paper we present several ODE systems encoding the most essential observations and assumptions about the complex hierarchical inter- active processes of tumor neo-vascularization (angiogenesis). From experi- mental results we infer that a significant marker of tumor aggressiveness is the oscillatory behavior of tumor size, which is driven by its vascularization dynamics. To study the forces underlying these oscillations we perform a Hopf point analysis of the angiogenesis models. In the analyzed models Hopf points appear if and only if a nontrivial set of time-delays is introduced into the tu- mor proliferation or the neo-vascularization process. We suggest to examine in laboratory experiments how to employ these results for containing cancer growth.
Published in 2004.
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    • ...Rational and systematic planning of anticancer therapy may be directed by mathematical modeling and computer-aided analysis, which provides a better understanding of the involved dynamics. Over the past 25 years, mathematical modeling strategies have been applied in oncology-focused studies investigating tumor progression, angiogenesis and interactions with the immune system ...

    Moran Elishmereniet al. An Integrated Disease/Pharmacokinetic/Pharmacodynamic Model Suggests I...

    • ...For example, in [33], we have introduced and analysed several alternative general tumor angiogenesis models (with/without vascular maturation or with/without the effects of growth factors)...
    • ...While it is recognized that timedelay will often elicit Hopf points, in [33], it was shown that the latter were to be found in any angiogenesis model with time-delay...

    IJ Stamperet al. Oscillatory dynamics in a model of vascular tumour growth - implicatio...

    • ...The continuous models generally deflne three major processes: tumor growth, vascularization and protein proliferation which have been fully analyzed in [1,9].The dynamics of vessel compartments is assumed to be a superposition of four processes, some are contrasting some of the others: formation of immature vessels, regression of immature vessels, maturation of immature vessels, and destabilization of mature vessels into immature vessels...
    • ...It was shown in [1,4,7] that for models without time delays, no Hopf points occur...
    • ...The time dependent variables of angiogenesis models are the number of tumor cells or tumor size denoted N, the amount of growth factors like proteins P supplying the tumor, the volume of blood in mature and immature vessels feeding the tumor denoted V . A three-dimensional model with no time delay is given by the following equations [1,2,4,9]...
    • ...1 1+ek(x+s) [1,4] of Michaels-Menten type dx 2...
    • ...This model is modifled in [1] by assuming that the protein is consumed by growing vessels...
    • ...It was shown in [1,4] that there exist no Hopf points in these models and the oscillations observed in the experiments can not be detected in the models without time delays like (1)...
    • ...After introducing time delays, the three-dimensional model (1) becomes [1,4,7,9]...
    • ...In order to make the models more realistic, more compartments are introduced for representing complex vascularity and protein efiects[1,9]...
    • ...The protein denoted by P is replaced by two speciflc proteins, namely vascular endothelial growth factor(VGEF), denoted P1 and angiopoietin-1(Ang1), denoted P2. It was assumed in [1,9] that VEGF is produced by the tumor at a rate sigmoidally dependent on the efiective vessel density and decays at a constant rate ‐1, and that Ang1 is produced by the tumor at a constant rate fi and decays at a constant rate ‐2. The extension of the three ...
    • ... vascular endothelial growth factor(VGEF), denoted P1 and angiopoietin-1(Ang1), denoted P2. It was assumed in [1,9] that VEGF is produced by the tumor at a rate sigmoidally dependent on the efiective vessel density and decays at a constant rate ‐1, and that Ang1 is produced by the tumor at a constant rate fi and decays at a constant rate ‐2. The extension of the three dimensional (5) to flve dimensional models with time delays in [1,9] are ...
    • ...It was demonstrated in [1,9] that the ode systems (5) and (6) admit a Hopf bifurcation point if and only if at least one of the delays is non-zero...
    • ...The basic assumptions taken from the literature to build our model are as follows [1,2]:...
    • ...We flrst compared the our hybrid piecewise linear model(HDS) with the three variable delayed difierential equation(DDE) model in [1] for the tumor size...
    • ...In Figure 2 we have the simulation results for the tumor size plotted, where dashed lines represents the numerical results for the DDE model in [1], the solid line the results according to our model HDS...

    Bülent Karasözenet al. A model of angiogenesis by hybrid systems with delay on the piecewise ...

    • ...Mathematical models for tumor progression (4-6), cancer therapy (7-10), and related...
    • ...assumptions is described in detail elsewhere (4, 5, 24)...

    Boris Goreliket al. Efficacy of Weekly Docetaxel and Bevacizumab in Mesenchymal Chondrosar...

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