Bifurcation analysis on a hybrid systems model of intermittent hormonal therapy for prostate cancer

Tanaka, Gouhei; Tsumoto, Kunichika; Tsuji, Shigeki; Aihara, Kazuyuki

doi:10.1016/j.physd.2008.03.044

Cited by 40 publications

(30 citation statements)

References 39 publications

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“…This is an idea which has been used in different fields, in order to control physical [12], physiological [13], and clinical [14] systems. In Gibsonian terms [15] we may say that, in spite of the instability of the saddle-type equilibrium, the stable manifold of the saddle is an “affordance” that a smart agent is supposed to exploit in order to simplify the control problem and provide a more economical solution: indeed part of the job can be performed by the “environment” not the controlling agent.…”

Section: Discussionmentioning

confidence: 99%

A Model of Postural Control in Quiet Standing: Robust Compensation of Delay-Induced Instability Using Intermittent Activation of Feedback Control

et al. 2009

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The main purpose of this study is to compare two different feedback controllers for the stabilization of quiet standing in humans, taking into account that the intrinsic ankle stiffness is insufficient and that there is a large delay inducing instability in the feedback loop: 1) a standard linear, continuous-time PD controller and 2) an intermittent PD controller characterized by a switching function defined in the phase plane, with or without a dead zone around the nominal equilibrium state. The stability analysis of the first controller is carried out by using the standard tools of linear control systems, whereas the analysis of the intermittent controllers is based on the use of Poincaré maps defined in the phase plane. When the PD-control is off, the dynamics of the system is characterized by a saddle-like equilibrium, with a stable and an unstable manifold. The switching function of the intermittent controller is implemented in such a way that PD-control is ‘off’ when the state vector is near the stable manifold of the saddle and is ‘on’ otherwise. A theoretical analysis and a related simulation study show that the intermittent control model is much more robust than the standard model because the size of the region in the parameter space of the feedback control gains (P vs. D) that characterizes stable behavior is much larger in the latter case than in the former one. Moreover, the intermittent controller can use feedback parameters that are much smaller than the standard model. Typical sway patterns generated by the intermittent controller are the result of an alternation between slow motion along the stable manifold of the saddle, when the PD-control is off, and spiral motion away from the upright equilibrium determined by the activation of the PD-control with low feedback gains. Remarkably, overall dynamic stability can be achieved by combining in a smart way two unstable regimes: a saddle and an unstable spiral. The intermittent controller exploits the stabilizing effect of one part of the saddle, letting the system evolve by alone when it slides on or near the stable manifold; when the state vector enters the strongly unstable part of the saddle it switches on a mild feedback which is not supposed to impose a strict stable regime but rather to mitigate the impending fall. The presence of a dead zone in the intermittent controller does not alter the stability properties but improves the similarity with biological sway patterns. The two types of controllers are also compared in the frequency domain by considering the power spectral density (PSD) of the sway sequences generated by the models with additive noise. Different from the standard continuous model, whose PSD function is similar to an over-damped second order system without a resonance, the intermittent control model is capable to exhibit the two power law scaling regimes that are typical of physiological sway movements in humans.

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Section: Discussionmentioning

confidence: 99%

A Model of Postural Control in Quiet Standing: Robust Compensation of Delay-Induced Instability Using Intermittent Activation of Feedback Control

et al. 2009

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“…With considerable data on the time course of serum prostate-specific antigen (PSA) levels during therapy at our disposal (Bruchovsky et al, 2006(Bruchovsky et al, , 2007(Bruchovsky et al, , 2008, we have used this information to test the utility of our mathematical model in forecasting the outcome of androgen ablation therapy focusing especially on IAS. The effects of hormone treatment of prostate cancer have been described mathematically in previous publications (Jackson, 2004a, b;Ideta et al, 2008;Tanaka et al, 2008;Shimada and Aihara, 2008) but this is the first model to be validated by clinical data.…”

Section: Introductionmentioning

confidence: 99%

Development of a mathematical model that predicts the outcome of hormone therapy for prostate cancer

Hirata

Bruchovsky

Aihara

2010

Journal of Theoretical Biology

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114

150

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“…Deterministic hybrid systems that integrate continuous dynamics with discrete events have been used to model a wide array of biological processes that exhibit switching behaviour (Belta et al 2004;Ghosh & Tomlin 2004;Lincoln & Tiwari 2004;Tanaka et al 2008). However, deterministic frameworks often fail to capture biochemical processes within living cells where low population counts of mRNAs and proteins can set the stage for significant stochastic effects (Raj & van Oudenaarden 2008).…”

Section: Introductionmentioning

confidence: 99%

Stochastic hybrid systems for studying biochemical processes

Singh

Hespanha

2010

Phil. Trans. R. Soc. A.

126

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Many protein and mRNA species occur at low molecular counts within cells, and hence are subject to large stochastic fluctuations in copy numbers over time. Development of computationally tractable frameworks for modelling stochastic fluctuations in population counts is essential to understand how noise at the cellular level affects biological function and phenotype. We show that stochastic hybrid systems (SHSs) provide a convenient framework for modelling the time evolution of population counts of different chemical species involved in a set of biochemical reactions. We illustrate recently developed techniques that allow fast computations of the statistical moments of the population count, without having to run computationally expensive Monte Carlo simulations of the biochemical reactions. Finally, we review different examples from the literature that illustrate the benefits of using SHSs for modelling biochemical processes.

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Bifurcation analysis on a hybrid systems model of intermittent hormonal therapy for prostate cancer

Cited by 40 publications

References 39 publications

A Model of Postural Control in Quiet Standing: Robust Compensation of Delay-Induced Instability Using Intermittent Activation of Feedback Control

A Model of Postural Control in Quiet Standing: Robust Compensation of Delay-Induced Instability Using Intermittent Activation of Feedback Control

Development of a mathematical model that predicts the outcome of hormone therapy for prostate cancer

Stochastic hybrid systems for studying biochemical processes

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