Juvencio Alberto Betancourt-Mar

The Rössler system as a model for chronotherapy

The biological systems are opened and are kept far from thermodynamics equilibrium. For these reasons, biological systems are always exposed to external perturbations, which may produce alterations on these rhythms as a consequence of coupling synchronization of the autonomous oscillator with perturbation. Coupling of therapeutic perturbations, such as drugs and radiation, on biological systems delivery to biological rhythms is known as chronotherapy. We used the Rossler system as a theoretical model for chronotherapy, generalized this formalism for chaotic behaviour. We found that when the Rossler is more dissipative, such as c increase, the systems become more robust to the perturbations.

Theoretical models for chronotherapy: periodic perturbations in hyperchaos.

In this work, a hyperchaotic system was used as a model for chronotherapy. We applied a periodic perturbation to a variable, varying the period and amplitude of forcing. The system, five-dimensional, has until three positive Lyapunov exponents. As a result, we get small periodical windows, but it was possible to get large areas of hyperchaos of two positive Lyapunov exponents from a chaotic behavior. In this chronotherapy model, chaos could be considered as a dynamical disease, and therapy goal must be to restore the hyperchaotic state.

Theoretical models in chronotherapy: II. Periodic perturbations in a chaotic chemical reaction

Abstract In this work, the Rössler system is used as a model for chronotherapy. We applied a periodic perturbation to the y variable to take the Rössler system from a chaotic behaviour to a simple periodic one, varying the period and amplitude of forcing. Some periodical windows and period-doubling cascades are observed. The most important of them is the large period-1 area around T 0 = 6.154 (the period corresponding to the dominant frequency in the power spectrum). This is a wide region as is desired in a chronotherapy model, where the therapy is administered, with ample tolerance, to transform a chaotic behaviour into a periodic one.

Chronotherapy of cancer: periodic perturbations in vascular growth and metastasis

Abstract A non-autonomous model was developed for vascular tumor growth and cancer metastasis under periodic perturbations that simulate chronotherapy. It was found that for a critical amplitude and a perturbation frequency of twice the autonomous frequency in vascular growth, less complex and therefore less robust states are reached and the tumor population decreases. In metastasis, similar results were found for a perturbation frequency equal to the autonomous frequency. In metastasis, similar behaviors occur, for a perturbation’s frequency equal to the fundamental frequency of system.

Dose and frequency in cancer therapy. Theoretical non-autonomous model of p53 network

Abstract A non-autonomous model for the regulation of apoptosis by p53 is proposed as a model for cancer chronotherapy. A perturbation is introduced that simulates damage caused to DNA in a periodic regime. To characterize the resulting system dynamics, the techniques used were stroboscopic analysis, Poincare section and power spectrum. The complexity of the time series was determined using the LZ index. Periodic and quasi-periodic dynamics were obtained as the control parameters were varied. The less complex states are those corresponding to higher values of the amplitude which indicates a strict control of the dose is required on periodic treatments.

Mechanisms of bands and spirals formation during the drying of watery solutions of mercury (II) chloride with agar-agar

It is proposed two mechanisms to explain the formation of periodic and non periodic bands and spirals as thin films of gelatinous aqueous solutions of mercury (II) chloride are dried. The first mechanism supposes an homogeneous drying, where the height of the film decreases at constant rate, forming Liesegang bands. The second mechanism implies a non homogeneous drying where an evaporation front drives the formation of periodic bands and spirals.