There is, I think, something ethereal about i —the square root of minus one. I remember first hearing about it at school. It seemed an odd beast at that time—an intruder hovering on the edge of reality.

Usually familiarity dulls this sense of the bizarre, but in the case of i it was the reverse: over the years the sense of its surreal nature intensified. It seemed that it was impossible to write mathematics that described the real world in …

I and i

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Formulas For Natural Frequency And Mode Shape

The Viscosity can be imagined as the force that should be applied to a layer of fluid belonging to the plane fixed to the velocity of the layer placed at a fixed distance. For example a fluid flowing in a tube at different speeds: the minimum speed is in the edge of the section (due to friction) and the maximum speed is at the center. The viscosity in this example is that pressure which is exerted on the wall allows a constant speed on the whole section.

Non-newtonian fluids.

Explicit formula and graphs of few special functions are derived in the paper on the basis of various definitions of various fractional derivatives and various fractional integrals. Their applications are also reviewed in the paper.

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Applications of Fractional Calculus

Linear fractional order controllers; A survey in the frequency domain

In this paper we analyze the opportunity to provide a methodology and signal processing algorithm to model and quantify drug concentrations in blood. The properties of blood as a non-Newtonian fluid are used to develop a physiologically based model using fractional calculus tools, which provide natural solutions to derivatives which no longer limit their order to integer numbers. In vitro tests support our mathematical model development. A prospective use for continuous monitoring of drug concentration profiles during anesthesia or other interventions is proposed. The proposed solution is based on fractional order impedance models.

Fractional-order modeling of impedance measurements in a blood-resembling experimental setup *

Tissue impedance can be an evaluation factor for noxious stimulation in humans. It has been shown in several studies that bioimpedance can be used as a direct evaluation tool of pain. In this study, we evaluated the changes in perception of different stimuli (mechanical and thermal) applied to subjects in a safe laboratory environment. The measurements are performed with the in-house developed pain monitoring device ANSPEC-PRO, employing impedance evaluation as a function of time and frequency. In this paper the following hypotheses have been investigated: i) the time-frequency response is nonlinear and independent of stimuli type, ii) there are differences in energy between mechanical and thermal-induced pain as indicated by impedance changes.

Time-frequency varying tissue impedance properties from noxious stimulation protocols

Targeting a high-precision treatment in lung cancer assumes a good knowledge of the mechanisms that underlie tumor dynamics and drug kinetics. In this work, we consider a mathematical model for tumor radiotherapy and chemotherapy as an optimal solution for local tumor control. The prediction model encompasses the understanding of cancer biology and radiation response, incorporating dose fractionation, tumor volume, and therapies effect. The results are commonly found in practice and stress the importance of computer simulations as a tool for adjusting the therapy in the pre-treatment phase if needed. Significant gain in curative cancer therapy can be achieved by applying mathematically based treatment planning and obtaining a predicted outcome for a specific treatment schedule.

Optimizing radiotherapy with chemotherapy using PKPD modeling for lung cancer *

Computer-based control of anesthesia is a puzzle requiring multiple other states to be suitably monitored and controlled. Among these, hemodynamic stabilization is a must. Very few studies have been reported regarding the automatic control of the combined anesthesia-hemodynamic systems. Hemodynamic states are strongly affected by hypnotic drugs. At the same time, drugs used for hemodynamic control influence the depth of hypnosis. In this paper, a solution to maintain anesthetic and hemodynamic variables within safe operating ranges is presented and consists in a multivariable fractional order controller. The robustness of the proposed solution is analysed by considering a benchmark patient model and data from 24 patients.

Robustness analysis of a fractional order control system for the hemodynamic variables in anesthetized patients

Recently, the fractional-order element has been integrated into the Robust Control Framework considering the Oustaloup method. As such, the resulting infinite impulse response approximation manages to satisfy the robust stability and the robust performance criteria according to a given uncertainty block. However, the recommended approximation order for each fractional-order element is the number of decades of the frequency range where the approximation is valid, which can lead to a high-order controller. The current paper describes an optimization-based technique to find a low-order approximation of a fractional-order controller such that the resulting controller maintains the robust stability and robust performance as well. A set of numerical experiments have also been performed in order to illustrate the proposed method.

Isabela Roxana Birs

Papers

Fractional-order modeling of impedance measurements in a blood-resembling experimental setup *

Time-frequency varying tissue impedance properties from noxious stimulation protocols

Optimizing radiotherapy with chemotherapy using PKPD modeling for lung cancer *

Robustness analysis of a fractional order control system for the hemodynamic variables in anesthetized patients

Reduced-Order Approximation of Fractional-Order Controllers by keeping Robust Stability and Robust Performance