# Fractional-Order Closed-Loop Model Reference Adaptive Control for Anesthesia
## 3 Anesthesia
### New Modeling Paradigm: Fractional Calculus  
Some researchers had proposed the necessity for a fractal view of physiology that explicitly takes into account the complexity of the living matter and its dynamics. Complexity in this context incorporates recent advances in physiology concerned with the application of concepts from fractal geometry, fractal statistics, and nonlinear dynamics, to the formation of a new kind of understanding within the life sciences.  
The complexity of the human body and the characterization of that complexity through fractal measures and their dynamics involve the use of fractional calculus. Not only anatomical structures are fractal [67], such as the convoluted surface of the brain, the lining of the bowel, neural networks, and the placenta, but the output of dynamical physiologic networks are fractal as well [68].  
The time series for the inter-beat intervals of the heart, inter-breath intervals, and inter-stride intervals have all been shown to be fractal or multifractal statistical phenomena. Consequently, the fractal dimension is a significantly better indicator of organismic functions in health and disease than the traditional average measures. The observation that human physiology is primarily fractal was first made in the 1980s [68].  
Also, in reference [69], it was suggested that from the point of view of fractal physiology, blood flow and ventilation are delivered in a fractal manner in both space and time in a healthy body.  
The control of physiologic variables is one of the goals of medicine, in particular, understanding and controlling physiological networks in order to ensure their proper operation.  
Therefore, it seems reasonable that one novel strategy for modeling the dynamics and control of complex physiologic phenomena is through the application of fractional calculus [69].  
A fundamental mechanism in the absorption of a drug in the human body is the process of diffusion. In reference [17], the authors present the relationship between the diffusion process and fractional-order models. Also, the introduction of fractional-order pharmacokinetic models are presented in references [15; 16; 18; 19] which represent the experimental data in a more precise way, thanks to the \(t^{-\alpha}\) decay of the fractional operators, and with this, a new line of investigation was opened ion the area of drug delivery systems.  
In [15; 70; 71], it was suggested that biological systems (like in pharmacology and bioengineering) could be represented with a fractional-order model with a more simple structure compared with the integer-order counterpart, simplifying the control design by using a less complex model.  
Other interesting applications of fractional calculus are shown in reference [72], where a fractional-order impedance model was used to model the respiratory mechanics. In reference [62] a fractional-order integro-derivative function to model the pain perception in analgesia is presented, and in reference [73] a review of the recent results reported on the model of biological phenomena using fractional calculus is presented.  
Therefore, we can see that fractional calculus can offer us a new point of view to understand certain physical phenomena, especially those which, from the point of view of integer-order systems, seem too complex.