Fractional Dynamics and Control of Distributed Parameter Systems

Fractional Calculus (FC) goes back to the beginning of the theory of differential calculus. Nevertheless, the application of FC just emerged in the last two decades, due to the progress in the area of chaos that revealed subtle relationships with the FC concepts. In the field of dynamical systems theory some work has been carried out but the proposed models and algorithms are still in a preliminary stage of establishment. Having these ideas in mind, the paper discusses a FC perspective in the study of the dynamics and control of some distributed parameter systems.

Symmetry and order parameter dynamics of the human odometer

Bipedal gaits have been classified on the basis of the group symmetry of the minimal network of identical differential equations (alias cells) required to model them. Primary bipedal gaits (e.g., walk, run) are characterized by dihedral symmetry, whereas secondary bipedal gaits (e.g., gallop-walk, gallop- run) are characterized by a lower, cyclic symmetry. This fact has been used in tests of human odometry (e.g., Turvey et al. in P Roy Soc Lond B Biol 276:4309–4314, 2009, J Exp Psychol Hum Percept Perform 38:1014–1025, 2012). Results suggest that when distance is measured and reported by gaits from the same symmetry class, primary and secondary gaits are comparable. Switching symmetry classes at report compresses (primary to secondary) or inflates (secondary to primary) measured distance, with the compression and inflation equal in magnitude. The present research (a) extends these findings from overground locomotion to treadmill locomotion and (b) assesses a dynamics of sequentially coupled measure and report phases, with relative velocity as an order parameter, or equilibrium state, and difference in symmetry class as an imperfection parameter, or detuning, of those dynamics. The results suggest that the symmetries and dynamics of distance measurement by the human odometer are the same whether the odometer is in motion relative to a stationary ground or stationary relative to a moving ground.

Adhesive thickness effects of a ductile adhesive by optical measurement techniques

Adhesive bonding is an excellent alternative to traditional joining techniques such as welding, mechanical fastening or riveting. However, there are many factors that have to be accounted for during joint design to accurately predict the joint strength. One of these is the adhesive layer thickness (tA). Most of the results are for epoxy structural adhesives, tailored to perform best with small values of tA, and these show that the lap joint strength decreases with increase of tA (the optimum joint strength is usually obtained with tA values between 0.1 and 0.2 mm). Recently, polyurethane adhesives were made available in the market, designed to perform with larger tA values, and whose fracture behaviour is still not studied. In this work, the effect of tA on the tensile fracture toughness (View the MathML source) of a bonded joint is studied, considering a novel high strength and ductile polyurethane adhesive for the automotive industry. This work consists on the fracture characterization of the bond by a conventional and the J-integral techniques, which accurately account for root rotation effects. An optical measurement method is used for the evaluation of crack tip opening (δn) and adherends rotation at the crack tip (θo) during the test, supported by a Matlab® sub-routine for the automated extraction of these parameters. As output of this work, fracture data is provided in traction for the selected adhesive, enabling the subsequent strength prediction of bonded joints.

Tensile fracture characterization of adhesive joints by standard and optical techniques

The use of adhesive joints has increased in recent decades due to its competitive features compared with traditional methods. This work aims to estimate the tensile critical strain energy release rate (GIC) of adhesive joints by the Double-Cantilever Beam (DCB) test. The J-integral is used since it enables obtaining the tensile Cohesive Zone Model (CZM) law. An optical measuring method was developed for assessing the crack tip opening (δn) and adherends rotation (θo). The proposed CZM laws were best approximated by a triangular shape for the brittle adhesive and a trapezoidal shape for the two ductile adhesives.

Optical measurements of rat muscle samples under treatment with ethylene glycol and glucose

With the objective to study the variation of optical properties of rat muscle during optical clearing, we have performed a set of optical measurements from that kind of tissue. The measurements performed were total transmittance, collimated transmittance, specular reflectance and total reflectance. This set of measurements is sufficient to determine diffuse reflectance and absorbance of the sample, also necessary to estimate the optical properties. All the performed measurements and calculated quantities will be used later in inverse Monte Carlo (IMC) simulations to determine the evolution of the optical properties of muscle during treatments with ethylene glycol and glucose. The results obtained with the measurements already provide some information about the optical clearing treatments applied to the muscle and translate the mechanisms of turning the tissue more transparent and sequence of regimes of optical clearing.

The characteristic time of glucose diffusion measured for muscle tissue at optical clearing

The study of agent diffusion in biological tissues is very important to understand and characterize the optical clearing effects and mechanisms involved: tissue dehydration and refractive index matching. From measurements made to study the optical clearing, it is obvious that light scattering is reduced and that the optical properties of the tissue are controlled in the process. On the other hand, optical measurements do not allow direct determination of the diffusion properties of the agent in the tissue and some calculations are necessary to estimate those properties. This fact is imposed by the occurrence of two fluxes at optical clearing: water typically directed out of and agent directed into the tissue. When the water content in the immersion solution is approximately the same as the free water content of the tissue, a balance is established for water and the agent flux dominates. To prove this concept experimentally, we have measured the collimated transmittance of skeletal muscle samples under treatment with aqueous solutions containing different concentrations of glucose. After estimating the mean diffusion time values for each of the treatments we have represented those values as a function of glucose concentration in solution. Such a representation presents a maximum diffusion time for a water content in solution equal to the tissue free water content. Such a maximum represents the real diffusion time of glucose in the muscle and with this value we could calculate the corresponding diffusion coefficient.

Software development for estimation of optical clearing agent’s diffusion coefficients in biological tissues

The study of chemical diffusion in biological tissues is a research field of high importance and with application in many clinical, research and industrial areas. The evaluation of diffusion and viscosity properties of chemicals in tissues is necessary to characterize treatments or inclusion of preservatives in tissues or organs for low temperature conservation. Recently, we have demonstrated experimentally that the diffusion properties and dynamic viscosity of sugars and alcohols can be evaluated from optical measurements. Our studies were performed in skeletal muscle, but our results have revealed that the same methodology can be used with other tissues and different chemicals. Considering the significant number of studies that can be made with this method, it becomes necessary to turn data processing and calculation easier. With this objective, we have developed a software application that integrates all processing and calculations, turning the researcher work easier and faster. Using the same experimental data that previously was used to estimate the diffusion and viscosity of glucose in skeletal muscle, we have repeated the calculations with the new application. Comparing between the results obtained with the new application and with previous independent routines we have demonstrated great similarity and consequently validated the application. This new tool is now available to be used in similar research to obtain the diffusion properties of other chemicals in different tissues or organs.

Optical clearing mechanisms characterization in muscle

Optical immersion clearing is a technique that has been widely studied for more than two decades and that is used to originate a temporary transparency effect in biological tissues. If applied in cooperation with clinical methods it provides optimization of diagnosis and treatment procedures. This technique turns biological tissues more transparent through two main mechanisms — tissue dehydration and refractive index (RI) matching between tissue components. Such matching is obtained by partial replacement of interstitial water by a biocompatible agent that presents higher RI and it can be completely reversible by natural rehydration in vivo or by assisted rehydration in ex vivo tissues. Experimental data to characterize and discriminate between the two mechanisms and to find new ones are necessary. Using a simple method, based on collimated transmittance and thickness measurements made from muscle samples under treatment, we have estimated the diffusion properties of glucose, ethylene glycol (EG) and water that were used to perform such characterization and discrimination. Comparing these properties with data from literature that characterize their diffusion in water we have observed that muscle cell membrane permeability limits agent and water diffusion in the muscle. The same experimental data has allowed to calculate the optical clearing (OC) efficiency and make an interpretation of the internal changes that occurred in muscle during the treatments. The same methodology can now be used to perform similar studies with other agents and in other tissues in order to solve engineering problems at design of inexpensive and robust technologies for a considerable improvement of optical tomographic techniques with better contrast and in-depth imaging.