# Nonlinear mathematical relationship between concentration

Citation. Wang, Xuefeng. On concentration of positive bound states of nonlinear Schrödinger equations. Comm. Math. Phys. (), no. 2, Nawa, Hayato. ``Mass concentration'' phenomenon for the nonlinear Schrödinger equation with the critical power nonlinearity. II. Kodai Math. J. 13 (), no. Newton's law for a trajectory of concentration of solutions to nonlinear relation between wave and point dynamics is by means of the WKB method, see.

Originally, they considered only the extravascular component of the BOLD signal and concluded that if flow and metabolism were coupled, BOLD signal could be used as a robust estimator of flow changes. Later, Buxton Buxton et al. The balloon model used typical mass conservation properties in order to obtain a system of coupled differential equation linking blood volume and deoxyhemoglobin concentration. The temporal trend of blood flow entering the venous compartment was assumed as the typical boxcar function used for brain activation.

The blood flow exiting the venous compartment was modeled as a power law of the venous blood volume, extending improperly a static property, the Grubb's law Grubb et al. Mandeville Mandeville et al. The authors derived this system of partial differential equation by using more basic physiological principles, such as the fact that blood flow in the capillary and venous compartments is modulated by changes in blood pressure due to changes in arteriolar resistance, and that veins due to the presence of muscle cells respond to an increase in pressure with a delayed compliance, as observed in vivo Porciuncula et al.

In the original windkessel model Mandeville et al.

## The Scientific World Journal

The windkessel model also had a large impact in NIRS. A first example is found in the work of Boas Boas et al;where the authors used the windkessel model in order to assess the ratio between the changes in cerebral metabolic rate of oxygen CMRO2 and the changes in blood flow during a finger tapping task.

Their model used thirteen parameters to fit the changes of oxy- deoxy- and total hemoglobin concentrations measured on the subjects during a finger tapping task. Six of these parameters where held fixed, whereas the remaining seven were varied in an optimization scheme.

Among the parameters retrieved by the fit, four parameters characterized the dynamics of the arteriolar resistance and the windkessel transit time comprising the transit time in the capillary and venous compartments. Expanded three compartments windkessel models have also been proposed Zheng et al. Huppert Huppert et al. The authors used multimodal optical imaging, combining the information of laser speckle imaging and NIRS in order to measure the changes in blood flow and hemoglobin concentration in tissue during a rat whisker stimulation protocol.

### Nonlinear extension of a hemodynamic linear model for coherent hemodynamics spectroscopy

The coupled partial differential equations for blood volume and flow, and also for oxygen extraction, were solved based on the optimization of fourteen state variables that were retrieved by the inversion procedure. Seven of these parameters were used to model the dynamic arteriolar expansion and the dynamic change of CMRO2 in tissue. Other parameters retrieved by the model were the vascular transit time, the pial vessel transit time and the baseline hemoglobin saturations in the three compartments showing an interesting possibility of the model, to retrieve absolute baseline values from relative optical intensity measurements.

One of the results of this work was that a three compartment windkessel model could fit the experimental data better than a single compartment windkessel model. The windkessel model, together with a model of oxygen dynamics and exchange with tissue, has also been used for more complicated vascular networks, comprising thirty-two arterioles, thirty-two venules and sixty-four capillaries Boas et al.

This study was an important step toward the modeling of more realistic vascular networks. The changes in the concentration of oxy- deoxy- and total hemoglobin were also calculated and averaged in the arterioles, capillaries and venules.

Finally, we mention the work of Diamond Diamond et al. In summary, most of the hemodynamic models found in the literature, and described briefly above, are based on the solution of a system of partial differential equations in order to retrieve physiological information during different conditions. Recently, our group proposed a novel technique to study tissue hemodynamics, named coherent hemodynamic spectroscopy CHS Fantini, a ; Fantini, b.

The static nonlinear part uses static analysis of the controlled plant for introducing the mathematical nonlinear description of the relation between the controlled output and the change of the control input.

Proposed controller is tested by the simulations on the mathematical model of the continuous stirred-tank reactor with cooling in the jacket as a typical nonlinear system. Introduction The control of the chemical processes in the industry is always challenging because of the nonlinearity of the major group of systems.

The continuous stirred-tank reactor CSTR is one of the most common used types of chemical reactors because of easy controllability [ 1 ]. The adaptive control [ 2 ] is a control technique with good theoretical background and also practical implementations. It uses idea of the living organisms that adopts their behavior to the actual environmental conditions.

### Nonlinear versus Ordinary Adaptive Control of Continuous Stirred-Tank Reactor

There are also various adaptation techniques and variations described, for example, in [ 3 ]. The control method used here is based on the combination of the adaptive control and nonlinear control.

Theory of nonlinear control NC can be found, for example, in [ 45 ]. The DLP uses polynomial synthesis [ 7 ] with pole-placement method and spectral factorization and all these methods satisfy basic control requirements such as disturbance attenuation, stability, and reference signal tracking. The second, nonlinear, part uses measurements of the steady-state behavior of the system for mathematical description of the dependence between the controlled output variable and the control input variable.

The controlled system, CSTR, with originally nonlinear behavior could be mathematically described for the control purposes by the external linear model ELM [ 8 ], parameters of which could vary because of the nonlinearity of the system.

This problem could be overcome with the use of recursive identification which recomputes parameters of the ELM according to the actual state and the behavior of the system.