A New Numerical Approach for the Analysis of Variable Fractal and Fractional Order Differential Equations.

The variable fractional dimensions differential and integral operator overrides the phenomenon of the constant fractional order. This leads to exploring some new ideas in the proposed direction due to its varied applications in the recent era of science and engineering. The present papers deal with the replacement of the constant fractional order by variable fractional order in various fractal-fractional differential equations. An advanced numerical scheme is developed with the help of Lagrange three-point interpolation and further, it is employed for the solution of the proposed differential equations. However, the properties of these new operators are presented in detail. Finally, the error analysis is also conducted for the numerical scheme deployed. The results are validated by the suitable choice of applications to real-life problems. The well- known multi-step-Adams-Bashforth numerical scheme for classical differential equations is recovered when the non-integer order is one.

The first derivative of the transthoracic electrical impedance as an index of changes in myocardial contractility in the intact anaesthetised dog.

The suitability of the peak value of the first derivative of the thoracic electrical impedance dZ/dtmax has been investigated in dogs as a non-invasive index of changes in myocardial contracility by comparing if with the peak left ventricular dP/dt and the peak values of aortic blood velocity and acceleration. An increase in the inspired halothane concentration was used to produce changes in contracility. In 5 dogs the combined values for the correlation coefficient between dZ/dtmax peak velocity and acceleration were respectively 0.937, 0.954 and 0.950. In 14 out 15 comparisons p less than 0.001 and in one, p less than 0.01. On these grounds, dZ/dt max is proposed as a variable worthy of sconsideration in patient monitoring.

Blood resistivity and its implications for the calculation of cardiac output by the thoracic electrical impedance technique.

A relationship has been established between the haematocrit and the resistivity of whole blood at 37 degrees C for neonates and adults not suffering from renal failure. Values of resistivity obtained in this way were substituted into Kubicek's equation for stroke volume by the electrical impedance technique, the signal pick-up electrodes being placed in standard positions. The calculated cardiac outputs were then compared with those obtained simultaneously from other techniques such as the dye dilution method for adults and the measurement of pulmonary effective capillary blood flow by rebreathing nitrous oxide in neonates. It was found that the impedance method overrestimates under these circumstances, and other workers have shown that this is likely to be due to a contribution from the right heart. A haemotocrit-dependent correction factor has been employed in the case of neonates to align the impedance and nitrous oxide results.

The changes in blood resistivity with haematocrit and temperature.

The temperature dependence of the resistivity of blood samples with haematocrits from 16 to 52.5% has been investigated over the temperature range of 22 degrees ot 40 degrees C at a frequency of 100 kHz. The resistivity of whole blood increased with an increase in haematocrit and a decrease in temperature. The data fitted the relationship: Pohm-cm=(6.272 Hct + 75.176) - (0.104 Hct + 1.467) tC weree Hct is the percentage haematocrit.

The effect of a progressive decrease in the circulating blood volume of the dog on the transthoracic impedance.

The correlations between the haemodynamic and transthoracic electrial impedance changes resulting from a progressive reduction in the circulating blood volume were studied in four intact mongrel dogs artificially ventilated with a mixture of halothane in nitrous oxide-oxygen. The cardiac output of the dogs was measured by both the electrical impedance and the fibre optic dye dilution techniques. It was found that significant correlations existed between the blood lloss and the arterial blood pressure, the maximum first derivative of the transthoracic impedance, the Heather Index, the transthoracic impedance, the maximum rate of change of aortic pressure and the cardiac stroke work. There was also a good correlation between the dye and impedance cardiac output values, the impedance value always being higher than the corresponding dye value. The correlation between the Heather Index and the PEP/LVET ratio and 1/PEP2 varied markedly from dog to dog.

Motion artifact suppression technique (MAST) for MR imaging.

A technique has been developed that significantly improves the image resolution and reduces motion artifacts in conventional two-dimensional Fourier transform and three-dimensional Fourier transform magnetic resonance imaging sequences. Modifications on the gradient waveforms completely refocus the transverse magnetization at the echo time, regardless of the motion occurring between the time of the 90 degrees radiofrequency excitation and the echo time (within-view). This accomplishes suppression of motion artifacts and regains the signal from flowing blood and CSF. Images of the head, abdomen, chest, and spine are reproduced which show the increase in signal and anatomical detail that would otherwise be degraded and lost in artifact noise. This technique has reduced the practical difficulty of obtaining clinically diagnostic T2-weighted abdominal images. It also has allowed diagnostic quality T1- and T2-weighted images to be obtained with one acquisition per view, thus reducing the total scan time.