ObjectiveTo compare foam bubble size and bubble size distribution, stability, and degradation rate of commercially available polidocanol endovenous microfoam (Varithena®) and physician-compounded foams using a number of laboratory tests.MethodsFoam properties of polidocanol endovenous microfoam and physician-compounded foams were measured and compared using a glass-plate method and a Sympatec QICPIC image analysis method to measure bubble size and bubble size distribution, Turbiscan™ LAB for foam half time and drainage and a novel biomimetic vein model to measure foam stability. Physician-compounded foams composed of polidocanol and room air, CO2, or mixtures of oxygen and carbon dioxide (O2:CO2) were generated by different methods.ResultsPolidocanol endovenous microfoam was found to have a narrow bubble size distribution with no large (>500 µm) bubbles. Physician-compounded foams made with the Tessari method had broader bubble size distribution and large bubbles, which have an impact on foam stability. Polidocanol endovenous microfoam had a lower degradation rate than any physician-compounded foams, including foams made using room air (p < 0.035). The same result was obtained at different liquid to gas ratios (1:4 and 1:7) for physician-compounded foams. In all tests performed, CO2 foams were the least stable and different O2:CO2 mixtures had intermediate performance. In the biomimetic vein model, polidocanol endovenous microfoam had the slowest degradation rate and longest calculated dwell time, which represents the length of time the foam is in contact with the vein, almost twice that of physician-compounded foams using room air and eight times better than physician-compounded foams prepared using equivalent gas mixes.ConclusionBubble size, bubble size distribution and stability of various sclerosing foam formulations show that polidocanol endovenous microfoam results in better overall performance compared with physician-compounded foams. Polidocanol endovenous microfoam offers better stability and cohesive properties in a biomimetic vein model compared to physician-compounded foams. Polidocanol endovenous microfoam, which is indicated in the United States for treatment of great saphenous vein system incompetence, provides clinicians with a consistent product with enhanced handling properties.

In the present article, fractional-order heat and wave equations are solved by using the natural transform decomposition method. The series form solutions are obtained for fractional-order heat and wave equations, using the proposed method. Some numerical examples are presented to understand the procedure of natural transform decomposition method. The natural transform decomposition method procedure has shown that less volume of calculations and a high rate of convergence can be easily applied to other nonlinear problems. Therefore, the natural transform decomposition method is considered to be one of the best analytical techniques, in order to solve fractional-order linear and nonlinear Partial deferential equations, particularly fractional-order heat and wave equation.

A divergence measure plays a crucial part in discriminating two probability distributions and drawing inferences constructed on such discrimination. The intention of this study is to propose such a divergence measure based on Jensen inequality and exponential entropy in the settings of probability theory. Further, the idea has been generalized to fuzzy sets to familiarize a novel picture fuzzy divergence measure. Besides proposing the validity, some of its key properties are also deliberated. Finally, two illustrative examples are solved based on the proposed picture fuzzy divergence measure which shows the expediency and effectiveness of the proposed approach.

In the present article, we related the analytical solution of the fractional-order dispersive partial differential equations, using the Laplace–Adomian decomposition method. The Caputo operator is used to define the derivative of fractional-order. Laplace–Adomian decomposition method solutions for both fractional and integer orders are obtained in series form, showing higher convergence of the proposed method. Illustrative examples are considered to confirm the validity of the present method. The fractional order solutions that are convergent to integer order solutions are also investigated.