Mathematical Modeling of Oxygen Diffusion from Capillary to Tissues during Hypoxia through Multiple Points Using Fractional Balance Equations with Memory.

The diffusion of oxygen through capillary to surrounding tissues through multiple points along the length has been addressed in many clinical studies, largely motivated by disorders including hypoxia. However relatively few analytical or numerical studies have been communicated. In this paper, as a compliment to physiological investigations, a novel mathematical model is developed which incorporates the multiple point diffusion of oxygen from different locations in the capillary to tissues, in the form of a fractional dynamical system of equations using the concept of system of balance equations with memory. Stability analysis of the model has been conducted using the well known Routh-Hurwitz stability criterion. Comprehensive analytical solutions for the differntial equation problem in the new proposed model are obtained using Henkel transformations. Both spatial and temporal variation of concentration of oxygen is visualized graphically for different control parameters. Close correlation with simpler models is achieved. Diffusion is shown to arise from different points of the capillary in decreasing order along the length of the capillary i.e. for the different values of z. The concentration magnitudes at low capillary length far exceed those further along the capillary. Furthermore with progrssive distance along the capillary, the radial distance of diffusion decreases, such that oxygen diffuses only effectively in very close proximity to tissues. The simulations provide a useful benchmark for more generalized mass diffusion computations with commercial finite element and finite volume software including ANSYS FLUENT.

Analysis of the concurrent validity and reliability of five common clinical goniometric devices.

Measurement errors play an important role in the development of goniometric equipment, devices used to measure range of motion. Reasonable validity and reliability are critical for both the device and examiner before and after to testing in human subjects. The objective is to evaluate the concurrent validity and reliability of five different clinical goniometric devices for the purpose of establishing an acceptable measurement error margin for a novel device. We explored the validity and inter- and intrarater reliability scores of five goniometric devices namely (i) the universal goniometer (UG), a two-armed hand-held goniometer, (ii) the inclinometer (IC), featuring a single base, fluid level, and gravity-weighted inclinometer, (iii) the digital inclinometer (DI), functioning as both a DI and dynamometer, (iv) the smartphone application (SA), employing gyroscope-based technology within a smartphone platform application and (v) the modified inclinometer (MI), a gravity pendulum-based inclinometer equipped with a specialized fixing apparatus. Measurements were obtained at 12 standard angles and 8 human shoulder flexion angles ranging from 0° to 180°. Over two testing sessions, 120 standardized angle measurements and 160 shoulder angle measurements from 20 shoulders were repetitively taken by three examiners for each device. The intraclass correlation coefficient (ICC), standard error of measurement (SEM), and minimal detectable change (MDC) were calculated to assess reliability and validity. Concurrent validity was also evaluated through the execution of the 95% limit of agreement (95% LOA) and Bland-Altman plots, with comparisons made to the UG. The concurrent validity for all device pairs was excellent in both study phases (ICC > 0.99, 95% LOA - 4.11° to 4.04° for standard angles, and - 10.98° to 11.36° for human joint angles). Inter- and intrarater reliability scores for standard angles were excellent across all devices (ICC > 0.98, SEM 0.59°-1.75°, MDC 1°-4°), with DI showing superior reliability. For human joint angles, device reliability ranged from moderate to excellent (ICC 0.697-0.975, SEM 1.93°-4.64°, MDC 5°-11° for inter-rater reliability; ICC 0.660-0.996, SEM 0.77°-4.06°, MDC 2°-9° for intra-rater reliability), with SA demonstrating superior reliability. Wider angle measurement however resulted in reduced device reliability. In conclusion, our study demonstrates that it is essential to assess measurement errors independently for standard and human joint angles. The DI is the preferred reference for standard angle testing, while the SA is recommended for human joint angle testing. Separate evaluations across the complete 0°-180° range offer valuable insights.