A modified Kacanov iteration scheme with application to quasilinear diffusion models

The classical Kacanov scheme for the solution of nonlinear variational problems can be interpreted as a fixed point iteration method that updates a given approximation by solving a linear problem in each step. Based on this observation, we introduce a modified Kacanov method, which allows for (adaptive) damping, and, thereby, to derive a new convergence analysis under more general assumptions and for a wider range of applications. For instance, in the specific context of quasilinear diffusion models, our new approach does no longer require a standard monotonicity condition on the nonlinear diffusion coefficient to hold. Moreover, we propose two different adaptive strategies for the practical selection of the damping parameters involved.

Study of Continuous and Intermittent Drying of Pear Through Mathematical and Diffusion Models

The methods for drying agricultural products are necessary to check the quality of the final product and feasibility of the process. The objective of this study was to perform the drying of pear by continuous and intermittent methods (α = 2/3) at two different temperatures, compare the two methods and describe the processes using mathematical and diffusion models to evaluate the saving of time and effective processing energy. For continuous drying, Page model was the one which best fitted to the experimental data, with highest coefficient of determination and lowest chi-square. Regarding intermittent drying Page model was also the one which best fitted to the experimental data for the two temperatures studied. When comparing the drying methods studied, it was found that intermittent drying produced a similar and considerably better effect on the energy saving aspect, compared with continuous drying: 41.7% at 50 °C and 25.8% at 70 °C. The effect of time intervals with the product out of the dryer, during the intermittent drying process, was reflected in the increase in the diffusion coefficient, so that the transport of moisture occurred more quickly for this type of drying, which contributes to justify the reduction of effective operating time and increased dry product quality.

A Comparative Study of Diffusion Fiber Reconstruction Models for Pyramidal Tract Branches

Currently, comparative studies evaluating the quantification accuracy of pyramidal tracts (PT) and PT branches that were tracked based on four mainstream diffusion models are deficient. The present study aims to evaluate four mainstream models using the high-quality Human Connectome Project (HCP) dataset. Diffusion tensor imaging (DTI), diffusion spectral imaging (DSI), generalized Q-space sampling imaging (GQI), and Q-ball imaging (QBI) were used to construct the PT and PT branches in 50 healthy volunteers from the HCP. False and true PT fibers were identified based on anatomic information. One-way repeated measure analysis of variance and post hoc paired-sample t-test were performed to identify the best PT and PT branch quantification model. The number, percentage, and density of true fibers of PT obtained based on GQI and QBI were significantly larger than those based on DTI and DSI (all p < 0.0005, Bonferroni corrected), whereas false fibers yielded the opposite results (all p < 0.0005, Bonferroni corrected). More trunk branches (PTtrunk) were present in the four diffusion models compared with the upper limb (PTUlimb), lower limb (PTLlimb), and cranial (PTcranial) branches. In addition, significantly more true fibers were obtained in PTtrunk, PTUlimb, and PTLlimb based on the GQI and QBI compared with DTI and DSI (all p < 0.0005, Bonferroni corrected). Finally, GQI-based group probabilistic maps showed that the four PT branches exhibited relatively unique spatial distributions. Therefore, the GQI and QBI represent better diffusion models for the PT and PT branches. The group probabilistic maps of PT branches have been shared with the public to facilitate more precise studies on the plasticity of and the damage to the motor pathway.

Evolution of Hemodynamic Parameters Simulated by Means of Diffusion Models

Diffusion is a well-known physical phenomenon governing such processes as movement of particles or transportation of heat. In this paper, we prove that a close analogy to those processes exists in medical data behavior, and that changes in the values of medical parameters measured while treating patients may be described using diffusion models as well. The medical condition of a patient is usually described by a set of discrete values. The evolution of that condition and, consequently, of the disease has the form of a transition of that set of discrete values, which correspond to specific parameters. This is a typical medical diagnosis scheme. However, disease evolution is a phenomenon that is characterized by continuously varying, temporal characteristics. A mathematical disease evolution model is, in fact, a continuous diffusion process from one discrete slot of the diagnosed parameter value to another inside the mentioned set. The ability to predict such diffusion-related properties offer precious support in diagnostic decision-making. We have examined several hundred patients while conducting a medical research project. All patients were under treatment to stabilize their hemodynamic parameters. A diffusion model relied upon simulating the results of treatment is proposed here. Time evolution of thoraric fluid content (TFC) has been used as the illustrative example. The objective is to prove that diffusion models are a proper and convenient solution for predicting disease evolution processes. We applied the Fokker-Planck equation (FPE), considering it to be most adequate for examining the treatment results by means of diffusion. We confirmed that the phenomenon of diffusion explains the evolution of the heart disease parameters observed. The evolution of TFC has been chosen as an example of a hemodynamic parameter.