In this work, we propose a solution to the problem of identification of sources in the brain from measurements of the electrical potential, recorded on the scalp EEG (electroencephalogram), where boundary problems are used to model the skull, brain region, and scalp, solving the inverse problem from the EEG measurements, the so-called Electroencephalographic Inverse Problem (EIP), which is ill-posed in the Hadamard sense since the problem has numerical instability. We focus on the identification of volumetric dipolar sources of the EEG by constructing and modeling a simplification to reduce the multilayer conductive medium (two layers or regions Ω1 and Ω2) to a problem of a single layer of a homogeneous medium with a null Neumann condition on the boundary. For this simplification purpose, we consider the Cauchy problem to be solved at each time. We compare the results we obtained solving the multiple layers problem with those obtained by our simplification proposal. In both cases, we solve the direct and inverse problems for two different sources, as synthetic results for dipolar sources resembling epileptic foci, and a similar case with an external stimulus (intense light, skin stimuli, sleep problems, etc). For the inverse problem, we use the Tikhonov regularization method to handle its numerical instability. Additionally, we build an algorithm to solve both models (multiple layers problem and our simplification) in time, showing optimization of the problem when considering 128 divisions in the time interval [0,1] s, solving the inverse problem at each time (interval division) and comparing the recovered source with the initial one in the algorithm. We observed a significant decrease in the computation times when simplifying the numerical calculations, resulting in a decrease up to 50% in the execution times, between the EIP multilayer model and our simplification proposal, to a single layer homogeneous problem of a homogeneous medium, which translates into a numerical efficiency in this type of problem.
We simulate a liquid crystal spatial light modulator (LCSLM), previously validated by Fraunhofer diffraction to observe super-Gaussian periodic profiles and analyze the wavefront of optical surfaces applying the transport-of-intensity equation (TIE). The LCSLM represents an alternative to the Ronchi Rulings, allowing to avoid all the related issues regarding diffractive and refractive properties, and noise. To this aim, we developed and numerically simulated a LCSLM resembling a fractal from a generating base. Such a base is constituted by an active square (values equal to one) and surrounded by eight switched-off pixels (zero-valued). We replicate the base in order to form 1 ×N-pixels and the successive rows to build the 1024×1024 LCSLM of active pixels. We visually test the LCSLM with calibration images as a diffractive object that is mathematically inducible, using mathematical induction over the N×N-shape (1×1, 2×2, 3×3, …, n×n pixels for the generalization). Finally, we experimentally generate periodic super-Gaussian profiles to be visualized in the LCSLM (transmission SLM, 1024×768-pixels LC 2012 Translucent SLM), modifying the TIE as an optical test in order to analyze the optical elements by comparing the results with ZYGO/APEX.
The Irradiance Transport Equation is considered as an elliptical differential equation that determines the phase as a poorly stated problem. The instability of such a problem is handled as Tikhonov regularization method to determine aberrations.
In this work we introduce a technique to speed up the interpretation of bone scans with the aim of determining the presence or absence of metastatic disease. We use gray tone histograms, resembling the use of band-pass filters, in order to ensure a reliable interpretation of the bone scan, therefore providing an accurate diagnosis. We draw particular attention to three cases. The first case corresponds to shifted histograms. If the histogram is shifted toward the origin, the bone scan is free of metastasis. If it is shifted to the right and slightly broadened, this indicates the presence of a bone scan anomaly other than metastasis. On the other hand, if the histogram is broadened and shifted to the left, this suggests the presence of metastatic disease. The second case corresponds to a histogram with noticeable fluctuations, indicating the presence of metastasis. Such fluctuations could become local maxima peaks, indicating the advance of the metastasis. The third case corresponds to the false color results, displayed in terms of the gray tones, observed in the histogram. Such false color is assigned from the construction of a 7-color palette and is selected in terms of the gray tones range. This eases the ad hoc false color assignation for visualization purposes. The final diagnosis is carried out in terms of the color, geometry, extension, and location of the region of interest in the images. Our proposed technique has the potential to be used in high-demand oncology centers due to its simplicity and diagnostic efficiency, confirmed and tested by specialists in the Centro Medico Siglo XXI (XXI Century Medical Center), CDMX-México.
Objectives:We introduce a software tool to optimize the visualization of bone gammagrams, with the aim of differentiating (substantially) the pathologies from the radiotracer accumulation. The main objective of the mentioned software tool is to determine the presence of early-stage metastatic disease. The software contains routines capable of finding other types of anomalies and bone pathologies. Materials and methods: We analyzed a sample of 43 whole-body bone gammagrams, diagnosed with prostate cancer with ages ranging from 23 to 87 years (with a mean age of 55 years). We use several quantitative methods, such as histograms analysis, to analyze the gammagrams in gray tones. Results: Our proposed software is friendly, practical, and based on methods statistically robust, suitable for critical cases. We obtained optimal results in the search of degenerative changes, infections, or malign bone anomalies. Conclusions: The gray tone values in a bone gammagram depend on the bone health, with low order gray tones (around 20) corresponding to a healthy bone whereas values larger than 50 indicate the presence of bone disease. The gray tones distribution (besides their location) in the bone system is considered by the software for determining the bone disease type in the analyzed image.
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