Juan Navas

Fractal dimension and white matter changes in multiple sclerosis

The brain white matter (WM) in multiple sclerosis (MS) suffers visible and non-visible (normal-appearing WM (NAWM)) changes in conventional magnetic resonance (MR) images. The fractal dimension (FD) is a quantitative parameter that characterizes the morphometric variability of a complex object. Our aim was to assess the usefulness of FD analysis in the measurement of WM abnormalities in conventional MR images in patients with MS, particularly to detect NAWM changes. First, we took on a voxel-based morphometry approach optimized for MS to obtain the segmented brain. Then, the FD of the whole grey-white matter interface (WM border) and skeletonized WM was calculated in patients with MS and healthy controls. To assess the FD of the NAWM, we focused our analysis on single sections without lesions at the centrum semiovale level. We found that patients with MS had a significant decrease in the FD of the entire brain WM compared with healthy controls. Such a decrease of the FD was detected not only on MR image sections with MS lesions but also on single sections with NAWM. Taken together, the results showed that FD identifies changes in the brain of patients with MS, including in NAWM, even at an early phase of the disease. Thus, FD might become a useful marker of diffuse damage of the central nervous system in MS.

Fractal dimension analysis of grey matter in multiple sclerosis

The fractal dimension (FD) is a quantitative parameter that characterizes the morphometric variability of a complex object. Among other applications, FD has been used to identify abnormalities of the human brain in conventional magnetic resonance imaging (MRI), including white matter abnormalities in patients with Multiple Sclerosis (MS). Extensive grey matter (GM) pathology has been recently identified in MS and it appears to be a key factor in long-term disability. The aim of the present work was to assess whether FD measurement of GM in T1 MRI sequences can identify GM abnormalities in patients with MS in the early phase of the disease. A voxel-based morphometry approach optimized for MS was used to obtain the segmented brain, where we later calculated the three-dimensional FD of the GM in MS patients and healthy controls. We found that patients with MS had a significant increase in the FD of the GM compared to controls. Such differences were present even in patients with short disease durations, including patients with first attacks of MS. In addition, the FD of the GM correlated with T1 and T2 lesion load, but not with GM atrophy or disability. The FD abnormalities of the GM here detected differed from the previously published FD of the white matter in MS, suggesting that different pathological processes were taking place in each structure. These results indicate that GM morphology is abnormal in patients with MS and that this alteration appears early in the course of the disease.

A Web platform for the interactive visualization and analysis of the 3D fractal dimension of MRI data

This study presents a Web platform (http://3dfd.ujaen.es) for computing and analyzing the 3D fractal dimension (3DFD) from volumetric data in an efficient, visual and interactive way. The Web platform is specially designed for working with magnetic resonance images (MRIs) of the brain. The program estimates the 3DFD by calculating the 3D box-counting of the entire volume of the brain, and also of its 3D skeleton. All of this is done in a graphical, fast and optimized way by using novel technologies like CUDA and WebGL. The usefulness of the Web platform presented is demonstrated by its application in a case study where an analysis and characterization of groups of 3D MR images is performed for three neurodegenerative diseases: Multiple Sclerosis, Intrauterine Growth Restriction and Alzheimers disease. To the best of our knowledge, this is the first Web platform that allows the users to calculate, visualize, analyze and compare the 3DFD from MRI images in the cloud.

UJA-3DFD: A program to compute the 3D fractal dimension from MRI data

This work presents a computer program for computing the 3D fractal dimension (3DFD) from magnetic-resonance images of the brain. The program is based on an algorithm that calculates the 3D box counting of the entire volume of the brain, and also of its 3D skeletonization. The validity and accuracy of the software has been confirmed using solids with well-known 3DFD values. The usefulness of the program developed is demonstrated by its successful characterization of several neurodegenerative diseases.

Rate of Convergence of the Linear Discrete Polya Algorithm

In this paper, we consider the problem of best approximation in lp(n),1 ≤ p ≤ ∞. If hp, 1 ≤ p ≤ ∞, denotes the best lp-approximation of the element h ∈ Rn from a proper affine subspace K of Rn, h ∉ K, then limp→1 hp =h*1, where h*1 is a best l1-approximation of h from K, the so-called natural l1-approximation. Our aim is to give a complete description of the rate of convergence of hp to h*1 as p → 1.

Asymptotic behaviour of best l p -approximations from affine subspaces

In this paper we consider the problem of best approximation in lpn, 1 < p ≤ ∞. If hp, 1 < p < ∞, denotes the best lp-approximation of the element h ∈ Rn from a proper affine subspace K of Rn, h ∉ K, then limp→∞hp = h*∞ where h*∞ is a best uniform approximation of h from K, the so-called strict uniform approximation. Our aim is to prove that for all r ∈ N there are αj ∈ Rn, 1 ≤ j ≤ r, such that hp = h*∞ + α1/p-1 + α2/(p-1)2 + ... + αr/(p-1)r + γpr, with γp(r) ∈ Rn and ||γp(r)|| = O(p-r-1).

A system dynamics model of the population dynamics of Oestrus sp. (Diptera: Oestridae) infesting Iberian ibex, Capra pyrenaica

Abstract We modelled the population biology of Oestrus sp. parasitising the Iberian ibex (Capra pyrenaica), using a system dynamics approach. Levels included in the model were as follows: first-instar larvae, third-instar larvae, pupae and resilient pupae. The values used for flows and auxiliary variables were obtained from the scientific literature, including research on both Oestrus ovis and O. sp.; we assumed that these two species are closely related, both morphologically and from a biological point of view. Simulations provided by our model allowed us to estimate the minimum monthly production of first-instar larvae and thus to establish periodic population dynamics, together with predictions for the number of larval generations. The model predicted 3–4 larval generations per year. The model proved to be very sensitive to minimal changes in a number of variables, especially the first-instar larval production rate. Despite its limitations, this methodology could be a versatile tool for studying the population dynamics of this kind of parasites, and for simulating the effects of control programmes.

Rate of convergence of the Pólya algorithm from polyhedral sets

In this paper we consider a problem of best approximation in lp, 1 < p ≤ ∞. Lethp denote the best p-approximation of h ∈ Rn from a closed, convex set K of Rn 1 < p < ∞, h ∉ K, and let h∞* be the strict uniform approximation of h from K. We prove that if K satisfies locally a geometrical property, fulfilled by any polyhedral set of Rn, then lim sup p→∞ p||hp - h∞*|| < ∞.

Taylor expansion of best lp-approximations about p = 1

Abstract In this paper, we consider the problem of best approximation in lp (n), 1 ≤ p ≤ ∞. If hp, 1 ≤ p lim p→1 h p = h 1 ∗ , where h 1 ∗ , is a best l1-approximation of h from K, the so-called natural best k1-approximation. We prove that, for every r ϵ n, the best kp-approximations have a Taylor expansion of order r of the form h p =h 1 ∗ + ∑ l=1 r αl(p−1) l +γ p (r) for some αl ϵ rn, 1 ≤ l ≤ r, and γp(r) ϵ rn with |γp(r)|=o((p−1)r+1).