Pierluigi Maponi

MR Imaging and Osteoporosis: Fractal Lacunarity Analysis of Trabecular Bone

We develop a method of magnetic resonance (MR) image analysis able to provide parameter(s) sensitive to bone microarchitecture changes in aging, and to osteoporosis onset and progression. The method has been built taking into account fractal properties of many anatomic and physiologic structures. Fractal lacunarity analysis has been used to determine relevant parameter(s) to differentiate among three types of trabecular bone structure (healthy young, healthy perimenopausal, and osteoporotic patients) from lumbar vertebra MR images. In particular, we propose to approximate the lacunarity function by a hyperbola model function that depends on three coefficients, alpha,beta, and gamma, and to compute these coefficients as the solution of a least squares problem. This triplet of coefficients provides a model function that better represents the variation of mass density of pixels in the image considered. Clinical application of this preliminary version of our method suggests that one of the three coefficients, beta, may represent a standard for the evaluation of trabecular bone architecture and a potentially useful parametric index for the early diagnosis of osteoporosis

Fractal characterisation of boundary irregularity in skin pigmented lesions

A growing literature shows researchers interest in fractal analysis, arising from its ability to describe and characterise quantitatively the complexity of several tumour profiles. The aim of the work was to investigate the fractal properties of skin pigmented lesion boundaries. Although melanoma is one of the most aggressive tumours, early detection and a high rate of diagnostic accuracy, followed by timely excision, can allow complete recovery in melanoma patients. A modified approach to fractal dimension estimation was performed that was able to consider, in a data fit procedure, the range in which lesions show fractal properties. Identification of this zone is the most important step towards a correct fractal analysis procedure. The method was checked against a known fractal dimension object (Kochs curve) with an error of 0.007. The fractal dimension was estimated in 110 skin pigmented lesions and showed a significantly increasing linear regression (p<0.05), from common naevi to naevi with dysplasia to melanomas. This result is important for screening, as it can inform the decision to excise precociously malignant lesions or to avoid unnecessary removal of benign ones. The limitations of the method are discussed.

An inverse problem for the three dimensional vector Helmholtz equation for a perfectly conducting obstacle

Abstract A numerical algorithm for a three-dimensional inverse electromagnetic scattering problem is considered. For time-harmonic waves the Maxwells equations are reduced to the vector Helmholtz equation. From the knowledge of several far fields generated by an obstacle D when hit by incoming linearly polarized plane waves the boundary ∂D of the obstacle is reconstructed. The obstacle D is supposed to be bounded, connected, with smooth boundary and perfectly conducting. The reconstruction procedure proposed here generalizes the “Herglotz function method” introduced by Colton and Monk [1] for the acoustic problem and is effective in the so-called resonance region.

Medical imaging and osteoporosis: fractal's lacunarity analysis of trabecular bone in MR images

The aim of this study was to develop a method of MR image analysis able to provide parameter(s) sensitive to bone microarchitecture changes in aging and osteoporosis onset and progression. The method has been built taking into account fractal properties of many anatomic and physiologic structures. Fractal lacunarity analysis has been used to determine relevant parameter(s) to differentiate among three types of trabecular bone structure (healthy young, healthy perimenopaused, and osteoporotic patients) from lumbar vertebra MR images. In particular, we propose to approximate the lacunarity function by a hyperbola model function, that depends on three different coefficients, /spl alpha/, /spl beta/, /spl gamma/, and to compute these coefficients as the solution of a least squares problem. This term of coefficients provides the model function that better represents the variation of mass density of pixels in the image considered. Clinical application of this preliminary version of our method suggests that one of the three coefficients, namely /spl beta/, may represent a standard for an evaluation of trabecular bone architecture and a potential useful parametric index in early diagnosis of osteoporosis.

A numerical method to solve the inverse medium problem: an application to the Ipswich data. II

We consider the propagation of electromagnetic waves in an inhomogeneous medium. The inhomogeneity is contained in a vertical cylinder with a compact base. When an incident wave meets the inhomogeneity, a scattered wave is generated. We consider the problem of the reconstruction of the inhomogeneous refractive index from some knowledge of the scattered waves generated by the interaction of the inhomogeneity with several known incident waves. We give a mathematical formulation of the problem, a numerical method to solve it, and some numerical results obtained using the numerical method proposed on the Ipswich data. The Ipswich data are experimental data. The inverse-scattering problem considered above is of great theoretical and practical interest. It is an ill-posed problem.

Adaptation of the Canadian Fire Weather Index to Mediterranean forests

The Canadian Fire Weather Index (FWI) is one of the most used tools to forecast forest fire hazard. In this paper, we propose an adaptation of the FWI to take into account the Mediterranean vegetation and climate, to improve FWI performances for areas with these characteristics. In particular, the FWI has been calibrated for two Mediterranean regions (Algarve, Portugal and Peloponnese, Greece) by fitting the experimental data, collected in the field, for the forest fuel moisture content, with the values expected by the application of the original Canadian FWI. Field data were collected during the 2010 and 2011 fire seasons. The satisfactory results obtained by the adaptation of parameters from the original FWI, in spite of the small sample areas and limited time of collection, allowed us to verify the usefulness of the method in describing the fuel moisture dynamics and obtaining indexes closer to the characteristics of the Mediterranean forests considered. The adapted FWI, if further tested and extended to other sample areas, may help in a more detailed and precise application of the FWI index by improving the daily forest fire hazard forecast, to become also a better support for the Civil Protection Agency of Mediterranean countries.

Exploding solutions of the complex two-dimensional Burgers equations: Computer simulations

We study by computer simulations the complex solutions of the two-dimensional Burgers equations in the whole plane in absence of external forces. For such model the existence of singularities, corresponding to a divergence of the total energy at a finite time, is proved by Li and Sinai [“Singularities of complex-valued solutions of the two-dimensional Burgers system,” J. Math. Phys. 51, 015205 (2010)]10.1063/1.3276099 for a large class of initial data. The simulations show that the blow-up takes place in a very short time, of the order of 10−5 time units. Moreover near the blow-up time the support of the solution in Fourier space moves out to infinity along a straight line. In x-space the solutions are concentrated in a finite region, with large space derivatives, as one would expect for physical phenomena such as tornadoes.

The Use of Optimization in the Reconstruction of Obstacles from Acoustic or Electromagnetic Scattering Data

We consider some three dimensional time harmonic acoustic and electromagnetic scattering problems for bounded simply connected obstacles. We consider the following inverse problem: from the knowledge of several far field patterns generated by the obstacle when hit by known incoming waves and from the knowledge of some a-priori information about the obstacle, i.e. boundary impedance, shape symmetry, etc., reconstruct the shape or the shape and the impedance of the obstacle. There are a large number of effective numerical methods to solve the direct problem associated with this inverse problem, but techniques to solve the inverse problem are still in their infancy. We reformulate the inverse problem as two different unconstrained optimization problems. We present a review of results obtained by the authors on the inverse problem and we give some ideas concerning the solution of the direct problem by efficient parallel algorithms.

Three-dimensional time harmonic electromagnetic inverse scattering: The reconstruction of the shape and the impedance of an obstacle

Abstract A numerical method for the reconstruction of the shape and the impedance of an obstacle from time harmonic electromagnetic scattering data is presented. Let D be a bounded, simply connected domain contained in the three-dimensional Euclidean space R 3 , with smooth boundary ∂D . The three-dimensional Euclidean space is filled with an isotropic homogeneous medium. We assume that D contains the origin, and D is regarded as an obstacle whose electric properties are given by a boundary impedance χ(x), x ∈ ∂D . From the knowledge of the electric far fields generated by the obstacle D when hit by known time harmonic electromagnetic waves, the shape ∂D , and the boundary impedance χ(x) of the obstacle are reconstructed. The reconstruction algorithm is based on the “Herglotz function method” introduced by Colton and Monk [1] in acoustic scattering.

Three dimensional time harmonic inverse electromagnetic scattering

A numerical method for the three dimensional inverse electromagnetic time harmonic scattering is presented. We consider an obstacle D that is a bounded, simply connected domain with smooth boundary ∂D contained in the three dimensional euclidean space IR3. The far field patterns of the vector Helmholtz equation generated by a known electromagnetic wave incident on the obstacle D are used as data. From these data the boundary of the obstacle ∂D is reconstructed. The reconstruction procedure proposed here generalizes the “Herglotz function method” introduced by Colton and Monk [1] in the acoustic problem and is effective in theso called resonance region.