Improved reconstruction methodology of clinical electron energy spectra based on Tikhonov regularization and generalized simulated annealing

Electron beam radiotherapy is the most widespread treatment modality todeal with superficial cancers. In electron radiotherapy, the energy spectrum isimportant for electron beam modelling and accurate dose calculation. Since thepercentage depth-dose (PDD) is a function of the beam’s energy, the reconstruction of the spectrum from the depth-dose curve represents an inverse problem.Thus, the energy spectrum can be related to the depth-dose by means of anappropriate mathematical model as the Fredholm equation of the first kind.Since the Fredholm equation of the first kind is ill-posed, some regularizationmethod has to be used to achieve a useful solution. In this work the Tikhonovregularization function was solved by the generalized simulated annealing optimization method. The accuracy of the reconstruction was verified by thegamma index passing rate criterion applied to the simulated PDD curves forthe reconstructed spectra compared to experimental PDD curves. Results showa good coincidence between the experimental and simulated depth-dose curvesaccording to the gamma passing rate better than 95% for 1% dose difference(DD)/1 mm distance to agreement (DTA) criteria. Moreover, the results showimprovement from previous works not only in accuracy but also in calculationtime. In general, the proposed method can help in the accuracy of dosimetryprocedures, treatment planning and quality control in radiotherapy.

Mapping the Intensity Function of a Non-stationary Point Process in Unobserved Areas

Seismic networks provide data that are used as basis both for public safety decisions and for scientific research. Their configuration affects the data completeness, which in turn, critically affects several seismological scientific targets (e.g., earthquake prediction, seismic hazard...). In this context, a key aspect is how to map earthquakes density in seismogenic areas from censored data or even in areas that are not covered by the network. We propose to predict the spatial distribution of earthquakes from the knowledge of presence locations and geological relationships, taking into account any interactions between records. Namely, in a more general setting, we aim to estimate the intensity function of a point process, conditional to its censored realization, as in geostatistics for continuous processes. We define a predictor as the best linear unbiased combination of the observed point pattern. We show that the weight function associated to the predictor is the solution of a Fredholm equation of second kind. Both the kernel and the source term of the Fredholm equation are related to the first- and second-order characteristics of the point process through the intensity and the pair correlation function. Results are presented and illustrated on simulated non-stationary point processes and real data for mapping Greek Hellenic seismicity in a region with unreliable and incomplete records.

Integral Equations for Problems on Wave Propagation in Near-Earth Plasma

We consider the solutions of two integrodifferential equations in this work. These equations describe the ultra-low frequency waves in the dipol-like model of the magnetosphere in the gyrokinetic framework. The first one is reduced to the homogeneous, second kind Fredholm equation. This equation describes the structure of the parallel component of the magnetic field of drift-compression waves along the Earth’s magnetic field. The second equation is reduced to the inhomogeneous, second kind Fredholm equation. This equation describes the field-aligned structure of the parallel electric field potential of Alfvén waves. Both integral equations are solved numerically.

On the Zap Integral Operators over Fourier Transforms

We devote the current chapter to describe a class of integral operators with properties equivalent to a killer operator of the quantum mechanics theory acting over a determined state, literally killing the state but now operating over some kind of Fourier integral transforms that satisfies a certain Fredholm integral equation, we call this operators Zap Integral Operators (ZIO). The result of this action is to eliminate the inhomogeneous term and recover a homogeneous integral equation. We show that thanks to this class of operators we can explain the presence of two extremely different solutions of the same Generalized Inhomogeneous Fredholm equation. So we can regard the Generalized Inhomogeneous Fredholm Equation as a Super-Equation with two kinds of solutions, the resonant and the conventional but coexisting simultaneously. Also, we remember the generalized projection operators and we show they are the precursors of the ZIO. We present simultaneous academic examples for both kinds of solutions.