Piecewise fractal interpolation models to reconstruct the three-dimensional tumor perfusion [MRI data analysis

Abstract

Fractal interpolation functions (FIFs) are used to reconstruct the 3D tumor perfusion data from 2D slices obtained using DE-MRI, as an alternative to the classical interpolation means (spline, linear and polynomial). Two FIFs techniques are used: the piece-wise self-affine fractal interpolation model (PSAFIM) and the piece-wise hidden variable fractal interpolation model (PHVFIM). The fractal dimension of the original 2D slices is considered piece-wise smooth, such that the distance between the fractal dimension of the reconstructed 2D slices and the fractal dimension of the 2D original slices it is used as a measure of the accuracy of the data fit. The conclusion is that the FIFs methods conserve the fractal-like structure of the perfusion data observed in the 2D analysis. With the drawback of algebraic complexity there appear advantages in pursuing multiple variable fractal interpolation (the PHVFI method) for both increasing flexibility in matching data and in constructing joint properties of data (cross correlation between the data in the 2D slices and the data along the reconstructed direction). The 3D structure resulted with PHVFIM has a fractal dimension within 3-5% of the one reported for the 3D percolation, and a minimum path dimension larger than one. It is thus concluded that the reconstructed 3D perfusion data has a percolation-like scaling as was previously assumed. As the perfusion term from the bio-heat equation is better described by reconstruction via fractal interpolation, a more suitable computation of the temperature field induced during hyperthermia treatments is expected.