@article{ZienerKurzBuschleetal.2015,
  author    = {Ziener, Christian H. and Kurz, Felix T. and Buschle, Lukas R. and Kampf, Thomas},
  title     = {Orthogonality, Lommel integrals and cross product zeros of linear combinations of Bessel functions},
  series = {SpringerPlus},
  volume    = {4},
  journal   = {SpringerPlus},
  number    = {390},
  doi       = {10.1186/s40064-015-1142-0},
  url       = {http://nbn-resolving.de/urn:nbn:de:bvb:20-opus-151432},
  year      = {2015},
  abstract  = {The cylindrical Bessel differential equation and the spherical Bessel differential equation in the interval R\(\leq\)r\(\leq\)\(\gamma\)R with Neumann boundary conditions are considered. The eigenfunctions are linear combinations of the Bessel function \(\Phi\)\(_{n,ν}\)(r) = Y'\(_{ν}\) (\(\lambda\)\(_{n,ν}\))J\(_{ν}\)(\(\lambda\)\(_{n,ν}\) r/R) - J'\(_{ν}\)(\(\lambda\)\(_{n,ν}\))Y\(_{ν}\)(\(\lambda\)\(_{n,ν}\)r/R) or linear combinations of the spherical Bessel functions \(\psi\)\(_{m,ν}\)(r) = y'\(_{ν}\)(\(\lambda\)\(_{m,ν}\))j\(_{ν}\)(\(\lambda\)\(_{m,ν}\)r/R) - j'\(_{ν}\)(\(\lambda\)\(_{m,ν}\))y\(_{ν}\)(\(\lambda\)\(_{m,ν}\)r/R). The orthogonality relations with analytical expressions for the normalization constant are given. Explicit expressions for the Lommel integrals in terms of Lommel functions are derived. The cross product zeros Y'\(_{ν}\)\(\lambda\)\(_{n,ν}\))J'\(_{ν}\)(\(\gamma\)\(\lambda\)\(_{n,ν}\))- J'\(_{ν}\)(\(\lambda\)\(_{n,ν}\))Y'\(_{ν}\)(\(\gamma\)\(\lambda\)\(_{n,ν}\)) = 0 and y'\(_{ν}\)(\(\lambda\)\(_{m,ν}\))j'\(_{ν}\)(\(\gamma\)\(\lambda\)\(_{m,ν}\)) - j'\(_{ν}\)(\(\lambda\)\(_{m,ν}\))y'\(_{ν}\)(\(\gamma\)\(\lambda\)\(_{m,ν}\)) = 0 are considered in the complex plane for real as well as complex values of the index ν and approximations for the exceptional zero \(\lambda\)\(_{1,ν}\) are obtained. A numerical scheme based on the discretization of the twodimensional and three-dimensional Laplace operator with Neumann boundary conditions is presented. Explicit representations of the radial part of the Laplace operator in form of a tridiagonal matrix allow the simple computation of the cross product zeros.},
  language  = {en}
}
@article{KurzKampfBuschleetal.2015,
  author    = {Kurz, Felix T. and Kampf, Thomas and Buschle, Lukas R. and Schlemmer, Heinz-Peter and Heiland, Sabine and Bendszus, Martin and Ziener, Christian H.},
  title     = {Microstructural Analysis of Peripheral Lung Tissue through CPMG Inter-Echo Time R2 Dispersion},
  series = {PLoS One},
  volume    = {10},
  journal   = {PLoS One},
  number    = {11},
  doi       = {10.1371/journal.pone.0141894},
  url       = {http://nbn-resolving.de/urn:nbn:de:bvb:20-opus-138345},
  pages     = {e0141894},
  year      = {2015},
  abstract  = {Since changes in lung microstructure are important indicators for (early stage) lung pathology, there is a need for quantifiable information of diagnostically challenging cases in a clinical setting, e.g. to evaluate early emphysematous changes in peripheral lung tissue. Considering alveoli as spherical air-spaces surrounded by a thin film of lung tissue allows deriving an expression for Carr-Purcell-Meiboom-Gill transverse relaxation rates R-2 with a dependence on inter-echo time, local air-tissue volume fraction, diffusion coefficient and alveolar diameter, within a weak field approximation. The model relaxation rate exhibits the same hyperbolic tangent dependency as seen in the Luz-Meiboom model and limiting cases agree with Brooks et al. and Jensen et al. In addition, the model is tested against experimental data for passively deflated rat lungs: the resulting mean alveolar radius of RA = 31.46 \(\pm\) 13.15 \(\mu\)m is very close to the literature value (similar to 34 \(\mu\)m). Also, modeled radii obtained from relaxometer measurements of ageing hydrogel foam (that mimics peripheral lung tissue) are in good agreement with those obtained from mu CT images of the same foam (mean relative error: 0.06 \(\pm\) 0.01). The model's ability to determine the alveolar radius and/or air volume fraction will be useful in quantifying peripheral lung microstructure.},
  language  = {en}
}