TY  - JOUR
A1  - Ziener, Christian H.
A1  - Kurz, Felix T.
A1  - Buschle, Lukas R.
A1  - Kampf, Thomas
T1  - Orthogonality, Lommel integrals and cross product zeros of linear combinations of Bessel functions
JF  - SpringerPlus
N2  - 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.
KW  - magnetic field relaxation
KW  - time inhomogeneities
KW  - model Bessel function
KW  - linear combination
KW  - integral
Y1  - 2015
U6  - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:bvb:20-opus-151432
VL  - 4
IS  - 390
ER  - 
TY  - JOUR
A1  - Kurz, Felix T.
A1  - Kampf, Thomas
A1  - Buschle, Lukas R.
A1  - Schlemmer, Heinz-Peter
A1  - Heiland, Sabine
A1  - Bendszus, Martin
A1  - Ziener, Christian H.
T1  - Microstructural Analysis of Peripheral Lung Tissue through CPMG Inter-Echo Time R2 Dispersion
JF  - PLoS One
N2  - 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.
KW  - transverse relaxation
KW  - number
KW  - HE-3 diffusion MRI
KW  - magnetic-resonance behavior
KW  - hyperpolarized HE-3
KW  - Bessle functions
KW  - self-diffusion
KW  - alveolar
KW  - field
KW  - morphometry
Y1  - 2015
U6  - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:bvb:20-opus-138345
VL  - 10
IS  - 11
ER  -