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Orthogonality, Lommel integrals and cross product zeros of linear combinations of Bessel functions
(2015)

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.

Microstructural Analysis of Peripheral Lung Tissue through CPMG Inter-Echo Time R2 Dispersion
(2015)

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.

It is poorly understood how progressive brain swelling in experimental cerebral malaria (ECM) evolves in space and over time, and whether mechanisms of inflammation or microvascular sequestration/obstruction dominate the underlying pathophysiology. We therefore monitored in the Plasmodium berghei ANKA-C57BL/6 murine ECM model, disease manifestation and progression clinically, assessed by the Rapid-Murine-Coma-and-Behavioral-Scale (RMCBS), and by high-resolution in vivo MRI, including sensitive assessment of early blood-brain-barrier-disruption (BBBD), brain edema and microvascular pathology. For histological correlation HE and immunohistochemical staining for microglia and neuroblasts were obtained. Our results demonstrate that BBBD and edema initiated in the olfactory bulb (OB) and spread along the rostral-migratory-stream (RMS) to the subventricular zone of the lateral ventricles, the dorsal-migratory-stream (DMS), and finally to the external capsule (EC) and brainstem (BS). Before clinical symptoms (mean RMCBS = 18.5±1) became evident, a slight, non-significant increase of quantitative T2 and ADC values was observed in OB+RMS. With clinical manifestation (mean RMCBS = 14.2±0.4), T2 and ADC values significantly increased along the OB+RMS (p = 0.049/p = 0.01). Severe ECM (mean RMCBS = 5±2.9) was defined by further spread into more posterior and deeper brain structures until reaching the BS (significant T2 elevation in DMS+EC+BS (p = 0.034)). Quantitative automated histological analyses confirmed microglial activation in areas of BBBD and edema. Activated microglia were closely associated with the RMS and neuroblasts within the RMS were severely misaligned with respect to their physiological linear migration pattern. Microvascular pathology and ischemic brain injury occurred only secondarily, after vasogenic edema formation and were both associated less with clinical severity and the temporal course of ECM. Altogether, we identified a distinct spatiotemporal pattern of microglial activation in ECM involving primarily the OB+RMS axis, a distinct pathway utilized by neuroblasts and immune cells. Our data suggest significant crosstalk between these two cell populations to be operative in deeper brain infiltration and further imply that the manifestation and progression of cerebral malaria may depend on brain areas otherwise serving neurogenesis.