On the geometry of rolling and interpolation curves on S-n, SOn, and Grassmann manifolds

We present a procedure to generate smooth interpolating curves on submanifolds, which are given in closed form in terms of the coordinates of the embedding space. In contrast to other existing methods, this approach makes the corresponding algorithm eas

Imaging the kidney using magnetic resonance techniques: structure to function

Purpose of review Magnetic resonance imaging (MRI) offers the possibility to non-invasively assess both the structure and function of the kidney in a single MR scan session. This review summarises recent advancements in functional renal MRI techniques, with a particular focus on their clinical relevance. Recent findings A number of MRI techniques have been developed that provide non-invasive measures of relevance to the pathophysiology of kidney disease. Diffusion-weighted imaging (DWI) has been used in chronic kidney disease (CKD) and renal transplantation, and appears promising as a measure of renal impairment and fibrosis. Longitudinal relaxation time (T1) mapping has been utilised in cardiac MRI to measure fibrosis and oedema; recent work suggests its potential for assessment of the kidney. Blood oxygen level dependent (BOLD) MRI to measure renal oxygenation has been extensively studied, but a number of other factors affect results making it hard to draw definite conclusions as to its utility as an independent measure. Phase contrast and arterial spin labelling (ASL) can measure renal artery blood flow and renal perfusion respectively without exogenous contrast, in contrast to dynamic contrast enhanced (DCE) studies. Current data on clinical use of such functional renal MR measures is largely restricted to cross-sectional studies. Summary Renal MRI has seen significant recent interest and advances. Current evidence demonstrates its potential, and next steps include wider evaluation of its clinical application