We study 3D-multidirectional images, using Finsler geometry. The application considered here is in medical image analysis, specifically in High Angular Resolution Diffusion Imaging (HARDI) (Tuch et al. in Magn. Reson. Med. 48(6):1358-1372) of the brain. The goal is to reveal the architecture of the neural fibers in brain white matter. To the variety of existing techniques, we wish to add novel approaches that exploit differential geometry and tensor calculus.In Diffusion Tensor Imaging (DTI), the diffusion of water is modeled by a symmetric positive definite second order tensor, leading naturally to a Riemannian geometric framework. A limitation is that it is based on the assumption that there exists a single dominant direction of fibers restricting the thermal motion of water molecules. Using HARDI data and higher order tensor models, we can extract multiple relevant directions, and Finsler geometry provides the natural geometric generalization appropriate for multi-fiber analysis. In this paper we provide an exact criterion to determine whether a spherical function satisfies the strong convexity criterion essential for a Finsler norm. We also show a novel fiber tracking method in Finsler setting. Our model incorporates a scale parameter, which can be beneficial in view of the noisy nature of the data. We demonstrate our methods on analytic as well as simulated and real HARDI data.L. Astola ( ) · L. Florack

Abstract. In this paper we discuss new measures for connectivity analysis of brain white matter, using MR diffusion tensor imaging. Our approach is based on Riemannian geometry, the viability of which has been demonstrated by various researchers in foregoing work. In the Riemannian framework bundles of axons are represented by geodesics on the manifold. Here we do not discuss methods to compute these geodesics, nor do we rely on the availability of geodesics. Instead we propose local measures which are directly computable from the local DTI data, and which enable us to preselect viable or exclude uninteresting seed points for the potentially time consuming extraction of geodesics. If geodesics are available, our measures can be readily applied to these as well.We consider two types of geodesic measures. One pertains to the connectivity saliency of a geodesic, the second to its stability with respect to local spatial perturbations. For the first type of measure we consider both differential as well as integral measures for characterizing a geodesic's saliency either locally or globally. (In the latter case one needs to be in possession of the geodesic curve, in the former case a single tangent vector suffices.) The second type of measure is intrinsically local, and turns out to be related to a well known tensor in Riemannian geometry.

• A submitted manuscript is the author's version of the article upon submission and before peer-review. There can be important differences between the submitted version and the official published version of record. People interested in the research are advised to contact the author for the final version of the publication, or visit the DOI to the publisher's website.• The final author version and the galley proof are versions of the publication after peer review.• The final published version features the final layout of the paper including the volume, issue and page numbers. Link to publication Citation for published version (APA):Gomez-Roldan, M. V., Engel, B., Vos, de, R. C. H., Vereijken, P., Astola, L. J., Groenenboom, M., ... Hall, R. D. (2014). Metabolomics reveals organ-specic metabolic rearrangements during early tomato seedling development. Metabolomics, 10(5), 958-974. DOI: 10.1007/s11306-014-0625-2 General rightsCopyright and moral rights for the publications made accessible in the public portal are retained by the authors and/or other copyright owners and it is a condition of accessing publications that users recognise and abide by the legal requirements associated with these rights.• Users may download and print one copy of any publication from the public portal for the purpose of private study or research.• You may not further distribute the material or use it for any profit-making activity or commercial gain • You may freely distribute the URL identifying the publication in the public portal ? Take down policyIf you believe that this document breaches copyright please contact us providing details, and we will remove access to the work immediately and investigate your claim. Abstract Tomato seedlings (Solanum lycopersicum cv. MoneyMaker), grown under strictly controlled conditions, have been used to study alterations occurring in secondary metabolite biosynthetic pathways following developmental and environmental perturbations. Robustness and reproducibility of the system were confirmed using detailed statistical analyses of the metabolome. LCMS profiling was applied using whole germinated seeds as well as cotyledons, hypocotyls and roots from 3 to 9 days old seedlings to generate relative levels of 433 metabolites, of which 62 were annotated. Initial focus was given to the polyphenol pathway and several additional mass signals have been putatively annotated using high mass resolution fragmentation. Clear organ and developmental stage-specific differences were observed. Seeds accumulated saponin-like compounds; roots accumulated mainly alkaloids; cotyledons contained mainly glycosylated flavonols and; hypocotyls contained mainly anthocyanins. For each organ, the developmental changes in metabolite profiles were described by using linear mixed models. Across three independent experiments, 85 % of the metabolites showed similar developmental trends. This tomato seedling system has given us valuable additional insights into the complexity of seedling secondary metabolism. How metabolic profiles refle...

Flavonoids are secondary metabolites present in all terrestrial plants. The flavonoid pathway has been extensively studied, and many of the involved genes and metabolites have been described in the literature. Despite this extensive knowledge, the functioning of the pathway in vivo is still poorly understood. Here, we study the flavonoid pathway using both experiments and mathematical models. We measured flavonoid metabolite dynamics in two tissues, hypocotyls and cotyledons, during tomato seedling development. Interestingly, the same backbone of interactions leads to very different accumulation patterns in the different tissues. Initially, we developed a mathematical model with constant enzyme concentrations that described the metabolic networks separately in both tissues. This model was unable to fit the measured flavonoid dynamics in the hypocotyls, even if we allowed unrealistic parameter values. This suggested us to investigate the effect of transcript abundance on flavonoid accumulation. We found that the expression of candidate flavonoid genes varies considerably with time. Variation in transcript abundance results in enzymatic variation, which could have a large effect on metabolite accumulation. Candidate transcript abundance was included in the mathematical model as representative for enzyme concentration. We fitted the resulting model to the flavonoid dynamics in the cotyledons, and tested it by applying it to the data from hypocotyls. When transcript abundance is included, we are indeed able to explain flavonoid dynamics in both tissues. Importantly, this is possible under the biologically relevant restriction that the enzymatic properties estimated by the model are conserved between the tissues.