An intrinsic characterization of 2+2 warped spacetimes

We give several equivalent conditions that characterize the 2+2 warped spacetimes: imposing the existence of a Killing-Yano tensor $A$ subject to complementary algebraic restrictions; in terms of the projector $v$ (or of the canonical 2-form $U$) associated with the 2-planes of the warped product. These planes are principal planes of the Weyl and/or Ricci tensors and can be explicitly obtained from them. Therefore, we obtain the necessary and sufficient (local) conditions for a metric tensor to be a 2+2 warped product. These conditions exclusively involve explicit concomitants of the Riemann tensor. We present a similar analysis for the conformally 2+2 product spacetimes and give an invariant classification of them. The warped products correspond to two of these invariant classes. The more degenerate class is the set of product metrics which are also studied from an invariant point of view.Comment: 18 pages; submitted to Class. Quantum Grav

Vacuum type I spacetimes and aligned Papapetrou fields: symmetries

We analyze type I vacuum solutions admitting an isometry whose Killing 2--form is aligned with a principal bivector of the Weyl tensor, and we show that these solutions belong to a family of type I metrics which admit a group $G_3$ of isometries. We give a classification of this family and we study the Bianchi type for each class. The classes compatible with an aligned Killing 2--form are also determined. The Szekeres-Brans theorem is extended to non vacuum spacetimes with vanishing Cotton tensor.Comment: 19 pages; a reference adde

Type I vacuum solutions with aligned Papapetrou fields: an intrinsic characterization

We show that Petrov type I vacuum solutions admitting a Killing vector whose Papapetrou field is aligned with a principal bivector of the Weyl tensor are the Kasner and Taub metrics, their counterpart with timelike orbits and their associated windmill-like solutions, as well as the Petrov homogeneous vacuum solution. We recover all these metrics by using an integration method based on an invariant classification which allows us to characterize every solution. In this way we obtain an intrinsic and explicit algorithm to identify them.Comment: 14 pages; v2: added new section, references and tabl

On the classification of type D spacetimes

We give a classification of the type D spacetimes based on the invariant differential properties of the Weyl principal structure. Our classification is established using tensorial invariants of the Weyl tensor and, consequently, besides its intrinsic nature, it is valid for the whole set of the type D metrics and it applies on both, vacuum and non-vacuum solutions. We consider the Cotton-zero type D metrics and we study the classes that are compatible with this condition. The subfamily of spacetimes with constant argument of the Weyl eigenvalue is analyzed in more detail by offering a canonical expression for the metric tensor and by giving a generalization of some results about the non-existence of purely magnetic solutions. The usefulness of these results is illustrated in characterizing and classifying a family of Einstein-Maxwell solutions. Our approach permits us to give intrinsic and explicit conditions that label every metric, obtaining in this way an operational algorithm to detect them. In particular a characterization of the Reissner-Nordstr\"{o}m metric is accomplished.Comment: 29 pages, 0 figure

Painlev\'e-Gullstrand synchronizations in spherical symmetry

A Painlev\'e-Gullstrand synchronization is a slicing of the space-time by a family of flat spacelike 3-surfaces. For spherically symmetric space-times, we show that a Painlev\'e-Gullstrand synchronization only exists in the region where $(dr)^2 \leq 1$, $r$ being the curvature radius of the isometry group orbits ($2$-spheres). This condition says that the Misner-Sharp gravitational energy of these 2-spheres is not negative and has an intrinsic meaning in terms of the norm of the mean extrinsic curvature vector. It also provides an algebraic inequality involving the Weyl curvature scalar and the Ricci eigenvalues. We prove that the energy and momentum densities associated with the Weinberg complex of a Painlev\'e-Gullstrand slice vanish in these curvature coordinates, and we give a new interpretation of these slices by using semi-metric Newtonian connections. It is also outlined that, by solving the vacuum Einstein's equations in a coordinate system adapted to a Painlev\'e-Gullstrand synchronization, the Schwarzschild solution is directly obtained in a whole coordinate domain that includes the horizon and both its interior and exterior regions.Comment: 16 page

The Cotton tensor in Riemannian spacetimes

Recently, the study of three-dimensional spaces is becoming of great interest. In these dimensions the Cotton tensor is prominent as the substitute for the Weyl tensor. It is conformally invariant and its vanishing is equivalent to conformal flatness. However, the Cotton tensor arises in the context of the Bianchi identities and is present in any dimension. We present a systematic derivation of the Cotton tensor. We perform its irreducible decomposition and determine its number of independent components for the first time. Subsequently, we exhibit its characteristic properties and perform a classification of the Cotton tensor in three dimensions. We investigate some solutions of Einstein's field equations in three dimensions and of the topologically massive gravity model of Deser, Jackiw, and Templeton. For each class examples are given. Finally we investigate the relation between the Cotton tensor and the energy-momentum in Einstein's theory and derive a conformally flat perfect fluid solution of Einstein's field equations in three dimensions.Comment: 27 pages, revtex

Robust association between vascular habitats and patient prognosis in glioblastoma: an international retrospective multicenter study

This is the peer reviewed version of the following article: del Mar Álvarez-Torres, M., Juan-Albarracín, J., Fuster-Garcia, E., Bellvís-Bataller, F., Lorente, D., Reynés, G., Font de Mora, J., Aparici-Robles, F., Botella, C., Muñoz-Langa, J., Faubel, R., Asensio-Cuesta, S., García-Ferrando, G.A., Chelebian, E., Auger, C., Pineda, J., Rovira, A., Oleaga, L., Mollà-Olmos, E., Revert, A.J., Tshibanda, L., Crisi, G., Emblem, K.E., Martin, D., Due-Tønnessen, P., Meling, T.R., Filice, S., Sáez, C. and García-Gómez, J.M. (2020), Robust association between vascular habitats and patient prognosis in glioblastoma: An international multicenter study. J Magn Reson Imaging, 51: 1478-1486, which has been published in final form at https://doi.org/10.1002/jmri.26958. This article may be used for non-commercial purposes in accordance with Wiley Terms and Conditions for Self-Archiving.[EN] Background Glioblastoma (GBM) is the most aggressive primary brain tumor, characterized by a heterogeneous and abnormal vascularity. Subtypes of vascular habitats within the tumor and edema can be distinguished: high angiogenic tumor (HAT), low angiogenic tumor (LAT), infiltrated peripheral edema (IPE), and vasogenic peripheral edema (VPE). Purpose To validate the association between hemodynamic markers from vascular habitats and overall survival (OS) in glioblastoma patients, considering the intercenter variability of acquisition protocols. Study Type Multicenter retrospective study. Population In all, 184 glioblastoma patients from seven European centers participating in the NCT03439332 clinical study. Field Strength/Sequence 1.5T (for 54 patients) or 3.0T (for 130 patients). Pregadolinium and postgadolinium-based contrast agent-enhanced T-1-weighted MRI, T-2- and FLAIR T-2-weighted, and dynamic susceptibility contrast (DSC) T-2* perfusion. Assessment We analyzed preoperative MRIs to establish the association between the maximum relative cerebral blood volume (rCBV(max)) at each habitat with OS. Moreover, the stratification capabilities of the markers to divide patients into "vascular" groups were tested. The variability in the markers between individual centers was also assessed. Statistical Tests Uniparametric Cox regression; Kaplan-Meier test; Mann-Whitney test. Results The rCBV(max) derived from the HAT, LAT, and IPE habitats were significantly associated with patient OS (P < 0.05; hazard ratio [HR]: 1.05, 1.11, 1.28, respectively). Moreover, these markers can stratify patients into "moderate-" and "high-vascular" groups (P < 0.05). The Mann-Whitney test did not find significant differences among most of the centers in markers (HAT: P = 0.02-0.685; LAT: P = 0.010-0.769; IPE: P = 0.093-0.939; VPE: P = 0.016-1.000). Data Conclusion The rCBV(max) calculated in HAT, LAT, and IPE habitats have been validated as clinically relevant prognostic biomarkers for glioblastoma patients in the pretreatment stage. This study demonstrates the robustness of the hemodynamic tissue signature (HTS) habitats to assess the GBM vascular heterogeneity and their association with patient prognosis independently of intercenter variability. Technical Efficacy Stage: 2 J. Magn. Reson. Imaging 2019.This work was partially supported by: MTS4up project (National Plan for Scientific and Technical Research and Innovation 2013-2016, No. DPI2016-80054-R) (to J.M.G.G.); H2020-SC1-2016-CNECT Project (No. 727560) (to J.M.G.G.) and H2020-SC1-BHC-2018-2020 (No. 825750) (to J.M.G.G.); M.A.T was supported by DPI2016-80054-R (Programa Estatal de Promocion del Talento y su Empleabilidad en I + D + i). The data acquisition and curation of the Oslo University Hospital was supported by: the European Research Council (ERC) under the European Union's Horizon 2020 (Grant Agreement No. 758657), the South-Eastern Norway Regional Health Authority Grants 2017073 and 2013069, and the Research Council of Norway Grants 261984 (to K.E.E.). We gratefully acknowledge the support of NVIDIA Corporation with the donation of the Titan V GPU used for this research. E.F.G. was supported by the European Union's Horizon 2020 research and innovation programme under the Marie Sklodowska-Curie grant agreement No. 844646. Figure 1 was designed by the Science Artist Elena Poritskaya.Álvarez-Torres, MDM.; Juan-Albarracín, J.; Fuster García, E.; Bellvís-Bataller, F.; Lorente, D.; Reynés, G.; Font De Mora, J.... (2020). Robust association between vascular habitats and patient prognosis in glioblastoma: an international retrospective multicenter study. Journal of Magnetic Resonance Imaging. 51(5):1478-1486. https://doi.org/10.1002/jmri.2695814781486515Louis, D. N., Perry, A., Reifenberger, G., von Deimling, A., Figarella-Branger, D., Cavenee, W. K., … Ellison, D. W. (2016). The 2016 World Health Organization Classification of Tumors of the Central Nervous System: a summary. Acta Neuropathologica, 131(6), 803-820. doi:10.1007/s00401-016-1545-1Gately, L., McLachlan, S., Dowling, A., & Philip, J. (2017). Life beyond a diagnosis of glioblastoma: a systematic review of the literature. Journal of Cancer Survivorship, 11(4), 447-452. doi:10.1007/s11764-017-0602-7Bae, S., Choi, Y. S., Ahn, S. S., Chang, J. H., Kang, S.-G., Kim, E. H., … Lee, S.-K. (2018). Radiomic MRI Phenotyping of Glioblastoma: Improving Survival Prediction. Radiology, 289(3), 797-806. doi:10.1148/radiol.2018180200Akbari, H., Macyszyn, L., Da, X., Wolf, R. L., Bilello, M., Verma, R., … Davatzikos, C. (2014). Pattern Analysis of Dynamic Susceptibility Contrast-enhanced MR Imaging Demonstrates Peritumoral Tissue Heterogeneity. Radiology, 273(2), 502-510. doi:10.1148/radiol.14132458Weis, S. M., & Cheresh, D. A. (2011). Tumor angiogenesis: molecular pathways and therapeutic targets. Nature Medicine, 17(11), 1359-1370. doi:10.1038/nm.2537De Palma, M., Biziato, D., & Petrova, T. V. (2017). Microenvironmental regulation of tumour angiogenesis. Nature Reviews Cancer, 17(8), 457-474. doi:10.1038/nrc.2017.51Jain, R., Poisson, L. M., Gutman, D., Scarpace, L., Hwang, S. N., Holder, C. A., … Flanders, A. (2014). Outcome Prediction in Patients with Glioblastoma by Using Imaging, Clinical, and Genomic Biomarkers: Focus on the Nonenhancing Component of the Tumor. Radiology, 272(2), 484-493. doi:10.1148/radiol.14131691Jensen, R. L., Mumert, M. L., Gillespie, D. L., Kinney, A. Y., Schabel, M. C., & Salzman, K. L. (2013). Preoperative dynamic contrast-enhanced MRI correlates with molecular markers of hypoxia and vascularity in specific areas of intratumoral microenvironment and is predictive of patient outcome. Neuro-Oncology, 16(2), 280-291. doi:10.1093/neuonc/not148Jena, A., Taneja, S., Gambhir, A., Mishra, A. K., D’souza, M. M., Verma, S. M., … Sogani, S. K. (2016). Glioma Recurrence Versus Radiation Necrosis. Clinical Nuclear Medicine, 41(5), e228-e236. doi:10.1097/rlu.0000000000001152Price, S. J., Young, A. M. H., Scotton, W. J., Ching, J., Mohsen, L. A., Boonzaier, N. R., … Larkin, T. J. (2015). Multimodal MRI can identify perfusion and metabolic changes in the invasive margin of glioblastomas. Journal of Magnetic Resonance Imaging, 43(2), 487-494. doi:10.1002/jmri.24996Chang, Y.-C. C., Ackerstaff, E., Tschudi, Y., Jimenez, B., Foltz, W., Fisher, C., … Stoyanova, R. (2017). Delineation of Tumor Habitats based on Dynamic Contrast Enhanced MRI. Scientific Reports, 7(1). doi:10.1038/s41598-017-09932-5Cui, Y., Tha, K. K., Terasaka, S., Yamaguchi, S., Wang, J., Kudo, K., … Li, R. (2016). Prognostic Imaging Biomarkers in Glioblastoma: Development and Independent Validation on the Basis of Multiregion and Quantitative Analysis of MR Images. Radiology, 278(2), 546-553. doi:10.1148/radiol.2015150358Juan-Albarracín, J., Fuster-Garcia, E., Pérez-Girbés, A., Aparici-Robles, F., Alberich-Bayarri, Á., Revert-Ventura, A., … García-Gómez, J. M. (2018). Glioblastoma: Vascular Habitats Detected at Preoperative Dynamic Susceptibility-weighted Contrast-enhanced Perfusion MR Imaging Predict Survival. Radiology, 287(3), 944-954. doi:10.1148/radiol.2017170845Fuster-Garcia, E., Juan-Albarracín, J., García-Ferrando, G. A., Martí-Bonmatí, L., Aparici-Robles, F., & García-Gómez, J. M. (2018). Improving the estimation of prognosis for glioblastoma patients by MR based hemodynamic tissue signatures. NMR in Biomedicine, 31(12), e4006. doi:10.1002/nbm.4006Abramson, R. G., Burton, K. R., Yu, J.-P. J., Scalzetti, E. M., Yankeelov, T. E., Rosenkrantz, A. B., … Subramaniam, R. M. (2015). Methods and Challenges in Quantitative Imaging Biomarker Development. Academic Radiology, 22(1), 25-32. doi:10.1016/j.acra.2014.09.001Stupp, R., Mason, W. P., van den Bent, M. J., Weller, M., Fisher, B., Taphoorn, M. J. B., … Mirimanoff, R. O. (2005). Radiotherapy plus Concomitant and Adjuvant Temozolomide for Glioblastoma. New England Journal of Medicine, 352(10), 987-996. doi:10.1056/nejmoa043330Wetzel, S. G., Cha, S., Johnson, G., Lee, P., Law, M., Kasow, D. L., … Xue, X. (2002). Relative Cerebral Blood Volume Measurements in Intracranial Mass Lesions: Interobserver and Intraobserver Reproducibility Study. Radiology, 224(3), 797-803. doi:10.1148/radiol.2243011014Schnack, H. G., van Haren, N. E. M., Hulshoff Pol, H. E., Picchioni, M., Weisbrod, M., Sauer, H., … Kahn, R. S. (2004). Reliability of brain volumes from multicenter MRI acquisition: A calibration study. Human Brain Mapping, 22(4), 312-320. doi:10.1002/hbm.20040De Guio, F., Jouvent, E., Biessels, G. J., Black, S. E., Brayne, C., Chen, C., … Chabriat, H. (2016). Reproducibility and variability of quantitative magnetic resonance imaging markers in cerebral small vessel disease. Journal of Cerebral Blood Flow & Metabolism, 36(8), 1319-1337. doi:10.1177/0271678x16647396Hirai, T., Murakami, R., Nakamura, H., Kitajima, M., Fukuoka, H., Sasao, A., … Yamashita, Y. (2008). Prognostic Value of Perfusion MR Imaging of High-Grade Astrocytomas: Long-Term Follow-Up Study. American Journal of Neuroradiology, 29(8), 1505-1510. doi:10.3174/ajnr.a1121Sawlani, R. N., Raizer, J., Horowitz, S. W., Shin, W., Grimm, S. A., Chandler, J. P., … Carroll, T. J. (2010). Glioblastoma: A Method for Predicting Response to Antiangiogenic Chemotherapy by Using MR Perfusion Imaging—Pilot Study. Radiology, 255(2), 622-628. doi:10.1148/radiol.10091341Hambardzumyan, D., & Bergers, G. (2015). Glioblastoma: Defining Tumor Niches. Trends in Cancer, 1(4), 252-265. doi:10.1016/j.trecan.2015.10.009Artzi, M., Bokstein, F., Blumenthal, D. T., Aizenstein, O., Liberman, G., Corn, B. W., & Ben Bashat, D. (2014). Differentiation between vasogenic-edema versus tumor-infiltrative area in patients with glioblastoma during bevacizumab therapy: A longitudinal MRI study. European Journal of Radiology, 83(7), 1250-1256. doi:10.1016/j.ejrad.2014.03.02