Examples of signature (2,2) manifolds with commuting curvature operators

We exhibit Walker manifolds of signature (2,2) with various commutativity properties for the Ricci operator, the skew-symmetric curvature operator, and the Jacobi operator. If the Walker metric is a Riemannian extension of an underlying affine structure A, these properties are related to the Ricci tensor of A

Immunotherapy in patients with brain metastasis: advances and challenges for the treatment and the application of circulating biomarkers

The central nervous system (CNS) is one of the most frequent metastatic sites of various cancers, including lung cancer, breast cancer and melanoma. The development of brain metastases requires a specific therapeutic approach and is associated with high mortality and morbidity in cancer patients. Advances in precision medicine and the introduction in recent years of new drugs, such as immunotherapy, have made it possible to improve the prognosis of these patients by improving survival and quality of life. New diagnostic techniques such as liquid biopsy allow real-time monitoring of tumor evolution, providing molecular information on prognostic and predictive biomarkers of response to treatment in blood or other fluids. In this review, we perform an exhaustive update of the clinical trials that demonstrate the utility of immunotherapy in patients with brain metastases and the potential of circulating biomarkers to improving the results of efficacy and toxicity in this subgroup of patients