Eigenvalue asymptotics for weighted Laplace equations on rough Riemannian manifolds with boundary

Our topological setting is a smooth compact manifold of dimension two or higher with smooth boundary. Although this underlying topological structure is smooth, the Riemannian metric tensor is only assumed to be bounded and measurable. This is known as a rough Riemannian manifold. For a large class of boundary conditions we demonstrate a Weyl law for the asymptotics of the eigenvalues of the Laplacian associated to a rough metric. Moreover, we obtain eigenvalue asymptotics for weighted Laplace equations associated to a rough metric. Of particular novelty is that the weight function is not assumed to be of fixed sign, and thus the eigenvalues may be both positive and negative. Key ingredients in the proofs were demonstrated by Birman and Solomjak nearly fifty years ago in their seminal work on eigenvalue asymptotics. In addition to determining the eigenvalue asymptotics in the rough Riemannian manifold setting for weighted Laplace equations, we also wish to promote their achievements which may have further applications to modern problems

Pretransplant C3D-fixing donor-specific anti-HLA antibodies are not associated with increased risk for kidney graft failure

Background Complement-fixing antibodies against donor HLA are considered a contraindication for kidney transplant. A modification of the IgG single-antigen bead (SAB) assay allows detection of anti-HLA antibodies that bind C3d. Because early humoral graft rejection is considered to be complement mediated, this SAB-based technique may provide a valuable tool in the pretransplant risk stratification of kidney transplant recipients. Methods Previously, we established that pretransplant donor-specific anti-HLA antibodies (DSAs) are associated with increased risk for long-term graft failure in complement-dependent cytotoxicity crossmatch-negative transplants. In this study, we further characterized the DSA-positive serum samples using the C3d SAB assay. Results Among 567 pretransplant DSA-positive serum samples, 97 (17%) contained at least one C3d-fixing DSA, whereas 470 (83%) had non-C3d-fixing DSA. At 10 years after transplant, patients with C3d-fixing antibodies had a death-censored, covariate-adjusted graft survival of 60%, whereas patients with non-C3d-fixing DSA had a graft survival of 64% (hazard ratio, 1.02; 95% confidence interval, 0.70 to 1.48 for C3d-fixing DSA compared with non-C3d-fixing DSA; P50.93). Patients without DSA had a 10-year graft survival of 78%. Conclusions The C3d-fixing ability of pretransplant DSA is not associated with increased risk for graft failure

Effect of initial immunosuppression on long-term kidney transplant outcome in immunological low-risk patients

BACKGROUND: Few studies have evaluated the effect of different immunosuppressive strategies on long-term kidney transplant outcomes. Moreover, as they were usually based on historical data, it was not possible to account for the presence of pretransplant donor-specific human-leukocyte antigen antibodies (DSA), a currently recognized risk marker for impaired graft survival. The aim of this study was to evaluate to what extent frequently used initial immunosuppressive therapies increase graft survival in immunological low-risk patients. METHODS: We performed an analysis on the PROCARE cohort, a Dutch multicentre study including all transplantations performed in the Netherlands between 1995 and 2005 with available pretransplant serum (n = 4724). All sera were assessed for the presence of DSA by a luminex single-antigen bead assay. Patients with a previous kidney transplantation, pretransplant DSA or receiving induction therapy were excluded from the analysis. RESULTS: Three regimes were used in over 200 patients: cyclosporine (CsA)/prednisolone (Pred) (n = 542), CsA/mycophenolate mofetil (MMF)/Pred (n = 857) and tacrolimus (TAC)/MMF/Pred (n = 811). Covariate-adjusted analysis revealed no significant differences in 10-year death-censored graft survival between patients on TAC/MMF/Pred therapy (79%) compared with patients on CsA/MMF/Pred (82%, P = 0.88) or CsA/Pred (79%, P = 0.21). However, 1-year rejection-free survival censored for death and failure unrelated to rejection was significantly higher for TAC/MMF/Pred (81%) when compared with CsA/MMF/Pred (67%, P < 0.0001) and CsA/Pred (64%, P < 0.0001). CONCLUSION: These results suggest that in immunological low-risk patients excellent long-term kidney graft survival can be achieved irrespective of the type of initial immunosuppressive therapy (CsA or TAC; with or without MMF), despite differences in 1-year rejection-free survival