External validation of a model to predict the survival of patients presenting with a spinal epidural metastasis

The surgical treatment of spinal metastases is evolving. The major problem is the selection of patients who may benefit from surgical treatment. One of the criteria is an expected survival of at least 3 months. A prediction model has been previously developed. The present study has been performed in order to validate externally the model and to demonstrate that this model can be generalized to other institutions and other countries than the Netherlands. Data of 356 patients from five centers in Germany, Spain, Sweden, and the Netherlands who were treated for metastatic epidural spinal cord compression were collected. Hazard ratios in the test population corresponded with those of the developmental population. However, the observed and the expected survival were different. Analysis revealed that the baseline hazard function was significantly different. This tempted us to combine the data and develop a new prediction model. Estimating iteratively, a baseline hazard was composed. An adapted prediction model is presented. External validation of a prediction model revealed a difference in expected survival, although the relative contribution of the specific hazard ratios was the same as in the developmental population. This study emphasized the need to check the baseline hazard function in external validation. A new model has been developed using an estimated baseline hazar

We consider integrals as they arise in the boundary element method. Let us consider

We introduce a method for the computation of singular integrals arising in the discretization of integral equations. The basic method is based on the concept of admissible subdomains, known, e.g., from panel clustering techniques and H-matrices: We split the domain of integration into a hierarchy of subdomains and perform standard quadrature on those subdomains that are amenable to it. By using additional properties of the integrand, we can significantly reduce the algorithmic complexity of our approach. The method works also well for hypersingular integrals

H²-matrix approximation of integral operators by interpolation

Typical panel clustering methods for the fast evaluation of integral operators are based on the Taylor expansion of the kernel function and therefore usually require the user to implement the evaluation of the derivatives of this function up to an arbitrary degree. We propose an alternative approach that replaces the Taylor expansion by simple polynomial interpolation. By applying the interpolation idea to the approximating polynomials on different levels of the cluster tree, the matrix vector multiplication can be performed in only O(np d ) operations for a polynomial order of p and an n-dimensional trial space. The main advantage of our method, compared to other methods, is its simplicity: Only pointwise evaluations of the kernel and of simple polynomials have to be implemented.

An introduction to hierarchical matrices

We give a short introduction to a method for the data-sparse approximation of matrices resulting from the discretisation of non-local operators occurring in boundary integral methods or as the inverses of partial differential operators. The result of the approximation will be so-called hierarchical matrices (or short H-matrices). These matrices form a subset of the set of all matrices and have a data-sparse representation. The essential operations for these matrices (matrix-vector and matrix-matrix multiplication, addition and inversion) can be performed in, up to logarithmic factors, optimal complexity