We analyze, through linear algebra, the topography of the cost functional in linear and nonlinear inverse problems with the aim of illuminating general characteristics. To a first-order approximation, the local data misfit function in any inverse problem is valley-shaped and elongated in the directions of the null space of the Jacobian and/or in the directions of the smallest singular values. In nonlinear inverse problems, valleys persist; however, local minima might also coexist in the misfit space and might be related to nonlinear effects ignored by the Gauss-Newton approximation to the Hessian, the regularization term designed to provide convexity to the misfit function, or to noise in the data. Furthermore, noise perturbs the size of the equivalence region making location of solutions easier but finding a global minimum harder (in the case of existence). Understanding the behavior of the cost functional is an important step in the developing techniques to appraise inverse solutions and estimate uncertainties caused by noise, incomplete sampling, regularization, and more fundamentally, simplified physical models.
Aims
It is known that matrix metalloproteinase (MMP)‐11 has a role in tumour development and progression, and also that immune cells can influence cancer cells to increase their proliferative and invasive properties. The aim of the present study was to propose the evaluation of MMP11 expression by intratumoral mononuclear inflammatory cells (MICs) as a useful biological marker for breast cancer prognosis.
Methods and results
This study comprised 246 women with invasive breast carcinoma, and a long follow‐up period. Patients were stratified with regard to nodal status and to the development of metastatic disease. The median follow‐up period in patients without metastasis was 146 months and in patients with metastatic disease 31 months. MMP11 was determined by immunohistochemistry. For relapse‐free survival (RFS) and overall survival (OS) analysis we used the Cox’s univariate method. Cox’s regression model was used to examine the interactions between different prognostic factors in a multivariate analysis.
Conclusions
Our results showed that MMP11 expression by stromal cells was significantly associated with prognosis. MMP11 expression by cancer‐associated fibroblasts (CAFs) was associated with both shortened RFS and OS, but MMP11 expression by MICs showed a stronger association with both shortened RFS and OS, therefore being the most potent and independent factor to predict RFS and OS.
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