Alessandro Borghi

Davide Pradovera, Alessandro Borghi, Lukas Pieronek et al.

The interior transmission eigenvalue problem (ITP) plays a central role in inverse scattering theory and in the spectral analysis of inhomogeneous media. Despite its smooth dependence on the refractive index at the PDE level, the corresponding spectral map from material parameters to eigenpairs may exhibit non-smooth or bifurcating behavior. In this work, we develop a theoretical framework identifying sufficient conditions for such non-smooth spectral behavior in the ITP on general domains. We further specialize our analysis to some radially symmetric geometries, enabling a more precise characterization of bifurcations in the spectrum. Computationally, we formulate the ITP as a parametric, discrete, nonlinear eigenproblem and use a match-based adaptive contour eigensolver to accurately and efficiently track eigenvalue trajectories under parameter variation. Numerical experiments confirm the theoretical predictions and reveal novel non-smooth spectral effects.

Itxasne Antúnez Sáenz, Ane Alberdi Aramendi, David Dunaway et al.

Craniosynostosis is a medical condition that affects the growth of babies' heads, caused by an early fusion of cranial sutures. In recent decades, surgical treatments for craniosynostosis have significantly improved, leading to reduced invasiveness, faster recovery, and less blood loss. At Great Ormond Street Hospital (GOSH), the main surgical treatment for patients diagnosed with sagittal craniosynostosis (SC) is spring assisted cranioplasty (SAC). This procedure involves a 15x15 mm2 osteotomy, where two springs are inserted to induce distraction. Despite the numerous advantages of this surgical technique for patients, the outcome remains unpredictable due to the lack of efficient preoperative planning tools. The surgeon's experience and the baby's age are currently relied upon to determine the osteotomy location and spring selection. Previous tools for predicting the surgical outcome of SC relied on finite element modeling (FEM), which involved computed tomography (CT) imaging and required engineering expertise and lengthy calculations. The main goal of this research is to develop a real-time prediction tool for the surgical outcome of patients, eliminating the need for CT scans to minimise radiation exposure during preoperative planning. The proposed methodology involves creating personalised synthetic skulls based on three-dimensional (3D) photographs, incorporating population average values of suture location, skull thickness, and soft tissue properties. A machine learning (ML) surrogate model is employed to achieve the desired surgical outcome. The resulting multi-output support vector regressor model achieves a R2 metric of 0.95 and MSE and MAE below 0.13. Furthermore, in the future, this model could not only simulate various surgical scenarios but also provide optimal parameters for achieving a maximum cranial index (CI).