Fachgruppe Mathematik

Deep Learning–Based SD-OCT Layer Segmentation Quantifies Outer Retina Changes in Patients With Biallelic RPE65 Mutations Undergoing Gene Therapy

Thu, 02 Jan 2025 00:00:00 GMT

Deep Learning–Based SD-OCT Layer Segmentation Quantifies Outer Retina Changes in Patients With Biallelic RPE65 Mutations Undergoing Gene Therapy Pinedo-Diaz, German; Lorenz, Birgit; Künzel, Sandrine H.; Thiele, Sarah; Ortega-Cisneros, Susana; Bayro Corrochano, Eduardo; Holz, Frank G.; Effland, Alexander PURPOSE. To quantify outer retina structural changes and define novel biomarkers of inherited retinal degeneration associated with biallelic mutations in RPE65 (RPE65-IRD) in patients before and after subretinal gene augmentation therapy with voretigene neparvovec (Luxturna).
METHODS. Application of advanced deep learning for automated retinal layer segmentation, specifically tailored for RPE65-IRD. Quantification of five novel biomarkers for the ellipsoid zone (EZ): thickness, granularity, reflectivity, and intensity. Estimation of the EZ_area in single and volume scans was performed with optimized segmentation boundaries. The control group was age similar and without significant refractive error. Spherical equivalent refraction and ocular length were evaluated in all patients.
RESULTS. We observed significant differences in the structural analysis of EZ biomarkers in 22 patients with RPE65-IRD compared with 94 healthy controls. Relative EZ intensities were already reduced in pediatric eyes. Reductions of EZ local granularity and EZ thickness were only significant in adult eyes. Distances of the outer plexiform layer, external limiting membrane, and Bruch’s membrane to EZ were reduced at all ages. EZ diameter and area were better preserved in pediatric eyes undergoing therapy with voretigene neparvovec and in patients with a milder phenotype.
CONCLUSIONS. Automated quantitative analysis of biomarkers within EZ visualizes distinct structural differences in the outer retina of patients including treatment-related effects. The automated approach using deep learning strategies allows big data analysis for distinct forms of inherited retinal degeneration. Limitations include a small dataset and potential effects on OCT scans from myopia at least −5 diopters, the latter considered nonsignificant for outer retinal layers.

Operator based multi-scale analysis of simulation bundles

Sun, 01 Nov 2015 00:00:00 GMT

Operator based multi-scale analysis of simulation bundles Iza-Teran, Rodrigo; Garcke, Jochen We propose a new mathematical data analysis approach, which is based on the mathematical principle of symmetry, for the post-processing of bundles of finite element data from computer-aided engineering. Since all those numerical simulation data stem from the numerical solution of the same partial differential equation, there exists a set of transformations, albeit unknown, which map simulation to simulation. The transformations can be obtained indirectly by constructing a transformation invariant positive definitive operator valid for all simulations.
The eigenbasis of such an operator turns out to be a convenient basis for the handled simulation set due to two reasons. First, the spectral coefficients decay very fast, depending on the smoothness of the function being represented, and therefore a reduced multi-scale representation of all simulations can be obtained, which depends on the employed operator. Second, at each level of the eigendecomposition the eigenvectors can be seen to recover different independent variation modes like rotation, translation or local deformation. Furthermore, this representation enables the definition of a new distance measure between simulations using the spectral coefficients. From a theoretical point of view the space of simulations modulo a transformation group can be expressed conveniently using the operator eigenbasis as orbits in the quotient space with respect to a specific transformation group.
Based on this mathematical framework we study several examples. We show that for time dependent datasets from engineering simulations only a few spectral coefficients are necessary to describe the data variability, while the coarse variations get separated from the finer ones. Low dimensional structures are obtained in this way, which are able to capture information about the underlying simulation space. An effective mechanism to deal effectively with the analysis of many numerical simulations is obtained, due to the achieved dimensionality reduction.

Suboptimal feedback control of PDEs by solving HJB equations on adaptive sparse grids

Thu, 01 Jan 2015 00:00:00 GMT

Suboptimal feedback control of PDEs by solving HJB equations on adaptive sparse grids Garcke, Jochen; Kröner, Axel An approach to solve finite time horizon suboptimal feedback control problems for partial differential equations is proposed by solving dynamic programming equations on adaptive sparse grids. A semi-discrete optimal control problem is introduced and the feedback control is derived from the corresponding value function. The value function can be characterized as the solution of an evolutionary Hamilton-Jacobi Bellman (HJB) equation which is defined over a state space whose dimension is equal to the dimension of the underlying semi-discrete system. Besides a low dimensional semi-discretization it is important to solve the HJB equation efficiently to address the curse of dimensionality. We propose to apply a semi-Lagrangian scheme using spatially adaptive sparse grids. Sparse grids allow the discretization of the value functions in (higher) space dimensions since the curse of dimensionality of full grid methods arises to a much smaller extent. For additional efficiency an adaptive grid refinement procedure is explored.
The approach is illustrated for the wave equation and an extension to equations of Schrödinger type is indicated. We present several numerical examples studying the effect the parameters characterizing the sparse grid have on the accuracy of the value function and the optimal trajectory. closed-loop suboptimal control of PDEs and HJB equations and sparse grids and curse of dimensionality.

Multiscale simulation of polymeric fluids using the sparse grid combination technique

Sun, 01 Oct 2017 00:00:00 GMT

Multiscale simulation of polymeric fluids using the sparse grid combination technique Rüttgers, Alexander; Griebel, Michael We present a computationally efficient sparse grid approach to allow for multiscale simulations of non-Newtonian polymeric fluids. Multiscale approaches for polymeric fluids often involve model equations of high dimensionality. A conventional numerical treatment of such equations leads to computing times in the order of months even on massively parallel computers.
For a reduction of this enormous complexity, we propose the sparse grid combination technique. Compared to classical full grid approaches, the combination technique strongly reduces the computational complexity of a numerical scheme but only slightly decreases its accuracy.
Here, we use the combination technique in a general formulation that balances not only different discretization errors but also considers the accuracy of the mathematical model. For an optimal weighting of these different problem dimensions, we employ a dimension-adaptive refinement strategy. We finally verify substantial cost reductions of our approach for simulations of non-Newtonian Couette and extensional flow problems.