Mixed finite element approximation of the vector Laplacian with Dirichlet boundary conditions

We consider the finite element solution of the vector Laplace equation on a domain in two dimensions. For various choices of boundary conditions, it is known that a mixed finite element method, in which the rotation of the solution is introduced as a second unknown, is advantageous, and appropriate choices of mixed finite element spaces lead to a stable, optimally convergent discretization. However, the theory that leads to these conclusions does not apply to the case of Dirichlet boundary conditions, in which both components of the solution vanish on the boundary. We show, by computational example, that indeed such mixed finite elements do not perform optimally in this case, and we analyze the suboptimal convergence that does occur. As we indicate, these results have implications for the solution of the biharmonic equation and of the Stokes equations using a mixed formulation involving the vorticity

Personalized Prediction of Future Lesion Activity and Treatment Effect in Multiple Sclerosis from Baseline MRI

Precision medicine for chronic diseases such as multiple sclerosis (MS) involves choosing a treatment which best balances efficacy and side effects/preferences for individual patients. Making this choice as early as possible is important, as delays in finding an effective therapy can lead to irreversible disability accrual. To this end, we present the first deep neural network model for individualized treatment decisions from baseline magnetic resonance imaging (MRI) (with clinical information if available) for MS patients. Our model (a) predicts future new and enlarging T2 weighted (NE-T2) lesion counts on follow-up MRI on multiple treatments and (b) estimates the conditional average treatment effect (CATE), as defined by the predicted future suppression of NE-T2 lesions, between different treatment options relative to placebo. Our model is validated on a proprietary federated dataset of 1817 multi-sequence MRIs acquired from MS patients during four multi-centre randomized clinical trials. Our framework achieves high average precision in the binarized regression of future NE-T2 lesions on five different treatments, identifies heterogeneous treatment effects, and provides a personalized treatment recommendation that accounts for treatment-associated risk (e.g. side effects, patient preference, administration difficulties).Comment: Accepted to MIDL 202

Spatial and temporal variability of cadmium in Gulf Stream warm-core rings and associated waters

Seawater samples were collected and analyzed for cadmium during four cruises studying Gulf Stream warm-core rings and associated waters. Warm-core ring (WCR) 82-B was sampled in April (approximately two months after formation), in June (after seasonal stratification), and in August (during its interaction with the Gulf Stream). The September–October cruise studied closure and separation of a meander that formed ring 82-H. The depth of the cadmium maximum varied with the depth of the main thermocline; the maximum occurred at a potential temperature of 7.8 ± 0.5°C and sigma-theta 27.4 ± 0.05 in the Slope Water, Gulf Stream, and Sargasso Sea stations. As the upper 100 m of the ring progressed from vertically well-mixed in April to seasonally stratified in June, the mole-ratios of cadmium/nutrient removal in the mixed layer were similar to the calculated slopes of the linear regressions of cadmium with phosphate, nitrate and silicate calculated from spatial distributions. Lateral mixing processes near the boundaries of WCR 82-B markedly influenced the vertical cadmium distribution via intrusions of Shelf/Slope water containing elevated levels of cadmium. Comparison of ASV-labile and total dissolvable cadmium from the August WCR 82-B station indicated essentially 100% ASV-labile cadmium in the waters within and below the main thermocline but non-detectable (\u3c0.010 nmol kg−1) ASV-labile cadmium in the waters above the thermocline

Supraglacial lakes on the Larsen B ice shelf, Antarctica, and at Paakitsoq, West Greenland:A Comparative Study

This is the accepted manuscript. The final version is available from Ingenta Connect at http://www.ingentaconnect.com/content/igsoc/agl/2014/00000055/00000066/art00001.Supraglacial meltwater lakes trigger ice-shelf break-up and modulate seasonal ice\ud sheet flow, and are thus agents by which warming is transmitted to the Antarctic\ud and Greenland ice sheets. To characterize supraglacial lake variability we perform a\ud comparative analysis of lake geometry and depth in two distinct regions, one on the\ud pre-collapse (2002) Larsen B Ice Shelf, and the other in the ablation zone of\ud Paakitsoq, a land-terminating region of the Greenland Ice Sheet. Compared to\ud Paakitsoq, lakes on the Larsen B Ice Shelf cover a greater proportion of surface area\ud (5.3% vs. 1%), but are shallower and more uniform in area. Other aspects of lake\ud geometry, such as eccentricity, degree of convexity (solidity) and orientation, are\ud relatively similar between the two regions. We attribute the notable difference in\ud lake density and depth between ice-shelf and grounded ice to the fact that ice shelves\ud have flatter surfaces and less distinct drainage basins. Ice shelves also possess more\ud stimuli to small-scale, localized surface elevation variability due to the various\ud structural features that yield small variations in thickness and which float at\ud different levels by Archimedes? principle.We acknowledge the support of the U.S. National Science Foundation under grant ANT-0944248

Counterfactual Image Synthesis for Discovery of Personalized Predictive Image Markers

The discovery of patient-specific imaging markers that are predictive of future disease outcomes can help us better understand individual-level heterogeneity of disease evolution. In fact, deep learning models that can provide data-driven personalized markers are much more likely to be adopted in medical practice. In this work, we demonstrate that data-driven biomarker discovery can be achieved through a counterfactual synthesis process. We show how a deep conditional generative model can be used to perturb local imaging features in baseline images that are pertinent to subject-specific future disease evolution and result in a counterfactual image that is expected to have a different future outcome. Candidate biomarkers, therefore, result from examining the set of features that are perturbed in this process. Through several experiments on a large-scale, multi-scanner, multi-center multiple sclerosis (MS) clinical trial magnetic resonance imaging (MRI) dataset of relapsing-remitting (RRMS) patients, we demonstrate that our model produces counterfactuals with changes in imaging features that reflect established clinical markers predictive of future MRI lesional activity at the population level. Additional qualitative results illustrate that our model has the potential to discover novel and subject-specific predictive markers of future activity.Comment: Accepted to the MIABID workshop at MICCAI 202