Fully Adaptive Newton--Galerkin Methods for Semilinear Elliptic Partial Differential Equations

In this paper we develop an adaptive procedure for the numerical solution of general, semilinear elliptic problems with possible singular perturbations. Our approach combines both prediction-type adaptive Newton methods and a linear adaptive finite element discretization (based on a robust a posteriori error analysis), thereby leading to a fully adaptive Newton–Galerkin scheme. Numerical experiments underline the robustness and reliability of the proposed approach for various examples

Coupling biomechanics to a cellular level model: An approach to patient-specific image driven multi-scale and multi-physics tumor simulation

Modeling of tumor growth has been performed according to various approaches addressing different biocomplexity levels and spatiotemporal scales. Mathematical treatments range from partial differential equation based diffusion models to rule-based cellular level simulators, aiming at both improving our quantitative understanding of the underlying biological processes and, in the mid- and long term, constructing reliable multi-scale predictive platforms to support patient-individualized treatment planning and optimization. The aim of this paper is to establish a multi-scale and multi-physics approach to tumor modeling taking into account both the cellular and the macroscopic mechanical level. Therefore, an already developed biomodel of clinical tumor growth and response to treatment is self-consistently coupled with a biomechanical model. Results are presented for the free growth case of the imageable component of an initially point-like glioblastoma multiforme tumor. The composite model leads to significant tumor shape corrections that are achieved through the utilization of environmental pressure information and the application of biomechanical principles. Using the ratio of smallest to largest moment of inertia of the tumor material to quantify the effect of our coupled approach, we have found a tumor shape correction of 20\% by coupling biomechanics to the cellular simulator as compared to a cellular simulation without preferred growth directions. We conclude that the integration of the two models provides additional morphological insight into realistic tumor growth behavior. Therefore, it might be used for the development of an advanced oncosimulator focusing on tumor types for which morphology plays an important role in surgical and/or radio-therapeutic treatment planning.

Optimized CUDA-Based PDE Solver for Reaction Diffusion Systems on Arbitrary Surfaces

Partial differential equation (PDE) solvers are commonly employed to study and characterize the parameter space for reaction-diffusion (RD) systems while investigating biological pattern formation. Increasingly, biologists wish to perform such studies with arbitrary surfaces representing ‘real’ 3D geometries for better insights. In this paper, we present a highly optimized CUDA-based solver for RD equations on triangulated meshes in 3D. We demonstrate our solver using a chemotactic model that can be used to study snakeskin pigmentation, for example. We employ a finite element based approach to perform explicit Euler time integrations. We compare our approach to a naive GPU implementation and provide an in-depth performance analysis, demonstrating the significant speedup afforded by our optimizations. The optimization strategies that we exploit could be generalized to other mesh based processing applications with PDE simulations.

Theoretical considerations to the verification of dynamic multileaf collimated fields with a SLIC-type portal image detector

The verification possibilities of dynamically collimated treatment beams with a scanning liquid ionization chamber electronic portal image device (SLIC-EPID) are investigated. The ion concentration in the liquid of a SLIC-EPID and therefore the read-out signal is determined by two parameters of a differential equation describing the creation and recombination of the ions. Due to the form of this equation, the portal image detector describes a nonlinear dynamic system with memory. In this work, the parameters of the differential equation were experimentally determined for the particular chamber in use and for an incident open 6 MV photon beam. The mathematical description of the ion concentration was then used to predict portal images of intensity-modulated photon beams produced by a dynamic delivery technique, the sliding window approach. Due to the nature of the differential equation, a mathematical condition for 'reliable leaf motion verification' in the sliding window technique can be formulated. It is shown that the time constants for both formation and decay of the equilibrium concentration in the chamber is in the order of seconds. In order to guarantee reliable leaf motion verification, these time constants impose a constraint on the rapidity of the image-read out for a given maximum leaf speed. For a leaf speed of 2 cm s(-1), a minimum image acquisition frequency of about 2 Hz is required. Current SLIC-EPID systems are usually too slow since they need about a second to acquire a portal image. However, if the condition is fulfilled, the memory property of the system can be used to reconstruct the leaf motion. It is shown that a simple edge detecting algorithm can be employed to determine the leaf positions. The method is also very robust against image noise.

Lions-type compactness and Rubik actions on the Heisenberg group

In this paper we prove a Lions-type compactness embedding result for symmetric unbounded domains of the Heisenberg group. The natural group action on the Heisenberg group TeX is provided by the unitary group U(n) × {1} and its appropriate subgroups, which will be used to construct subspaces with specific symmetry and compactness properties in the Folland-Stein’s horizontal Sobolev space TeX. As an application, we study the multiplicity of solutions for a singular subelliptic problem by exploiting a technique of solving the Rubik-cube applied to subgroups of U(n) × {1}. In our approach we employ concentration compactness, group-theoretical arguments, and variational methods.

Comparative theoretical analysis of continuous wave laser cutting of metals at 1 and 10 um wavelength

We present a derivation and, based on it, an extension of a model originally proposed by V.G. Niziev to describe continuous wave laser cutting of metals. Starting from a local energy balance and by incorporating heat removal through heat conduction to the bulk material, we find a differential equation for the cutting profile. This equation is solved numerically and yields, besides the cutting profiles, the maximum cutting speed, the absorptivity profiles, and other relevant quantities. Our main goal is to demonstrate the model’s capability to explain some of the experimentally observed differences between laser cutting at around 1 and 10 μm wavelengths. To compare our numerical results to experimental observations, we perform simulations for exactly the same material and laser beam parameters as those used in a recent comparative experimental study. Generally, we find good agreement between theoretical and experimental results and show that the main differences between laser cutting with 1- and 10-μm beams arise from the different absorptivity profiles and absorbed intensities. Especially the latter suggests that the energy transfer, and thus the laser cutting process, is more efficient in the case of laser cutting with 1-μm beams.

Computational Comparison of Continuous and Discontinuous Galerkin Time-Stepping Methods for Nonlinear Initial Value Problems

This article centers on the computational performance of the continuous and discontinuous Galerkin time stepping schemes for general first-order initial value problems in R n , with continuous nonlinearities. We briefly review a recent existence result for discrete solutions from [6], and provide a numerical comparison of the two time discretization methods.

Knowing the future: partial foreknowledge effects on the programming of prosaccades and antisaccades

Foreknowledge about the demands of an upcoming trial may be exploited to optimize behavioural responses. In the current study we systematically investigated the benefits of partial foreknowledge--that is, when some but not all aspects of a future trial are known in advance. For this we used an ocular motor paradigm with horizontal prosaccades and antisaccades. Predictable sequences were used to create three partial foreknowledge conditions: one with foreknowledge about the stimulus location only, one with foreknowledge about the task set only, and one with foreknowledge about the direction of the required response only. These were contrasted with a condition of no-foreknowledge and a condition of complete foreknowledge about all three parameters. The results showed that the three types of foreknowledge affected saccadic efficiency differently. While foreknowledge about stimulus-location had no effect on efficiency, task foreknowledge had some effect and response-foreknowledge was as effective as complete foreknowledge. Foreknowledge effects on switch costs followed a similar pattern in general, but were not specific for switching of the trial attribute for which foreknowledge was available. We conclude that partial foreknowledge has a differential effect on efficiency, most consistent with preparatory activation of a motor schema in advance of the stimulus, with consequent benefits for both switched and repeated trials.

Social cognition as a mediator variable between neurocognition and functional outcome in schizophrenia: empirical review and new results by structural equation modeling

Cognitive impairments are currently regarded as important determinants of functional domains and are promising treatment goals in schizophrenia. Nevertheless, the exact nature of the interdependent relationship between neurocognition and social cognition as well as the relative contribution of each of these factors to adequate functioning remains unclear. The purpose of this article is to systematically review the findings and methodology of studies that have investigated social cognition as a mediator variable between neurocognitive performance and functional outcome in schizophrenia. Moreover, we carried out a study to evaluate this mediation hypothesis by the means of structural equation modeling in a large sample of 148 schizophrenia patients. The review comprised 15 studies. All but one study provided evidence for the mediating role of social cognition both in cross-sectional and in longitudinal designs. Other variables like motivation and social competence additionally mediated the relationship between social cognition and functional outcome. The mean effect size of the indirect effect was 0.20. However, social cognitive domains were differentially effective mediators. On average, 25% of the variance in functional outcome could be explained in the mediation model. The results of our own statistical analysis are in line with these conclusions: Social cognition mediated a significant indirect relationship between neurocognition and functional outcome. These results suggest that research should focus on differential mediation pathways. Future studies should also consider the interaction with other prognostic factors, additional mediators, and moderators in order to increase the predictive power and to target those factors relevant for optimizing therapy effects.

Differential influence of arterial blood glucose on cerebral metabolism following severe traumatic brain injury

INTRODUCTION: Maintaining arterial blood glucose within tight limits is beneficial in critically ill patients. Upper and lower limits of detrimental blood glucose levels must be determined. METHODS: In 69 patients with severe traumatic brain injury (TBI), cerebral metabolism was monitored by assessing changes in arterial and jugular venous blood at normocarbia (partial arterial pressure of carbon dioxide (paCO2) 4.4 to 5.6 kPa), normoxia (partial arterial pressure of oxygen (paO2) 9 to 20 kPa), stable haematocrit (27 to 36%), brain temperature 35 to 38 degrees C, and cerebral perfusion pressure (CPP) 70 to 90 mmHg. This resulted in a total of 43,896 values for glucose uptake, lactate release, oxygen extraction ratio (OER), carbon dioxide (CO2) and bicarbonate (HCO3) production, jugular venous oxygen saturation (SjvO2), oxygen-glucose index (OGI), lactate-glucose index (LGI) and lactate-oxygen index (LOI). Arterial blood glucose concentration-dependent influence was determined retrospectively by assessing changes in these parameters within pre-defined blood glucose clusters, ranging from less than 4 to more than 9 mmol/l. RESULTS: Arterial blood glucose significantly influenced signs of cerebral metabolism reflected by increased cerebral glucose uptake, decreased cerebral lactate production, reduced oxygen consumption, negative LGI and decreased cerebral CO2/HCO3 production at arterial blood glucose levels above 6 to 7 mmol/l compared with lower arterial blood glucose concentrations. At blood glucose levels more than 8 mmol/l signs of increased anaerobic glycolysis (OGI less than 6) supervened. CONCLUSIONS: Maintaining arterial blood glucose levels between 6 and 8 mmol/l appears superior compared with lower and higher blood glucose concentrations in terms of stabilised cerebral metabolism. It appears that arterial blood glucose values below 6 and above 8 mmol/l should be avoided. Prospective analysis is required to determine the optimal arterial blood glucose target in patients suffering from severe TBI.

One-pot heterogeneous synthesis of Δ(3)-tetrahydrocannabinol analogues and xanthenes showing differential binding to CB(1) and CB(2) receptors

Δ(9)-tetrahydrocannabinol (Δ(9)-THC) is the major psychoactive cannabinoid in hemp (Cannabis sativa L.) and responsible for many of the pharmacological effects mediated via cannabinoid receptors. Despite being the major cannabinoid scaffold in nature, Δ(9)-THC double bond isomers remain poorly studied. The chemical scaffold of tetrahydrocannabinol can be assembled from the condensation of distinctly substituted phenols and monoterpenes. Here we explored a microwave-assisted one pot heterogeneous synthesis of Δ(3)-THC from orcinol (1a) and pulegone (2). Four Δ(3)-THC analogues and corresponding Δ(4a)-tetrahydroxanthenes (Δ(4a)-THXs) were synthesized regioselectively and showed differential binding affinities for CB1 and CB2 cannabinoid receptors. Here we report for the first time the CB1 receptor binding of Δ(3)-THC, revealing a more potent receptor binding affinity for the (S)-(-) isomer (hCB1Ki = 5 nM) compared to the (R)-(+) isomer (hCB1Ki = 29 nM). Like Δ(9)-THC, also Δ(3)-THC analogues are partial agonists at CB receptors as indicated by [(35)S]GTPγS binding assays. Interestingly, the THC structural isomers Δ(4a)-THXs showed selective binding and partial agonism at CB2 receptors, revealing a simple non-natural natural product-derived scaffold for novel CB2 ligands.

Diagnostic Accuracy of Copeptin in the Differential Diagnosis of the Polyuria-polydipsia Syndrome: A Prospective Multicenter Study

CONTEXT The polyuria-polydipsia syndrome comprises primary polydipsia (PP) and central and nephrogenic diabetes insipidus (DI). Correctly discriminating these entities is mandatory, given that inadequate treatment causes serious complications. The diagnostic "gold standard" is the water deprivation test with assessment of arginine vasopressin (AVP) activity. However, test interpretation and AVP measurement are challenging. OBJECTIVE The objective was to evaluate the accuracy of copeptin, a stable peptide stoichiometrically cosecreted with AVP, in the differential diagnosis of polyuria-polydipsia syndrome. DESIGN, SETTING, AND PATIENTS This was a prospective multicenter observational cohort study from four Swiss or German tertiary referral centers of adults >18 years old with the history of polyuria and polydipsia. MEASUREMENTS A standardized combined water deprivation/3% saline infusion test was performed and terminated when serum sodium exceeded 147 mmol/L. Circulating copeptin and AVP levels were measured regularly throughout the test. Final diagnosis was based on the water deprivation/saline infusion test results, clinical information, and the treatment response. RESULTS Fifty-five patients were enrolled (11 with complete central DI, 16 with partial central DI, 18 with PP, and 10 with nephrogenic DI). Without prior thirsting, a single baseline copeptin level >21.4 pmol/L differentiated nephrogenic DI from other etiologies with a 100% sensitivity and specificity, rendering a water deprivation testing unnecessary in such cases. A stimulated copeptin >4.9 pmol/L (at sodium levels >147 mmol/L) differentiated between patients with PP and patients with partial central DI with a 94.0% specificity and a 94.4% sensitivity. A stimulated AVP >1.8 pg/mL differentiated between the same categories with a 93.0% specificity and a 83.0% sensitivity. LIMITATION This study was limited by incorporation bias from including AVP levels as a diagnostic criterion. CONCLUSION Copeptin is a promising new tool in the differential diagnosis of the polyuria-polydipsia syndrome, and a valid surrogate marker for AVP. Primary Funding Sources: Swiss National Science Foundation, University of Basel.