Structure‐preserving integrators for constrained mechanical systems in the framework of the GGL principle

Simulating multi-body systems often requires an appropriate treatment of the differential-algebraic equations (DAEs). The recently proposed GGL principle considers constraints both on configuration and on velocity level and embodies an index-reduction technique in the spirit of the often-applied GGL stabilization. In sharp contrast to the original formulation, the Euler-Lagrange equations of the GGL principle, fit into the Hamiltonian framework of mechanics. Therefore, the GGL principle facilitates the design of structure-preserving integrators. Due to the close relationship of the GGL principle to optimal control, previously developed direct methods can be used to obtain variational integrators for constrained mechanical systems. Furthermore, slight modifications can be applied to obtain second-order energy-momentum consistent integrators emanating from the GGL principle, which represent another important class of structure-preserving time-stepping schemes. The newly devised schemes circumvent issues of standard methods and provide more realistic results by accounting for velocity level constraints

The longitudinal interplay between negative and positive symptom trajectories in patients under antipsychotic treatment: a post hoc analysis of data from a randomized, 1-year pragmatic trial

BACKGROUND: Schizophrenia is a highly heterogeneous disorder with positive and negative symptoms being characteristic manifestations of the disease. While these two symptom domains are usually construed as distinct and orthogonal, little is known about the longitudinal pattern of negative symptoms and their linkage with the positive symptoms. This study assessed the temporal interplay between these two symptom domains and evaluated whether the improvements in these symptoms were inversely correlated or independent with each other. METHODS: This post hoc analysis used data from a multicenter, randomized, open-label, 1-year pragmatic trial of patients with schizophrenia spectrum disorder who were treated with first- and second-generation antipsychotics in the usual clinical settings. Data from all treatment groups were pooled resulting in 399 patients with complete data on both the negative and positive subscale scores from the Positive and Negative Syndrome Scale (PANSS). Individual-based growth mixture modeling combined with interplay matrix was used to identify the latent trajectory patterns in terms of both the negative and positive symptoms. Pearson correlation coefficients were calculated to examine the relationship between the changes of these two symptom domains within each combined trajectory pattern. RESULTS: We identified four distinct negative symptom trajectories and three positive symptom trajectories. The trajectory matrix formed 11 combined trajectory patterns, which evidenced that negative and positive symptom trajectories moved generally in parallel. Correlation coefficients for changes in negative and positive symptom subscale scores were positive and statistically significant (P < 0.05). Overall, the combined trajectories indicated three major distinct patterns: (1) dramatic and sustained early improvement in both negative and positive symptoms (n = 70, 18%), (2) mild and sustained improvement in negative and positive symptoms (n = 237, 59%), and (3) no improvement in either negative or positive symptoms (n = 82, 21%). CONCLUSIONS: This study of symptom trajectories over 1 year shows that changes in negative and positive symptoms were neither inversely nor independently related with each other. The positive association between these two symptom domains supports the notion that different symptom domains in schizophrenia may depend on each other through a unified upstream pathological disease process

Identification of early changes in specific symptoms that predict longer-term response to atypical antipsychotics in the treatment of patients with schizophrenia

<p>Abstract</p> <p>Background</p> <p>To identify a simple decision tree using early symptom change to predict response to atypical antipsychotic therapy in patients with (Diagnostic and Statistical Manual, Fourth Edition, Text Revised) chronic schizophrenia.</p> <p>Methods</p> <p>Data were pooled from moderately to severely ill patients (n = 1494) from 6 randomized, double-blind trials (N = 2543). Response was defined as a ≥30% reduction in Positive and Negative Syndrome Scale (PANSS) Total score by Week 8 of treatment. Analyzed predictors were change in individual PANSS items at Weeks 1 and 2. A decision tree was constructed using classification and regression tree (CART) analysis to identify predictors that most effectively differentiated responders from non-responders.</p> <p>Results</p> <p>A 2-branch, 6-item decision tree was created, producing 3 distinct groups. First branch criterion was a 2-point score decrease in at least 2 of 5 PANSS positive items (Week 2). Second branch criterion was a 2-point score decrease in the PANSS excitement item (Week 2). "Likely responders" met the first branch criteria; "likely non-responders" did not meet first or second criterion; "not predictable" patients did not meet the first but did meet the second criterion. Using this approach, response to treatment could be predicted in most patients (92%) with high positive predictive value (79%) and high negative predictive value (75%). Predictive findings were confirmed through analysis of data from 2 independent trials.</p> <p>Conclusions</p> <p>Using a data-driven approach, we identified decision rules using early change in the scores of selected PANSS items to accurately predict longer-term treatment response or non-response to atypical antipsychotic therapy. This could lead to development of a simple quantitative evaluation tool to help guide early treatment decisions.</p> <p>Trial Registration</p> <p>This is a retrospective, non-intervention study in which pooled results from 6 previously published reports were analyzed; thus, clinical trial registration is not required.</p

Structure-preserving integrators based on a new variational principle for constrained mechanical systems

A new variational principle for mechanical systems subject to holonomic constraints is presented. The newly proposed GGL principle is closely related to the often used Gear-Gupta-Leimkuhler (GGL) stabilization of the differential–algebraic equations governing the motion of constrained mechanical systems. The GGL variational principle relies on an extension of the Livens principle (sometimes also referred to as Hamilton–Pontryagin principle) to mechanical systems subject to holonomic constraints. In contrast to the original GGL stabilization, the new approach facilitates the design of structure-preserving integrators. In particular, new variational integrators are presented, which result from the direct discretization of the GGL variational principle. These variational integrators are symplectic and conserve momentum maps in the case of systems with symmetry. In addition to that, a new energy–momentum scheme is developed, which results from the discretization of the Euler–Lagrange equations pertaining to the GGL variational principle. The numerical properties of the newly devised schemes are investigated in representative examples of constrained mechanical systems