Exceptional points in parity-time symmetric plasmonic Huygens metasurfaces

Parity-Time (PT) symmetric optical structures exhibit several unique and interesting characteristics with the most popular being exceptional points. While the emerging concept of PT-symmetry has been extensively investigated in bulky photonic designs, its exotic functionalities in nanophotonic non-Hermitian plasmonic systems still remain relatively unexplored. Towards this goal, in this work we analyze the unusual properties of a plasmonic Huygens metasurface composed of an array of active metal-dielectric core-shell nanoparticles. By calculating the reflection and transmission coefficients of the metasurface under various levels of gain, we demonstrate the existence of reflectionless transmission when an exceptional point is formed. The proposed new active metasurface design has subwavelength thickness and can be used to realize ultracompact perfect transmission optical filters

Log hazard rate function for the interaction between calendar time and duration of ESRD at the beginning of HEMO.

<p>This was estimated via a tensor based smooth in a stratified by center PGAM adjusting for all prespecified baseline covariates in HEMO, a general (not linear) interaction between baseline albumin concentration and observation time, the combined effects of albumin and disease duration (as a tensor product smooth) and the modification of the latter by high flux dialysis.</p

Standardized Bias (top row), Mean Square Error (MSE, second row), p-value coverage (third row) and average confidence interval (CIL) coverage (bottom row) of the Hazard Ratio (HR) from four different baselines of the Gompertz and Weibull.

<p>500 datasets of 300 individuals per arm (total 600 patients) were simulated for each combination of baseline hazard, parameters, HR and censoring percentage (either 30% or 70%). These were subsequently analyzed with the Cox proportional hazards model (Cox, blue) and the Poisson GAM (PGAM, red). The three horizontal black lines in the p-value coverage graph give the nominal coverage (0.95) and ± 2 SE(0.95). Coverage is considered acceptable if the p-values fall within the upper and lower horizontal lines.</p

Analysis of Time to Event Outcomes in Randomized Controlled Trials by Generalized Additive Models

<div><p>Background</p><p>Randomized Controlled Trials almost invariably utilize the hazard ratio calculated with a Cox proportional hazard model as a treatment efficacy measure. Despite the widespread adoption of HRs, these provide a limited understanding of the treatment effect and may even provide a biased estimate when the assumption of proportional hazards in the Cox model is not verified by the trial data. Additional treatment effect measures on the survival probability or the time scale may be used to supplement HRs but a framework for the simultaneous generation of these measures is lacking.</p><p>Methods</p><p>By splitting follow-up time at the nodes of a Gauss Lobatto numerical quadrature rule, techniques for Poisson Generalized Additive Models (PGAM) can be adopted for flexible hazard modeling. Straightforward simulation post-estimation transforms PGAM estimates for the log hazard into estimates of the survival function. These in turn were used to calculate relative and absolute risks or even differences in restricted mean survival time between treatment arms. We illustrate our approach with extensive simulations and in two trials: IPASS (in which the proportionality of hazards was violated) and HEMO a long duration study conducted under evolving standards of care on a heterogeneous patient population.</p><p>Findings</p><p>PGAM can generate estimates of the survival function and the hazard ratio that are essentially identical to those obtained by Kaplan Meier curve analysis and the Cox model. PGAMs can simultaneously provide multiple measures of treatment efficacy after a single data pass. Furthermore, supported unadjusted (overall treatment effect) but also subgroup and adjusted analyses, while incorporating multiple time scales and accounting for non-proportional hazards in survival data.</p><p>Conclusions</p><p>By augmenting the HR conventionally reported, PGAMs have the potential to support the inferential goals of multiple stakeholders involved in the evaluation and appraisal of clinical trial results under proportional and non-proportional hazards.</p></div

Estimates of the Hazard Ratio and the upper (UCI) and lower (LCI) confidence intervals of the treatment effect of Gefitinib in the IPASS trial.

<p>Cox: estimate based on a Cox proportional hazard model. GL: Gauss Lobatto rule. The number behind GL designates the number of nodes in the numerical integration rule. This number is equal to the number of sub-intervals used to split each observation time in the dataset.</p><p>Estimates of the Hazard Ratio and the upper (UCI) and lower (LCI) confidence intervals of the treatment effect of Gefitinib in the IPASS trial.</p

Standardized Bias (top row), Mean Square Error (MSE, second row), p-value coverage (third row) and average confidence interval (CIL) coverage (bottom row) for the survival probabilities from four different baselines of the Gompertz, Weibull and Lognormal distributions.

<p>500 datasets of 300 individuals were simulated for each combination of baseline hazard, parameters and censoring percentage (either 30% or 70%) and were subsequently analyzed with the Kaplan Meier method (blue) and the Poisson GAM (red). The three horizontal black lines in the p-value coverage graph give the nominal coverage (0.95) and ± 2 SE(0.95). Coverage is considered acceptable if the p-values fall within the upper and lower horizontal lines.</p

Log hazard rate function for the interaction between albumin and duration of ESRD at the beginning of HEMO and flux.

<p>This was estimated via a tensor based smooth in a stratified by center PGAM adjusting for all prespecified baseline covariates in HEMO, a general (not linear) smooth interaction between baseline albumin concentration and observation time, the interaction between albumin and disease duration (as a tensor smooth) and the triple interaction between albumin, disease duration and high flux arm assignment (shown in the Fig). A 10 node Gauss Lobatto quadrature rule was used to numerically integrate the cumulative hazard function.</p

Bounds of the Gauss Lobatto (GL) approximation error for the integration of survival data.

<p>A) relationship between (log) MST and the logarithm of the Maximum Hazard rate function for survival distributions with a cubic polynomial log baseline hazard function (B) Box plots of the GL error as a function of the number of nodes in the quadrature rule (C) GL error as a function of the length of the integration interval (taken equal to be equal to the MST for each distribution examined) for different orders of the quadrature rule (D) GL error as a function of the maximum value of the hazard rate for different orders of the quadrature rule.</p

Options for parametric, flexible modeling of survival data.

<p>^‡ Not pursued by the authors</p><p><i>Abbreviations</i>: ARR (Absolute Risk Reduction), CHR (Cumulative Hazard Ratio), GAM (Generalized Additive Model), GCV (Generalized Cross Validation), GPGPU (General Purpose Graphics Processing Unit) HR (Hazard Ratio), MCMC (Markov Chain Monte Carlo), ML (Maximum Likelihood), OpenMP (Open Multi-Processing), PGAM(Poisson Generalized Additive Model), R(Relative Survival), REML (Restricted Maximum Likelihood), RR (Relative Risk).</p><p>Options for parametric, flexible modeling of survival data.</p

Adjusted measures of treatment efficacy in the HEMO trial using the corrected group prognosis method in the subgroups patients with albumin less than 3.5 g/dl, greater than 3.5 g/dl and the entire HEMO sample.

<p>A 10 node Gauss Lobatto quadrature rule was used to numerically integrate the cumulative hazard function. ARD: Absolute Risk Difference, RR: Relative Risk, HR: Hazard Ratio, RMST: Restricted Mean Survival Time. Dotted lines are the associated 95% pointwise confidence interval of the GL10 PGAM.</p