To improve Multi-step Mathematical Reasoning (MsMR) of Large Language Models (LLMs), it is crucial to obtain scalable supervision from the corpus by automatically critiquing mistakes in the reasoning process of MsMR and rendering a final verdict of the problem-solution. Most existing methods rely on crafting high-quality supervised fine-tuning demonstrations for critiquing capability enhancement and pay little attention to delving into the underlying reason for the poor critiquing performance of LLMs. In this paper, we orthogonally quantify and investigate the potential reason — imbalanced evaluation preference, and conduct a statistical preference analysis. Motivated by the analysis of the reason, a novel perplexity-aware reinforcement learning algorithm is proposed to rectify the evaluation preference, elevating the critiquing capability. Specifically, to probe into LLMs' critiquing characteristics, a One-to-many Problem-Solution (OPS) benchmark is meticulously constructed to quantify the behavior difference of LLMs when evaluating the problem solutions generated by itself and others. Then, to investigate the behavior difference in depth, we conduct a statistical preference analysis oriented on perplexity and find an intriguing phenomenon — "LLMs incline to judge solutions with lower perplexity as correct", which is dubbed as imbalanced evaluation preference. To rectify this preference, we regard perplexity as the baton in the algorithm of Group Relative Policy Optimization, supporting the LLMs to explore trajectories that judge lower perplexity as wrong and higher perplexity as correct. Extensive experimental results on our built OPS and existing available critic benchmarks demonstrate the validity of our method.

Cardiac digital twins (CDTs) offer personalized in-silico cardiac representations for the inference of multi-scale properties tied to cardiac mechanisms. The creation of CDTs requires precise information about the electrode position on the torso, especially for the personalized electrocardiogram (ECG) calibration. However, current studies commonly rely on additional acquisition of torso imaging and manual/semi-automatic methods for ECG electrode localization. In this study, we propose a novel and efficient topology-informed model to fully automatically extract personalized ECG standard electrode locations from 2D clinically standard cardiac MRIs. Specifically, we obtain the sparse torso contours from the cardiac MRIs and then localize the standard electrodes of 12-lead ECG from the contours. Cardiac MRIs aim at imaging of the heart instead of the torso, leading to incomplete torso geometry within the imaging. To tackle the missing topology, we incorporate the electrodes as a subset of the keypoints, which can be explicitly aligned with the 3D torso topology. The experimental results demonstrate that the proposed model outperforms the time-consuming conventional model projection-based method in terms of accuracy (Euclidean distance: 1.24±0.293 cm vs. 1.48±0.362 cm) and efficiency (2 s vs. 30-35 min). We further demonstrate the effectiveness of using the detected electrodes for in-silico ECG simulation, highlighting their potential for creating accurate and efficient CDT models. The code is available at https://github.com/lileitech/12lead_ECG_electrode_localizer.

Electrocardiogram (ECG) recordings are affected by the heart's three-dimensional orientation within the thorax, i.e., the anatomical axis. Various cardiac conditions can cause the anatomical axis to shift and/or alter the pattern of electrical activation, leading to changes in the electrical axis. Nevertheless, there remains a lack of a formal, population-level study of the interplay between the cardiac anatomical and electrical axes and the factors that affect them. In this context, this study aimed to: (1) propose standardised definitions for the cardiac anatomical and electrical axes, (2) characterise their population-wide interplay in healthy conditions, (3) evaluate the impact of hypertension on their distribution and (4) identify associations with phenotypical and disease characteristics. Using cardiac magnetic resonance images and 12-lead ECGs from ~39,000 UK Biobank participants, patient-specific, paired biventricular geometries and vectorcardiograms were constructed. Five anatomical and four electrical axis definitions were computed, with the optimal pair of definitions selected based on their mutual alignment in 3D space within 28,000 healthy subjects. Accordingly, the anatomical axis was defined as the vector from the apex to the spatial centre of the four valves, and the electrical axis as the direction of the maximum QRS dipole. Mean angular separation in 3D, [Formula: see text], was 145.0° ± 16.8° in the healthy cohort. The electrical axes exhibited a much larger variability, and strong evidence of anatomical-electrical coupling was identified. Increasing BMI notably affected the anatomical axis, rotating the heart more horizontally-a pattern mirrored by the electrical axis. Both axes were also significantly influenced by sex and, to a lesser extent, age. The axes were then studied in the sub-cohort of ~3,500 UK BioBank participants with primary hypertension, where a similar rotational pattern as that with increasing BMI was revealed. Finally, phenome-wide association studies in the 39,000 participants reveal associations between the axes angular metrics and phenotypes signalling an increased afterload, and an association to hypertension among other clinical conditions. These findings underscore the complex anatomical-electrical interplay and highlight the potential of cardiac axes biomarkers for an improved clinical ECG interpretation and disease characterisation.