Wenbo Fei, Yuan Chen, Zexi Cai, Donglin Zeng, Yuanjia Wang
Implementing precision medicine for mental disorders presents challenges due to disease complexity and heterogeneity in patient responses. Empirical studies suggest that early indicators, such as interim measures (e.g., interim patient self-reports) of disease improvement or relapse, can predict longer-term outcomes, serving as proxies when final outcomes (e.g., in-clinic assessments) are less accessible. However, existing approaches for deriving individualized treatment rules (ITRs) often ignore these early signals, instead focusing only on a final outcome as the reward. In this work, we propose a new method incorporating intermediate outcomes from various domains into a personalized composite outcome, serving as the reward for learning ITRs. This composite is a weighted sum of inferred latent states from observed measures, with weights personalized for each patient, ensuring consistency with the long-term final response. Our simulations show that this approach not only provides early detection of non-responders but also improves long-term treatment outcomes. Applying our framework to a randomized clinical trial on major depressive disorder (MDD) demonstrates its effectiveness and advantages in ITR learning.
{"title":"Learning optimal early decision treatment rules with multi-domain intermediate outcomes.","authors":"Wenbo Fei, Yuan Chen, Zexi Cai, Donglin Zeng, Yuanjia Wang","doi":"10.1093/biomtc/ujaf167","DOIUrl":"https://doi.org/10.1093/biomtc/ujaf167","url":null,"abstract":"<p><p>Implementing precision medicine for mental disorders presents challenges due to disease complexity and heterogeneity in patient responses. Empirical studies suggest that early indicators, such as interim measures (e.g., interim patient self-reports) of disease improvement or relapse, can predict longer-term outcomes, serving as proxies when final outcomes (e.g., in-clinic assessments) are less accessible. However, existing approaches for deriving individualized treatment rules (ITRs) often ignore these early signals, instead focusing only on a final outcome as the reward. In this work, we propose a new method incorporating intermediate outcomes from various domains into a personalized composite outcome, serving as the reward for learning ITRs. This composite is a weighted sum of inferred latent states from observed measures, with weights personalized for each patient, ensuring consistency with the long-term final response. Our simulations show that this approach not only provides early detection of non-responders but also improves long-term treatment outcomes. Applying our framework to a randomized clinical trial on major depressive disorder (MDD) demonstrates its effectiveness and advantages in ITR learning.</p>","PeriodicalId":8930,"journal":{"name":"Biometrics","volume":"82 1","pages":""},"PeriodicalIF":1.7,"publicationDate":"2026-01-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145916679","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Randomized controlled trials are the gold standard for evaluating the efficacy of an intervention. However, there is often a trade-off between selecting the most scientifically relevant primary endpoint versus a less relevant, but more powerful, endpoint. For example, in the context of tobacco regulatory science many trials evaluate cigarettes per day as the primary endpoint instead of abstinence from smoking due to limited power. Additionally, it is often of interest to consider subgroup analyses to answer additional questions; such analyses are rarely adequately powered. In practice, trials often collect multiple endpoints. Intuitively, if multiple endpoints demonstrate a similar treatment effect, we would be more confident in the results of this trial. However, there is limited research on leveraging information from secondary endpoints besides using composite endpoints, which can be difficult to interpret. In this paper, we develop an estimator for the treatment effect on the primary endpoint based on a joint model for primary and secondary efficacy endpoints. This estimator gains efficiency over the standard treatment effect estimator when the model is correctly specified but is robust to model misspecification via model averaging. We illustrate our approach by estimating the effect of very low nicotine content cigarettes on the proportion of Black people who smoke who achieve abstinence and find our approach reduces the standard error by 27%.
{"title":"Jointly modeling multiple endpoints for efficient treatment effect estimation in randomized controlled trials.","authors":"Jack M Wolf, Joseph S Koopmeiners, David M Vock","doi":"10.1093/biomtc/ujaf166","DOIUrl":"10.1093/biomtc/ujaf166","url":null,"abstract":"<p><p>Randomized controlled trials are the gold standard for evaluating the efficacy of an intervention. However, there is often a trade-off between selecting the most scientifically relevant primary endpoint versus a less relevant, but more powerful, endpoint. For example, in the context of tobacco regulatory science many trials evaluate cigarettes per day as the primary endpoint instead of abstinence from smoking due to limited power. Additionally, it is often of interest to consider subgroup analyses to answer additional questions; such analyses are rarely adequately powered. In practice, trials often collect multiple endpoints. Intuitively, if multiple endpoints demonstrate a similar treatment effect, we would be more confident in the results of this trial. However, there is limited research on leveraging information from secondary endpoints besides using composite endpoints, which can be difficult to interpret. In this paper, we develop an estimator for the treatment effect on the primary endpoint based on a joint model for primary and secondary efficacy endpoints. This estimator gains efficiency over the standard treatment effect estimator when the model is correctly specified but is robust to model misspecification via model averaging. We illustrate our approach by estimating the effect of very low nicotine content cigarettes on the proportion of Black people who smoke who achieve abstinence and find our approach reduces the standard error by 27%.</p>","PeriodicalId":8930,"journal":{"name":"Biometrics","volume":"82 1","pages":""},"PeriodicalIF":1.7,"publicationDate":"2026-01-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.ncbi.nlm.nih.gov/pmc/articles/PMC12780332/pdf/","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145916693","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Correction to: Continuous-time mediation analysis for repeatedly measured mediators and outcomes.","authors":"","doi":"10.1093/biomtc/ujag029","DOIUrl":"https://doi.org/10.1093/biomtc/ujag029","url":null,"abstract":"","PeriodicalId":8930,"journal":{"name":"Biometrics","volume":"82 1","pages":""},"PeriodicalIF":1.7,"publicationDate":"2026-01-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"146084066","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
An important goal of environmental health research is to assess risks posed by mixtures of environmental exposures. Studies in different fields often group exposures based on their shared biological features. However, such grouping information has not been widely utilized in population-based environmental mixtures analyses due to the lack of appropriate statistical tools. Inspired by data from the National Health and Nutrition Examination Survey (NHANES), we propose a semiparametric multiple-index interaction model (MIIM) to explore the impact of three groups of persistent organic pollutants (POPs) on leukocyte telomere length (LTL). MIIM effectively addresses the challenge of high dimensionality by summarizing exposures into group-level indices, while allowing for nonlinear effects and interactions among exposures through these group indices. This formulation provides interpretable insights into both overall group effects and between-group interactions on the outcome, and allows for identification of key contributors within each group. MIIM can be applied to different types of health outcomes, including continuous, binary, and survival outcomes. We conducted Monte Carlo simulation studies to evaluate the performance of MIIM under various scenarios with high-dimensional and correlated exposure mixtures and illustrated its application to the NHANES data. By bridging biological insights with population-based epidemiological data, MIIM serves as a translational tool to explore the effects of environmental mixtures on health outcomes.
{"title":"Multiple-index interaction models to accommodate exposure grouping in environmental mixtures.","authors":"Myeonggyun Lee, Mengling Liu, Shanshan Zhao","doi":"10.1093/biomtc/ujaf175","DOIUrl":"10.1093/biomtc/ujaf175","url":null,"abstract":"<p><p>An important goal of environmental health research is to assess risks posed by mixtures of environmental exposures. Studies in different fields often group exposures based on their shared biological features. However, such grouping information has not been widely utilized in population-based environmental mixtures analyses due to the lack of appropriate statistical tools. Inspired by data from the National Health and Nutrition Examination Survey (NHANES), we propose a semiparametric multiple-index interaction model (MIIM) to explore the impact of three groups of persistent organic pollutants (POPs) on leukocyte telomere length (LTL). MIIM effectively addresses the challenge of high dimensionality by summarizing exposures into group-level indices, while allowing for nonlinear effects and interactions among exposures through these group indices. This formulation provides interpretable insights into both overall group effects and between-group interactions on the outcome, and allows for identification of key contributors within each group. MIIM can be applied to different types of health outcomes, including continuous, binary, and survival outcomes. We conducted Monte Carlo simulation studies to evaluate the performance of MIIM under various scenarios with high-dimensional and correlated exposure mixtures and illustrated its application to the NHANES data. By bridging biological insights with population-based epidemiological data, MIIM serves as a translational tool to explore the effects of environmental mixtures on health outcomes.</p>","PeriodicalId":8930,"journal":{"name":"Biometrics","volume":"82 1","pages":""},"PeriodicalIF":1.7,"publicationDate":"2026-01-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.ncbi.nlm.nih.gov/pmc/articles/PMC12862247/pdf/","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"146099568","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Nan Qiao, Wangcheng Li, Jingxiao Zhang, Canyi Chen
Synthesizing information from multiple data sources is crucial for constructing accurate individualized treatment rules (ITRs). However, privacy concerns often present significant barriers to the integrative analysis of such multicenter data. Classical meta-learning, which averages local estimates to derive the final ITR, is frequently suboptimal due to biases in these local estimates. To address these challenges, we propose a convolution-smoothed weighted support vector machine for learning the optimal ITR. The accompanying loss function is both convex and smooth, which allows us to develop an efficient multiround distributed learning procedure. Such distributed learning ensures optimal statistical performance with a fixed number of communication rounds, thereby minimizing coordination costs across data centers while preserving data privacy. Our method avoids pooling subject-level raw data and instead requires only sharing summary statistics. Additionally, we develop an efficient coordinate gradient descent algorithm, which guarantees at least linear convergence for the resulting optimization problem. Extensive simulations and an application to sepsis treatment across multiple intensive care units validate the effectiveness of the proposed method.
{"title":"Scalable and distributed individualized treatment rules for multicenter datasets.","authors":"Nan Qiao, Wangcheng Li, Jingxiao Zhang, Canyi Chen","doi":"10.1093/biomtc/ujag003","DOIUrl":"https://doi.org/10.1093/biomtc/ujag003","url":null,"abstract":"<p><p>Synthesizing information from multiple data sources is crucial for constructing accurate individualized treatment rules (ITRs). However, privacy concerns often present significant barriers to the integrative analysis of such multicenter data. Classical meta-learning, which averages local estimates to derive the final ITR, is frequently suboptimal due to biases in these local estimates. To address these challenges, we propose a convolution-smoothed weighted support vector machine for learning the optimal ITR. The accompanying loss function is both convex and smooth, which allows us to develop an efficient multiround distributed learning procedure. Such distributed learning ensures optimal statistical performance with a fixed number of communication rounds, thereby minimizing coordination costs across data centers while preserving data privacy. Our method avoids pooling subject-level raw data and instead requires only sharing summary statistics. Additionally, we develop an efficient coordinate gradient descent algorithm, which guarantees at least linear convergence for the resulting optimization problem. Extensive simulations and an application to sepsis treatment across multiple intensive care units validate the effectiveness of the proposed method.</p>","PeriodicalId":8930,"journal":{"name":"Biometrics","volume":"82 1","pages":""},"PeriodicalIF":1.7,"publicationDate":"2026-01-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"146123472","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Computational aspects of psychometric methods with R by Patricia Martinková and Adéla Hladká, Chapman & Hall/CRC, 2023, ISBN: 9781003054313, https://doi.org/10.1201/9781003054313.","authors":"Jinyuan Liu","doi":"10.1093/biomtc/ujaf132","DOIUrl":"https://doi.org/10.1093/biomtc/ujaf132","url":null,"abstract":"","PeriodicalId":8930,"journal":{"name":"Biometrics","volume":" ","pages":""},"PeriodicalIF":1.7,"publicationDate":"2025-10-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145273395","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Ivair R Silva, Joselito Montalban, Fernando L P de Oliveira
Ideally, the sequential monitoring of adverse events following post-licensed drugs and vaccines is correctly adjusted for confounding variables, such as gender and age, that may have an effect on the quality of the events. This is the idea behind the usual fully randomized, the placebo-control, and the self-control designs. Two prominent methods for conducting sequential analysis of the safety of post-market drugs and vaccines are the maximized sequential probability ratio test (MaxSPRT), and its conditional version, the CMaxSPRT. However, even when the assumption of sample homogeneity is realistic prior to the drug/vaccine administration, the effects caused by the drugs and vaccines on the risk of an adverse event, if any, can still vary according to observable covariates. For binomial and Poisson data, a straightforward sequential test method is introduced in order to accommodate a regression structure in the MaxSPRT. The proposed sequential regression test is also applicable for the CMaxSPRT, that is, the regression works for comparing historical and surveillance Poisson data with unknown heterogeneous baseline rates, taking into account seasonality and any other observable confounding covariates. To illustrate the usefulness of such a regression method, we describe the potential applications of the method to monitor vaccine-adverse events in Manitoba, Canada. The numeric results and examples were executed with the R Sequential package.
{"title":"Maximized sequential probability ratio test regression.","authors":"Ivair R Silva, Joselito Montalban, Fernando L P de Oliveira","doi":"10.1093/biomtc/ujaf170","DOIUrl":"10.1093/biomtc/ujaf170","url":null,"abstract":"<p><p>Ideally, the sequential monitoring of adverse events following post-licensed drugs and vaccines is correctly adjusted for confounding variables, such as gender and age, that may have an effect on the quality of the events. This is the idea behind the usual fully randomized, the placebo-control, and the self-control designs. Two prominent methods for conducting sequential analysis of the safety of post-market drugs and vaccines are the maximized sequential probability ratio test (MaxSPRT), and its conditional version, the CMaxSPRT. However, even when the assumption of sample homogeneity is realistic prior to the drug/vaccine administration, the effects caused by the drugs and vaccines on the risk of an adverse event, if any, can still vary according to observable covariates. For binomial and Poisson data, a straightforward sequential test method is introduced in order to accommodate a regression structure in the MaxSPRT. The proposed sequential regression test is also applicable for the CMaxSPRT, that is, the regression works for comparing historical and surveillance Poisson data with unknown heterogeneous baseline rates, taking into account seasonality and any other observable confounding covariates. To illustrate the usefulness of such a regression method, we describe the potential applications of the method to monitor vaccine-adverse events in Manitoba, Canada. The numeric results and examples were executed with the R Sequential package.</p>","PeriodicalId":8930,"journal":{"name":"Biometrics","volume":"81 4","pages":""},"PeriodicalIF":1.7,"publicationDate":"2025-10-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.ncbi.nlm.nih.gov/pmc/articles/PMC12745959/pdf/","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145848888","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Adaptive randomization is a clinical trial design feature used to modify treatment allocation probabilities during accrual. In time-to-event trials, the impact of adaptive randomization is less well understood for estimating treatment efficacy in the presence of time-varying effects [e.g., relative risk of progression to acquired immunodeficiency syndrome (AIDS) or death changes over time]. Here, we focus on time-to-event trials where the scientific estimand is a marginal hazard ratio in the absence of intermittent censoring over the support of observed times. We analytically show that adaptive randomization alters censoring patterns and illustrate via Monte Carlo simulations that the Cox proportional hazards estimator can yield biased estimates. As a remedy, we propose a censoring-robust estimator based on reweighting the partial likelihood score by treatment-specific censoring distributions that account for adaptive randomization. We derive the asymptotic properties of the proposed estimator and evaluate its finite sample operating characteristics via simulation. Finally, we apply our proposed method using data from the Community Programs for Clinical Research on AIDS Trial 002.
{"title":"Censoring-robust estimation in fixed sample time-to-event clinical trials with adaptive randomization.","authors":"Navneet R Hakhu, Daniel L Gillen","doi":"10.1093/biomtc/ujaf161","DOIUrl":"10.1093/biomtc/ujaf161","url":null,"abstract":"<p><p>Adaptive randomization is a clinical trial design feature used to modify treatment allocation probabilities during accrual. In time-to-event trials, the impact of adaptive randomization is less well understood for estimating treatment efficacy in the presence of time-varying effects [e.g., relative risk of progression to acquired immunodeficiency syndrome (AIDS) or death changes over time]. Here, we focus on time-to-event trials where the scientific estimand is a marginal hazard ratio in the absence of intermittent censoring over the support of observed times. We analytically show that adaptive randomization alters censoring patterns and illustrate via Monte Carlo simulations that the Cox proportional hazards estimator can yield biased estimates. As a remedy, we propose a censoring-robust estimator based on reweighting the partial likelihood score by treatment-specific censoring distributions that account for adaptive randomization. We derive the asymptotic properties of the proposed estimator and evaluate its finite sample operating characteristics via simulation. Finally, we apply our proposed method using data from the Community Programs for Clinical Research on AIDS Trial 002.</p>","PeriodicalId":8930,"journal":{"name":"Biometrics","volume":"81 4","pages":""},"PeriodicalIF":1.7,"publicationDate":"2025-10-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.ncbi.nlm.nih.gov/pmc/articles/PMC12715681/pdf/","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145792836","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We provide some comments about the recent paper by Yang et al. related to model estimation and hypothesis testing in segmented regression.
我们对Yang等人最近发表的关于分段回归中模型估计和假设检验的论文提供了一些评论。
{"title":"Letter to the Editors: Comments on \"Statistical inference on change points in generalized semiparametric segmented models\" by Yang et al. (2025).","authors":"Vito M R Muggeo","doi":"10.1093/biomtc/ujaf147","DOIUrl":"10.1093/biomtc/ujaf147","url":null,"abstract":"<p><p>We provide some comments about the recent paper by Yang et al. related to model estimation and hypothesis testing in segmented regression.</p>","PeriodicalId":8930,"journal":{"name":"Biometrics","volume":"81 4","pages":""},"PeriodicalIF":1.7,"publicationDate":"2025-10-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145666830","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Shengxian Ding, Debajyoti Sinha, Greg Hajcak, Roman Kotov, Chao Huang
Existing research in mental health has established that rising depressive symptoms in adolescents are associated with parental history of depression and other behavioral risk factors. Our goal is to investigate how these scalar variables, together with multiple functional covariates capturing neural responses to rewards, relate to future adolescent depression. Departing from prior studies that typically relied on simple linear regression to model all covariates, we propose a novel Bayesian quantile regression framework. This approach constructs a single-index summary of both scalar and functional covariates, coupled with a monotone link function that flexibly captures unknown nonlinear relationships and interactions. Our method addresses several limitations of existing approaches. It offers a clinically interpretable index akin to that of linear models, ensures that the estimated quantile remains within the response bounds, and jointly incorporates the registration of functional covariates within the quantile regression analysis. Our simulation studies demonstrate that our method outperforms existing unrestricted single-index-based methods, particularly when there are both scalar and preregistered functional covariates. Furthermore, we showcase the practical utility of our framework using data from a large-scale adolescent depression study, yielding a new, statistically principled summary of neural reward processing with direct relevance to future depression risk.
{"title":"Bayesian monotone single-index quantile regression model with bounded response and misaligned functional covariates.","authors":"Shengxian Ding, Debajyoti Sinha, Greg Hajcak, Roman Kotov, Chao Huang","doi":"10.1093/biomtc/ujaf145","DOIUrl":"10.1093/biomtc/ujaf145","url":null,"abstract":"<p><p>Existing research in mental health has established that rising depressive symptoms in adolescents are associated with parental history of depression and other behavioral risk factors. Our goal is to investigate how these scalar variables, together with multiple functional covariates capturing neural responses to rewards, relate to future adolescent depression. Departing from prior studies that typically relied on simple linear regression to model all covariates, we propose a novel Bayesian quantile regression framework. This approach constructs a single-index summary of both scalar and functional covariates, coupled with a monotone link function that flexibly captures unknown nonlinear relationships and interactions. Our method addresses several limitations of existing approaches. It offers a clinically interpretable index akin to that of linear models, ensures that the estimated quantile remains within the response bounds, and jointly incorporates the registration of functional covariates within the quantile regression analysis. Our simulation studies demonstrate that our method outperforms existing unrestricted single-index-based methods, particularly when there are both scalar and preregistered functional covariates. Furthermore, we showcase the practical utility of our framework using data from a large-scale adolescent depression study, yielding a new, statistically principled summary of neural reward processing with direct relevance to future depression risk.</p>","PeriodicalId":8930,"journal":{"name":"Biometrics","volume":"81 4","pages":""},"PeriodicalIF":1.7,"publicationDate":"2025-10-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.ncbi.nlm.nih.gov/pmc/articles/PMC12569519/pdf/","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145385616","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}