Ziyue Liu

We propose group‐specific nonlinear mixed effects models for longitudinal data whose indexing time is right‐aligned to specific events, modeled as the time preceding those events. These models are motivated by dementia cohort studies that examine nonlinear trajectories of longitudinal cognitive test scores in late life, with separate trajectories for individuals with and without dementia in the period leading up to death. The model framework includes three sub‐models: one for group membership, which represents dementia status in our motivating data, one for time to the indexing events, and one for the longitudinal outcome. Group membership is modeled using logistic regression or random forest. Conditional on group memberships, time to indexing events is modeled using accelerated failure time (AFT) models. Given both group membership and time to the indexing event, longitudinal data are modeled using nonlinear mixed effects models. By modeling and integrating out the time to indexing events, individuals whose indexing events have not been observed can be incorporated into the models. Stochastic approximation expectation maximization (SAEM) algorithms are adopted to maximize the likelihood for parameter estimation. In addition to modeling group‐specific longitudinal trajectories aligned to the index time, the proposed models can also be used to predict group memberships or improve the prediction of time to the indexing events. Simulation studies are conducted to evaluate finite sample performance. An application to data from a dementia study is used for illustration.

Autosomal dominant osteopetrosis (ADO) is a rare bone disorder caused by impaired osteoclastic resorption. Despite high bone mass, ADO is paradoxically associated with increased fracture risk. HR-pQCT provides unique, low-radiation assessments of bone, but prior studies in ADO are limited to small case series. Using baseline cross-sectional data from an ongoing natural history study, we characterized HR-pQCT phenotypes in adults with ADO and explored associations with self-reported fracture history and bone turnover markers. HR-pQCT outcomes at the distal ends and shafts of the radius and tibia in 48 adults with ADO were compared to 144 matched controls (1:3 ratio). In addition, z-scores for outcomes in ADO were calculated using established reference data. At trabecular-rich distal sites, ADO demonstrated markedly elevated total volumetric BMD (vBMD), trabecular vBMD, trabecular bone volume fraction, trabecular thickness, and estimated failure load (all p < .001), with values often approaching or exceeding twice those in controls. At cortical-rich shaft sites, total vBMD, bone area, bone area fraction, and cortical thickness were greater in ADO (all p < .05). However, there was considerable interindividual variability, with some individuals having normal z-scores while others exhibited z-scores exceeding +20. The phenotype reflected increased bone mass rather than mineralization—cortical vBMD was normal and tissue mineral density was normal-to-lower in ADO. Outcomes at both distal and shaft sites were strongly correlated with lifetime fracture number (Spearman ρ = 0.64-0.75, p < .001), indicating ADO individuals with the “strongest” bones (via micro-finite element modeling) reported most fractures. The outcomes were correlated inversely with serum C-telopeptide and positively with serum tartrate resistant acid phosphatase 5b, consistent with ADO being rich with dysfunctional osteoclasts. Overall, the data reveal the profound, yet variable, phenotype in ADO and indicate that HR-pQCT measures correlate with disease severity and represent potential surrogate endpoints for future therapeutic trials.

Background: Fracture nonunion remains a major clinical challenge. Although radiographic scores such as RUST (Radiographic Union Score for Tibial fractures) and mRUST (the modified version) are widely used to assess healing, there has been limited investigation into their longitudinal use. This study evaluated whether a log-logistic statistical model of radiographic healing scores over time could quantify differences between fractures that healed and those that went on to nonunion. Methods: This retrospective cohort analysis added serial radiographs of 49 femoral shaft nonunions to radiographs from previous cohorts. Tibiae and femora were scored with RUST and mRUST, respectively. Scores were plotted over time and fitted to a log-logistic curve to generate a healing (“position”) equation. Model-derived parameters (Y _inf [ultimate score], k [a curve shape parameter], and t _half [time to half of the score change]) were compared between fractures that did and did not unite. Secondarily, first and second derivatives were used to describe the fracture healing “velocity” and “acceleration” for each group. Results: A total of 613 fractures were analyzed: 234 tibial (196 healed, 38 nonunion) and 379 femoral (330 healed, 49 nonunion). In femoral fractures, Y _inf , k, and t _half all differed significantly (p < 0.008) between healed and nonunion cases. In tibial fractures, Y _inf and k differed (both p < 0.008), whereas t _half did not (p = 0.30). The first and second derivatives at the population level suggest early differences in healing kinematics between fractures that heal and those that go on to nonunion. Conclusions: This proof-of-concept study demonstrates that log-logistic modeling of radiographic scores can quantify differences in radiographic healing between femoral and tibial shaft fractures treated with intramedullary nails that will go on to union versus nonunion. Differences in healing velocity and acceleration may emerge early, supporting the potential of the longitudinal framework for earlier identification of fractures at risk for nonunion. Level of Evidence: Therapeutic Level III . See Instructions for Authors for a complete description of levels of evidence.

Autosomal dominant osteopetrosis (ADO) is a rare osteosclerotic disorder usually caused by missense variants in the CLCN7 gene, which results in impaired osteoclastic bone resorption. Penetrance is incomplete, and disease severity varies widely, even among relatives within the same family. Although ADO can cause visual loss, osteonecrosis, osteomyelitis, and bone marrow failure, the most common complication of ADO is fracture. We are conducting a natural history study to characterize disease progression and determinants of disease severity. We hypothesized that baseline BMD and bone turnover markers would correlate with self-reported fracture history. We report cross-sectional analysis of baseline data from the natural history study in 54 individuals (42 adults, 12 children). In adults, Z-scores for both volumetric (r = 0.87, p < .001) and areal BMD (aBMD) of the LS, and Z-scores for FN, and TH aBMD (r = 0.77 to 0.78; p < .001) were correlated with lifetime fracture number. Tartrate resistant acid phosphatase, a marker of osteoclast number, correlated positively with fracture (r = 0.52, p = .004) consistent with an adaptive response of higher numbers of osteoclasts among more severely affected individuals. However, fracture number correlated inversely with the bone resorption markers serum C-telopeptide (r = −0.60, p < .001) and urine N-telopeptide/creatinine ratio (r = −0.35, p = .047), suggesting that ADO subjects who have the most reduced osteoclast activity have a greater tendency to fracture. Correlation coefficients between fractures, BMD, and bone turnover markers were similar when limited to the 37 adults with disease-causing CLCN7 variants. There were no statistically significant differences between subjects with the most common CLCN7 variant (G215R), the most common variant in our cohort, compared to other CLCN7 variants with respect to fracture, bone density measures, or biochemical markers of bone turnover. These data demonstrate that bone density and biochemical bone turnover markers are indicators of ADO severity as defined by fracture number.

Background: There is increasing awareness of a role for muscle composition in sarcopenia and cachexia. Computed tomography (CT)–based measures of muscle density (MusD) are commonly used to indicate composition, with a decrease in MusD reflecting an increase in muscle fat infiltration. The current study explored predictors of MusD acquired using high‐resolution peripheral quantitative computed tomography (HR‐pQCT) and whether MusD predicted physical performance. In addition, reference data for MusD were generated and applied.Methods: HR‐pQCT scans performed in 1662 adults (aged 18–80 years) at 30% of bone length proximal from the distal end of the radius and tibia were analysed for forearm and leg MusD, respectively. Predictors of MusD were explored, and it was investigated whether MusD predicted physical performance. Centile curves were fit to the MusD data using the LMS approach to generate reference data, and a calculator was developed to enable computation of subject‐specific standardised outcomes. The utility of the calculator was explored in validation cohorts of female collegiate‐level athletes (n = 50) and individuals with chronic kidney disease (CKD) (n = 50).Results: Forearm and leg MusD were predicted by whole‐body percent fat, sex and age. Forearm and leg MusD were 0.46 (~1.9%) and 0.60 mgHA/cm³ (~2.6%) lower in females than in males, respectively (all p ≤ 0.002). For every decade of greater age, forearm and leg MusD were 0.28 (~1.2%) and 0.75 mgHA/cm³ (~3.3%) lower, respectively (all p < 0.001). These observations were independent of whole‐body percent fat and appendicular lean mass (ALM)/height². MusD predicted grip strength, fast gait speed and self‐reported physical function independent of ALM/height², body mass index and whole‐body percent fat. Grip strength was 0.756 kg (~2.4%) greater for every 1 mgHA/cm³ (~4.2%) greater forearm MusD (p < 0.001). Reference data were generated. Compared to the reference data, female athletes had above‐average leg MusD (z‐score = 0.20; 95% CI, 0.01–0.39), whereas those with CKD had z‐scores for forearm and leg MusD of −1.51 (95% CI, −1.95 to −1.08) and −1.70 (95% CI, −2.04 to −1.36), respectively.Conclusions: HR‐pQCT acquired MusD provides a novel indicator of muscle composition which predicts physical function independent of muscle quantity (i.e., ALM/height²). Whether the unique information provided by MusD has a role in quantifying health and the consequences of disease and illness (including sarcopenia and cachexia) requires further exploration. Studies in this area may be facilitated by the reference data generated which enable MusD in an individual or population of interest to be expressed relative to sex‐ and age‐matched norms.

Background: Researchers employ murine fracture models to study bone healing, but the temporal relationship between mouse and human fracture healing is poorly understood. The hypothesis of this study was that it was possible to quantify specific post-fracture time frames corresponding to the stages of endochondral ossification in both mice and humans. Methods: Radiographs of mice and human femoral fractures treated with intramedullary stabilization were reviewed. The study included 330 human femoral fractures (OTA/AO 32A, B, or C injuries) that ultimately healed without complications in patients aged 18 to 55 years and 309 surgically created midshaft femoral fractures in 3-month-old C57BL6/J mice. Multiple orthopaedic surgeons assessed the radiographs using the Modified Radiographic Union Score for Tibia (mRUST). A 4-parameter log-logistic curve was fit to describe fracture healing over time, with 3 parameters allowed to vary: Y_∞ (mRUST score at time = ∞), k (healing rate in [1/log(time)]), and X_0.5 (time to half-healing). Results: The values (and 95% confidence interval) for the mice were Y_∞ = 14.70 (14.54 to 14.87), k = 4.54/log(days) (4.30 to 4.77), and X_0.5 = 11.77 days (11.56 to 11.98). For the humans, the values were Y_∞ = 16.78 (16.21 to 17.36), k = 1.37/log(days) (1.28 to 1.45), and X_0.5 = 91 days (83 to 99). All parameters differed significantly between the mice and humans (p < 0.05). Conclusions: Using mRUST scoring and mathematical modeling, we were able to quantify and compare the temporal progression of fracture healing in mice and humans. Clinical Relevance: These data are relevant for designing and/or interpreting fracture healing studies of mice and humans to promote rational translation of fracture research between species.

The R47H variant of the Triggering Receptor Expressed on Myeloid Cells 2 (TREM2) is a risk factor for Alzheimer’s disease in humans and leads to lower bone mass accrual in female but not male 12-month-old mice. To determine whether, as with aging, gonadectomy results in sex-specific musculoskeletal effects, gonad removal or SHAM surgery were performed in 4-month-old TREM2^R47H/+ mice and wild type (WT) male and female littermates (n = 10-12/group), with sexes analyzed separately. Body weight was lower in males, but higher in females after gonadectomy, independently of their genotype. Gonadectomy also lead to decreased BMD in males at all sites and in whole body (total) and spine in female mice for both genotypes. Total and femur BMD was lower in gonadectomized male mice 6-weeks post-surgery, independently of the genotype. On the other hand, BMD was only lower in ovariectomized WT but not TREM2^R47H/+ mice in all sites measured at this time point. Bone formation and resorption markers levels were not affected by orchiectomy, whereas CTX was higher 3 weeks after surgery and P1NP showed a tendency towards lower values at the 6 weeks timepoint only in ovariectomized WT mice. μCT analyses showed no differences resulting from gonadectomy in structural parameters in femoral cortical bone for either sex, but lower tissue mineral density in males of either genotype 6 weeks post-surgery. Nevertheless, biomechanical properties were overall lower in gonadectomized males of either genotype, and only for WT ovariectomized mice. Distal femur cancellous bone structure was also affected by gonadectomy in a genotype- and sex-dependent manner, with genotype-independent changes in males, and only in WT female mice. Thus, expression of the TREM2 R47H variant minimally alters the impact of gonadectomy in the musculoskeletal system in males, whereas it partially ameliorates the consequences of ovariectomy in female mice. A link between dementia and loss of bone mass and risk of fractures (osteoporosis) has been suggested, but never demonstrated. Bone loss can be detected before cognitive decline and reduced mobility in Alzheimer’s disease (AD) patients. Since TREM2 gene mutations are associated with AD/dementia and bone fragility, we investigated whether expression of the TREM2-R47H mutated gene alters the response to removal of sex steroids in male and female mice. Expression of TREM2-R47H minimally alters the impact of sex steroid removal in the musculoskeletal system in males, whereas it partially ameliorates the consequences of removing the ovaries in female mice.

In this paper, we aim to both borrow information from existing units and incorporate the target unit's history data in time series forecasting. We consider a situation when we have time series data from multiple units that share similar patterns when aligned in terms of an internal time. The internal time is defined as an index according to evolving features of interest. When mapped back to the calendar time, these time series can span different time intervals that can include the future calendar time of the targeted unit, over which we can borrow the information from other units in forecasting the targeted unit. We first build a hierarchical state space model for the multiple time series data in terms of the internal time, where the shared components capture the similarities among different units while allowing for unit‐specific deviations. A conditional state space model is then constructed to incorporate the information of existing units as the prior information in forecasting the targeted unit. By running the Kalman filtering based on the conditional state space model on the targeted unit, we incorporate both the information from the other units and the history of the targeted unit. The forecasts are then transformed from internal time back into calendar time for ease of interpretation. A simulation study is conducted to evaluate the finite sample performance. Forecasting state‐level new COVID‐19 cases in United States is used for illustration.