A mathematical and computational framework to predict the time to and recovery from osteoporosis via serial DEXA scans: a proof-of-concept model for digital decision support

A mathematical and computational framework to predict the time to and recovery from osteoporosis via serial DEXA scans: a proof-of-concept model for digital decision support | Research Square window.SnipcartSettings = { analytics: { enabled: false } }; (function() { var accessVector = localStorage.getItem('access_vector') || ''; window.dataLayer = window.dataLayer || []; if (accessVector) { window.dataLayer.push({ user: { profile: { profileInfo: { snid: accessVector } } } }); } })(); (function(w,d,s,l,i){w[l]=w[l]||[];w[l].push({'gtm.start':new Date().getTime(),event:'gtm.js'});var f=d.getElementsByTagName(s)[0],j=d.createElement(s),dl=l!='dataLayer'?'&l='+l:'';j.async=true;j.src='https://www.googletagmanager.com/gtm.js?id='+i+dl;f.parentNode.insertBefore(j,f);})(window,document,'script','dataLayer','GTM-K279D39R'); Browse Preprints In Review Journals COVID-19 Preprints AJE Video Bytes Research Tools Research Promotion AJE Professional Editing AJE Rubriq About Preprint Platform In Review Editorial Policies Our Team Advisory Board Help Center Sign In Submit a Preprint Cite Share Download PDF Research Article A mathematical and computational framework to predict the time to and recovery from osteoporosis via serial DEXA scans: a proof-of-concept model for digital decision support P Scully, Azrin Muslim, T Sheehy, A Costello, N Asri, D Lyons This is a preprint; it has not been peer reviewed by a journal. https://doi.org/ 10.21203/rs.3.rs-8028195/v1 This work is licensed under a CC BY 4.0 License Status: Under Review Version 1 posted 9 You are reading this latest preprint version Abstract Background Dual energy X ray absorptiometry (DEXA) is the diagnostic standard for osteoporosis, yet its serial data remain underutilised in predictive analytics. To our knowledge, no published model provides explicit age based predictions of osteoporosis onset or recovery using serial DEXA T score trajectories. This study describes a mathematical framework for predicting time to osteoporosis (TTO), defined as the age at which a patient’s T score trajectory reaches negative 2 point five. Methods We developed a mathematical framework that converts serial DEXA T-scores into time-to-osteoporosis (TTO) and time-to-exit osteoporosis (TEO) predictions. Using hip DEXA results from 50 consecutive patients with ≥2 scans, T-scores were plotted against age, and we applied two models: (a) a two-point slope algorithm (TTOc) and (b) a multipoint least-squares regression (TTOt). Both were designed to estimate the age at which the T-score trajectory would cross the diagnostic threshold of negative 2 point five. Results Both algorithms successfully predict the age of entry into, or recovery from, osteoporosis. TTOc produced scan-pair–specific short-term projections, whereas TTOt provided smoothed cumulative trajectories. Worked examples demonstrated agreement between models in patients with monotonic decline and highlighted the stabilising effect of regression in fluctuating cases. Discussion This framework transforms static DEXA outputs into patient specific, age-based predictions, enhancing clinical interpretability. In addition to their immediate clinical use, deterministic equations can serve as the foundation for a hybrid machine-learning model that uses slope and intercept values as interpretable features within ensemble or deep learning architectures to improve temporal prediction accuracy across larger datasets. Conclusions This proof-of-concept demonstrates the feasibility of trajectory-based modelling for osteoporosis risk prediction. It represents a step toward AI-assisted DEXA interpretation systems that are transparent, explainable, and directly usable at the point of care. It also reframes DEXA outputs into an age-based measure that patients easily understand, offering clinicians a simplified parameter for monitoring therapy. The incorporation of these findings into DEXA reporting could strengthen patient engagement and adherence. DEXA Bone Mineral Density T-score Predictive Modelling Decision Support Machine Learning Figures Figure 1 Figure 2 SUMMARY BOX What was already known Serial DEXA measurements are difficult to interpret in clinical practice. Existing prediction tools rely on complex or opaque modelling approaches with limited explainability. The relationship between bone mineral density trajectories and future osteoporosis risk lacks intuitive, patient-facing metrics. What this study adds Introduces a transparent, deterministic algorithm (TTO/TEO) that converts serial DEXA data into age-based threshold predictions. Demonstrates a fully explainable method where slope, intercept, and threshold-crossing age map directly to measurable physiological parameters. Provides a scalable foundation for hybrid, explainable AI systems that incorporate non-linear bone changes while maintaining interpretability. DECISION MAKING IMPLICATIONS The proposed framework translates raw DEXA outputs into clinically intuitive, age-driven predictions that support individualised decision-making. By offering interpretable trajectories rather than isolated Z-scores, it can help clinicians identify high-risk patients earlier, communicate future fracture risk more effectively, and optimise the timing of interventions. Integration of this model into electronic DEXA reporting systems may improve adherence, facilitate shared decision-making, and promote algorithmic transparency across bone health services. INTRODUCTION Osteoporosis is the most common metabolic bone disease and is characterised by impaired bone strength and increased fragility fracture risk¹. In the United States, approximately 10 million adults aged ≥ 50 years have osteoporosis and an additional 43 million are at risk due to low bone mass, placing them at increased risk of fracture². With global ageing populations, the burden of osteoporosis and osteoporotic fractures is expected to rise further. Bone mineral density (BMD) testing is central to diagnosis, risk prediction, and monitoring. Among the available methods, dual-energy X-ray absorptiometry (DEXA) of the spine and hip is the accepted diagnostic standard and the most reliable technique for longitudinal assessment³. Osteoporosis is defined by the World Health Organization (WHO) as a T-score ≤ negative 2 point five (-2.5) at the hip or spine 4 . The International Society for Clinical Densitometry (ISCD) likewise recommends use of the NHANES III young-adult Caucasian reference database for hip T-scores across ethnic groups, with sex-specific reference data 5 . While DEXA remains the diagnostic standard for quantifying bone mineral density (BMD), its potential for longitudinal prediction is underexplored. Conventional reporting offers a numerical T-score, a statistical measure of deviation from peak bone mass, but provides no intuitive sense of temporal progression; when a patient is likely to become osteoporotic, or conversely, when improvement is expected under therapy. Existing tools such as FRAX estimate fracture probability but do not forecast the time to threshold. This temporal gap leaves clinicians without a way to visualise or communicate disease trajectories in age-based terms. The present study introduces a simple algorithmic framework that transforms serial DEXA data into interpretable age-based predictions: time to osteoporosis (TTO) and time to exit osteoporosis (TEO), thereby reframing densitometry into a decision-support metric that aligns with patient understanding and future digital automation. To our knowledge, no published model directly converts serial DEXA data into explicit time to threshold predictions using deterministic slope and intercept formulations, and this work therefore offers an initial framework for such trajectory-based estimation METHODS Study Design and Data Source We performed a retrospective proof-of-concept study using anonymised DEXA data from University Hospital Limerick. Fifty men and women with two or more hip DEXA scans were included, without exclusions for age, comorbidity, prior osteoporosis diagnosis, or treatment status. Scans were performed on a Lunar Prodigy densitometer (GE Medical Systems, USA; software version 13.60.033). BMD was measured at the total hip and lumbar spine (L2–L4), and T-scores were generated from the manufacturer’s reference database of sex-matched healthy young adults. Algorithm development The model was designed to answer a simple clinical question: Given a patient’s current trajectory, at what age will their T-score cross into the osteoporotic range (≤ − 2.5)? For analysis, only hip T-scores were used. At the proximal femur, measurements from the neck, trochanteric, and intertrochanteric regions were combined to yield a total hip value. Patient age and corresponding T-scores were entered into Microsoft Excel, where macros were developed to calculate slope and intercept values. Graphical outputs were generated via GraphPad Prism, and values were cross-checked against hand-written derivations to confirm accuracy. Two-point model (TTOc, Fig. ) For each pair of consecutive scans, a straight line was fitted between the two coordinates (x1,y1) and (x2,y2) where x = age and y = T-score. The slope was calculated as follows: $$\:m=\frac{{y}_{2}-{y}_{1}}{{x}_{2}-{x}_{1}}$$ The line equation was then: Solving for y = − 2.5 yielded the predicted age of entry into osteoporosis: $$\:x=\frac{-2.5-C}{m}$$ If the patient was already osteoporotic (y < − 2.5) and the slope was positive, the same calculation instead estimated the age at which the trajectory would exit the osteoporotic range ( time to exit osteoporosis, TEO ). Non-generating cases No TTO or TEO values were reported when the slope pointed away from the osteoporotic threshold (i.e. improving T-scores in non-osteoporotic patients or worsening scores in already osteoporotic patients), as such predictions would not be clinically meaningful. Multipoint regression model (TTOt, Figure 2a, 2b) For patients with three or more scans, a least-squares regression line was fitted through all available data points: The cumulative age at which the regression line crossed the osteoporotic threshold was as follows: $$\:x=\frac{-2.5-C}{m}$$ This regression approach provides a more stable trajectory by smoothing fluctuations that often occur in individual scan pairs. Full pseudocode for TTOc and TTOt is provided in Supplementary File 1 to facilitate reproducibility. RESULTS The algorithms successfully generated a predicted age of entry into the osteoporotic range on the basis of serial DEXA trajectories ( Table 1 ). Both the two-point model (TTOc) and the multipoint regression model (TTOt) produced estimates of time to osteoporosis (TTO) , whereas the two-point model also identified cases of time to exit osteoporosis (TEO) in patients showing recovery. Table 1 summarises outcomes from patients with two to five scans. In most cases, TTOc and TTOt produced consistent estimates; however, variability between consecutive scans occasionally led to divergence or non-generation of values, particularly when slopes were positive in non-osteoporotic patients or negative in already osteoporotic patients. Worked examples illustrate these patterns. In one subject with five scans over 13 years (Fig. 2a), T-scores declined steadily from –1.2 to –3.2. TTOc predicted entry into osteoporosis at 76.9 years from the first two scans and at 71.8 years after the third scan. Once the patient entered the osteoporotic range, no further TTOc was generated; however, improvement at the fifth scan (T-score –2.7 at age 79) allowed estimation of TEO at 82.5 years. The cumulative TTOt for this patient suggested overall entry into osteoporosis at 74.1 years, closely aligning with the two-point results. In contrast, a second patient (Fig. 2b) showed wide fluctuations in T-scores across four scans. Here, TTOc values varied markedly (57.4 to 94.0 years) or could not be generated, whereas the regression-based TTOt provided a more stable estimate of 73.6 years, smoothing the oscillations of individual measurements. DISCUSSION This study showed that simple mathematical models can likely estimate the age at which patients enter or exit the osteoporotic T score range, and that serial DEXA data can be converted into interpretable, time-based indicators of bone health trajectory. The framework aims to bridge quantitative densitometry and clinical reasoning by reframing T scores into predicted ages of onset or recovery. These findings are consistent with the initial hypothesis that trajectory-based modelling may convert numerical bone density data into a more intuitive, patient centred metric. Adherence to osteoporosis therapy remains a major challenge, and conventional reporting often provides limited guidance. Real world persistence with oral bisphosphonates varies widely, with medication possession ratios that are frequently low 6 . Systematic reviews suggest that multicomponent interventions involving education, counselling, and active patient involvement are more likely to improve adherence than single component approaches 7 . Evidence also indicates that personalised, numeric communication of osteoporosis risk may improve patient understanding and intent to treat when compared with qualitative descriptions 8 . In keeping with findings from the RICO study 8 , our model may help by translating abstract T scores into personalised, age-based predictions that patients can interpret more easily. Beyond communication, the usability of any predictive method depends on its transparency. Unlike black box models, the deterministic structure of this framework is likely to maintain interpretability. Informatics literature has highlighted that opaque predictive systems may reduce users’ trust, make validation more difficult, and obscure model failure modes. Rudin has argued that high stakes clinical decisions should rely on inherently interpretable models rather than post hoc methodological explanations, since transparency is essential for safety and auditability 9 . Lipton has similarly noted that parameter level interpretability allows end users to understand and challenge predictions when required 10 . Each of the proposed model parameters, including the slope, intercept, and the estimated threshold crossing age, is derived directly from measured DEXA T-scores, which are validated clinical indicators of bone mineral density. The framework also provides a transparent mathematical foundation for expansion. Hybrid systems could incorporate this deterministic layer within supervised learning pipelines. Gradient boosting regression or Gaussian process models may be trained on larger datasets using demographics, comorbidity profiles, and treatment histories as covariates, with TTO values used as interpretable targets. Recurrent neural networks may capture nonlinear BMD patterns while remaining anchored to the baseline logic of the slope-based trajectory. Such systems could enhance precision while remaining aligned with modern standards for explainable artificial intelligence. In clinical settings, age-based metrics may also offer practical support for communication and decision making. For example, a seventy-two-year-old individual with a TTO estimate of seventy-seven years could be informed that their current trajectory suggests a likely transition into the osteoporotic range within approximately five years. This type of framing may help clinicians time intervention and reinforce the rationale for follow up. Embedding TTO and TEO values directly into electronic DEXA reports may therefore support more effective shared decision making. This study also highlights methodological considerations. The two-point model is sensitive to fluctuations between consecutive scans, which may lead to divergent or non-generating values. The regression-based approach appears to provide a more stable long-term trajectory by smoothing measurement variability. These differences reflect the mathematical characteristics of the algorithms. TTOc derives slope and intercept from two serial measurements, whereas TTOt applies least squares fitting across all available data points. The predicted ages of entry into or exit from osteoporosis are obtained by solving these line equations at the diagnostic threshold of negative 2 point five. The worked examples support the internal consistency of these formulas and suggest that the overall method is valid at the proof-of-concept stage. Algorithm Limitations This work has several important limitations. 1. Assumption of linearity Bone loss trajectories are often non-linear, with periods of accelerated decline (menopause) and periods of stabilisation (treatment). The model therefore approximates rather than represents true biology. 2. Absence of confidence intervals Predictions are point estimates, and the model does not generate measures of uncertainty. 3. Effects of measurement error DEXA reproducibility error (1–2%) may influence slope estimation, particularly in two-point calculations. 4. Treatment heterogeneity Pharmacologic therapy was not modelled and may modify trajectories. 5. Non generating patterns For trajectories moving away from the osteoporotic threshold, TTO or TEO cannot be generated. Conclusions Despite these limitations, the proposed framework offers a novel and clinically meaningful method for transforming serial DEXA measurements into interpretable, age-based predictions. Its key strength lies in its transparency: each component of the model (slope, intercept, threshold-crossing age) corresponds directly to measurable properties of bone physiology. This aligns with the goals of explainable artificial intelligence (XAI), in which interpretability, reproducibility, and clarity are prioritised for safe clinical deployment. The deterministic structure of the model also provides a scalable foundation for future methodological extensions. As larger datasets become available, this framework can be expanded into hybrid informatics systems that retain interpretability while accommodating non-linear BMD changes and treatment effects. Candidates include Gaussian process regression, Bayesian hierarchical models, and interpretable neural ordinary differential equations, each of which could enhance predictive performance without compromising transparency. By converting abstract statistical information into patient-relevant metrics, the model has potential to improve clinical decision-making, strengthen communication around bone health, and support personalised risk assessment. Together, these findings suggest that time-based interpretation of serial DEXA data may offer a practical, transparent, and scalable foundation for future clinical decision support tools in osteoporosis care. Abbreviations AI Artificial Intelligence BMD Bone Mineral Density C Intercept (linear model) DEXA Dual-Energy X-ray Absorptiometry FRAX Fracture Risk Assessment Tool FRDP Fracture Risk Decision Point GE General Electric Medical Systems HSE Health Service Executive IOF International Osteoporosis Foundation ISCD International Society for Clinical Densitometry LSTM Long Short-Term Memory m Slope (linear model) NHANES National Health and Nutrition Examination Survey OP Osteoporosis TEO Time to Exit Osteoporosis TEOc Time to Exit Osteoporosis (two-point calculation) T-score Standardised BMD score relative to young-adult reference TTO Time to Osteoporosis TTOc Time to Osteoporosis (two-point calculation) TTOt Time to Osteoporosis (regression-based calculation) UHL University Hospital Limerick USA United States of America WHO World Health Organization XAI Explainable Artificial Intelligence Declarations Ethics approval and consent to participate The study was reviewed by the UL Hospital Group Research Ethics Committee, University Hospital Limerick. The Committee granted an ethics waiver on the basis that the study involved a retrospective analysis of anonymised clinical data and did not require full ethical review. This waiver was confirmed in writing by the Committee Chair, Prof. Colin Peirce, as documented in the approval letter (Supplementary File 2). All procedures were conducted in compliance with the Declaration of Helsinki, the General Data Protection Regulation (GDPR), the Data Protection Acts, and the Health Research Regulations, as outlined in the ethics correspondence. Because the dataset consisted solely of anonymised retrospective service data, the requirement for individual informed consent was waived by the UL Hospital Group Research Ethics Committee. Consent for publication Not applicable. The study uses anonymised retrospective data and contains no identifiable information. The UL Hospital Group Research Ethics Committee confirmed that consent for publication was not required. Availability of data and materials The clinical dataset analysed during this study is not publicly available owing to GDPR and HSE data-governance restrictions. De-identified data can be accessed upon reasonable request, contingent on approval from the UL Hospitals Group Data Controller and adherence to GDPR compliant safeguards. Competing interests The authors declare that they have no competing interests relevant to this work, including no commercial or proprietary involvement in the development or deployment of the algorithms described. Funding This research received no specific grant from any funding agency in the public, commercial, or not-for-profit sectors. Authors’ contribution PS conceived the study and designed the algorithm, AM wrote the manuscript and interpreted data, TS and AC involved in collecting and maintaining data, NA assisted with data curation and reference management, DL supervised and provided governance for the study. Acknowledgements The authors gratefully acknowledge the staff and nurses of the Clinical Age Assessment Unit, University Hospital Limerick, whose meticulous collection and maintenance of the DEXA dataset made this work possible. References Osteoporosis prevention, diagnosis, and therapy. JAMA. 2001;285:785–795. Looker AC, Frenk SM. Osteoporosis and low bone mass among adults aged 50 and over: United States, 2017–2018. NCHS Data Brief, no. 405. Hyattsville, MD: National Center for Health Statistics; 2021. Krugh M, Langaker MD. Dual-Energy X-Ray Absorptiometry . In: StatPearls [Internet]. Treasure Island (FL): StatPearls Publishing; 2024 May 20. Kanis JA, Cooper C, Rizzoli R, Reginster JY; on behalf of ESCEO and IOF. European guidance for the diagnosis and management of osteoporosis in postmenopausal women. Osteoporosis Int . 2019;30(1):3–44. International Society for Clinical Densitometry. ISCD Official Positions 2023 . Middletown, CT: ISCD; 2023. Fatoye F, Garcia-Tormo G, Gebrye T, et al. Real-world persistence and adherence with oral bisphosphonates in patients treated for osteoporosis: a retrospective cohort study. BMJ Open . 2019;9(4):e027049 Cornelissen D, de Kunder S, Si L, et al. Interventions to improve adherence to anti-osteoporosis medications: an updated systematic review. Osteoporosis Int . 2020;31(9):1645–1669. Sharma M, Beaudart C, Clark P, et al. Clinical and demographic factors determining patient fracture‑risk decision point (FRDP): the improving risk communication in osteoporosis (RICO) project. Osteoporosis Int . 2025;36:71–80. doi:10.1007/s00198-024-07264-5 Rudin C. Stop Explaining Black Box Machine Learning Models for High Stakes Decisions and Use Interpretable Models Instead. Nat Mach Intell. 2019 May;1(5):206-215. doi: 10.1038/s42256-019-0048-x. Epub 2019 May 13. PMID: 35603010; PMCID: PMC9122117. Lipton ZC. The mythos of model interpretability. Commun ACM. 2018;61(10):36–43. Table Table 1 is available in the supplementary files section Additional Declarations No competing interests reported. Supplementary Files Supp1ASCIISchematic.docx Supplementary File 1. ASCII method schematics for reproducibility. TableI.docx Cite Share Download PDF Status: Under Review Version 1 posted Editorial decision: Revision requested 17 Apr, 2026 Reviews received at journal 14 Apr, 2026 Reviews received at journal 11 Apr, 2026 Reviewers agreed at journal 09 Apr, 2026 Reviewers agreed at journal 09 Apr, 2026 Reviewers invited by journal 30 Dec, 2025 Editor assigned by journal 21 Nov, 2025 Submission checks completed at journal 20 Nov, 2025 First submitted to journal 20 Nov, 2025 You are reading this latest preprint version Research Square lets you share your work early, gain feedback from the community, and start making changes to your manuscript prior to peer review in a journal. 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05:55:28","extension":"html","order_by":22,"title":"","display":"","copyAsset":false,"role":"acdc-reference","size":100374,"visible":true,"origin":"","legend":"","description":"","filename":"earlyproof.html","url":"https://assets-eu.researchsquare.com/files/rs-8028195/v1/53c1ea1424954cdef35b07de.html"},{"id":99371524,"identity":"2e761f96-4a9b-4d2e-8f3e-876cb89836e8","added_by":"auto","created_at":"2026-01-02 05:55:28","extension":"png","order_by":1,"title":"Figure 1","display":"","copyAsset":false,"role":"figure","size":118519,"visible":true,"origin":"","legend":"\u003cp\u003e\u003cstrong\u003eWorkflow for the two-point (TTOc and TEOc) and regression based (TTOt) algorithms. \u003c/strong\u003eThis schematic illustrates how serial age and T score pairs are used to calculate slope, intercept, and predicted threshold crossing at T score negative 2 point five. The two-point model uses consecutive scan pairs, while the regression model fits a least squares trajectory across all available scans.\u003c/p\u003e","description":"","filename":"1.png","url":"https://assets-eu.researchsquare.com/files/rs-8028195/v1/b2fd9ab0a428a2406b3bf66c.png"},{"id":99789453,"identity":"a2d12465-a8e4-4cc7-9fe3-b7f974e87da7","added_by":"auto","created_at":"2026-01-08 12:49:42","extension":"png","order_by":2,"title":"Figure 2","display":"","copyAsset":false,"role":"figure","size":147672,"visible":true,"origin":"","legend":"\u003cp\u003e\u003cstrong\u003eTrajectory plots of two representative patients with serial hip DEXA measurements.\u003cbr\u003e\n \u003c/strong\u003e(a) A patient with five scans over thirteen years showing a monotonic decline in T score, including entry into and partial recovery from the osteoporotic range. The solid line represents the regression trajectory, and dotted lines show two-point estimates.\u003cbr\u003e\n(b) A patient with fluctuating T scores across four scans. The regression model provides a stable estimate of threshold crossing despite oscillations in individual measurements. The horizontal line indicates the osteoporosis threshold of T score negative 2 point five.\u003c/p\u003e","description":"","filename":"2.png","url":"https://assets-eu.researchsquare.com/files/rs-8028195/v1/cdec2f6e0fce2f02d152929e.png"},{"id":99789057,"identity":"3a47ea3a-810d-4fa5-b3da-b1e888295d4f","added_by":"auto","created_at":"2026-01-08 12:48:39","extension":"pdf","order_by":0,"title":"","display":"","copyAsset":false,"role":"manuscript-pdf","size":725069,"visible":true,"origin":"","legend":"","description":"","filename":"manuscript.pdf","url":"https://assets-eu.researchsquare.com/files/rs-8028195/v1/ac18d2a0-9012-448b-afe6-46a9a2605c9a.pdf"},{"id":99788916,"identity":"df80a4cb-9861-4560-bb89-f4c6123aef08","added_by":"auto","created_at":"2026-01-08 12:48:17","extension":"docx","order_by":1,"title":"","display":"","copyAsset":false,"role":"supplement","size":14582,"visible":true,"origin":"","legend":"\u003cp\u003e\u003cstrong\u003eSupplementary File 1\u003c/strong\u003e. ASCII method schematics for reproducibility.\u003c/p\u003e","description":"","filename":"Supp1ASCIISchematic.docx","url":"https://assets-eu.researchsquare.com/files/rs-8028195/v1/fb91aed2ba1f8fb7a7d4cf16.docx"},{"id":99371525,"identity":"4d1d43ea-0211-409f-bfe2-72c5a85101ab","added_by":"auto","created_at":"2026-01-02 05:55:28","extension":"docx","order_by":2,"title":"","display":"","copyAsset":false,"role":"supplement","size":22720,"visible":true,"origin":"","legend":"","description":"","filename":"TableI.docx","url":"https://assets-eu.researchsquare.com/files/rs-8028195/v1/8a22502279b32f041b14cb03.docx"}],"financialInterests":"No competing interests reported.","formattedTitle":"A mathematical and computational framework to predict the time to and recovery from osteoporosis via serial DEXA scans: a proof-of-concept model for digital decision support","fulltext":[{"header":"SUMMARY BOX","content":"\u003cp\u003e\u003cstrong\u003eWhat was already known\u003c/strong\u003e\u003c/p\u003e\n\u003cul type=\"disc\"\u003e\n \u003cli\u003eSerial DEXA measurements are difficult to interpret in clinical practice.\u003c/li\u003e\n \u003cli\u003eExisting prediction tools rely on complex or opaque modelling approaches with limited explainability.\u003c/li\u003e\n \u003cli\u003eThe relationship between bone mineral density trajectories and future osteoporosis risk lacks intuitive, patient-facing metrics.\u003c/li\u003e\n\u003c/ul\u003e\n\u003cp\u003e\u003cstrong\u003eWhat this study adds\u003c/strong\u003e\u003c/p\u003e\n\u003cul type=\"disc\"\u003e\n \u003cli\u003eIntroduces a transparent, deterministic algorithm (TTO/TEO) that converts serial DEXA data into age-based threshold predictions.\u003c/li\u003e\n \u003cli\u003eDemonstrates a fully explainable method where slope, intercept, and threshold-crossing age map directly to measurable physiological parameters.\u003c/li\u003e\n \u003cli\u003eProvides a scalable foundation for hybrid, explainable AI systems that incorporate non-linear bone changes while maintaining interpretability.\u003c/li\u003e\n\u003c/ul\u003e\n\u003cp\u003e\u003cu\u003eDECISION MAKING IMPLICATIONS\u003c/u\u003e\u003c/p\u003e\n\u003cp\u003eThe proposed framework translates raw DEXA outputs into clinically intuitive, age-driven predictions that support individualised decision-making. By offering interpretable trajectories rather than isolated Z-scores, it can help clinicians identify high-risk patients earlier, communicate future fracture risk more effectively, and optimise the timing of interventions. Integration of this model into electronic DEXA reporting systems may improve adherence, facilitate shared decision-making, and promote algorithmic transparency across bone health services.\u003c/p\u003e"},{"header":"INTRODUCTION","content":"\u003cp\u003eOsteoporosis is the most common metabolic bone disease and is characterised by impaired bone strength and increased fragility fracture risk\u0026sup1;. In the United States, approximately 10\u0026nbsp;million adults aged\u0026thinsp;\u0026ge;\u0026thinsp;50 years have osteoporosis and an additional 43\u0026nbsp;million are at risk due to low bone mass, placing them at increased risk of fracture\u0026sup2;. With global ageing populations, the burden of osteoporosis and osteoporotic fractures is expected to rise further.\u003c/p\u003e \u003cp\u003eBone mineral density (BMD) testing is central to diagnosis, risk prediction, and monitoring. Among the available methods, dual-energy X-ray absorptiometry (DEXA) of the spine and hip is the accepted diagnostic standard and the most reliable technique for longitudinal assessment\u0026sup3;. Osteoporosis is defined by the World Health Organization (WHO) as a T-score\u0026thinsp;\u0026le;\u0026thinsp;negative 2 point five (-2.5) at the hip or spine\u003csup\u003e\u003cspan citationid=\"CR4\" class=\"CitationRef\"\u003e4\u003c/span\u003e\u003c/sup\u003e. The International Society for Clinical Densitometry (ISCD) likewise recommends use of the NHANES III young-adult Caucasian reference database for hip T-scores across ethnic groups, with sex-specific reference data\u003csup\u003e\u003cspan citationid=\"CR5\" class=\"CitationRef\"\u003e5\u003c/span\u003e\u003c/sup\u003e.\u003c/p\u003e \u003cp\u003eWhile DEXA remains the diagnostic standard for quantifying bone mineral density (BMD), its potential for longitudinal prediction is underexplored. Conventional reporting offers a numerical T-score, a statistical measure of deviation from peak bone mass, but provides no intuitive sense of temporal progression; when a patient is likely to become osteoporotic, or conversely, when improvement is expected under therapy.\u003c/p\u003e \u003cp\u003eExisting tools such as FRAX estimate fracture probability but do not forecast the time to threshold. This temporal gap leaves clinicians without a way to visualise or communicate disease trajectories in age-based terms. The present study introduces a simple algorithmic framework that transforms serial DEXA data into interpretable age-based predictions: time to osteoporosis (TTO) and time to exit osteoporosis (TEO), thereby reframing densitometry into a decision-support metric that aligns with patient understanding and future digital automation.\u003c/p\u003e \u003cp\u003eTo our knowledge, no published model directly converts serial DEXA data into explicit time to threshold predictions using deterministic slope and intercept formulations, and this work therefore offers an initial framework for such trajectory-based estimation\u003c/p\u003e \u003cp\u003e \u003c/p\u003e"},{"header":"METHODS","content":"\u003cdiv id=\"Sec3\" class=\"Section2\"\u003e\n \u003ch2\u003eStudy Design and Data Source\u003c/h2\u003e\n \u003cp\u003eWe performed a retrospective proof-of-concept study using anonymised DEXA data from University Hospital Limerick. Fifty men and women with two or more hip DEXA scans were included, without exclusions for age, comorbidity, prior osteoporosis diagnosis, or treatment status. Scans were performed on a Lunar Prodigy densitometer (GE Medical Systems, USA; software version 13.60.033). BMD was measured at the total hip and lumbar spine (L2\u0026ndash;L4), and T-scores were generated from the manufacturer\u0026rsquo;s reference database of sex-matched healthy young adults.\u003c/p\u003e\n\u003c/div\u003e\n\u003ch3\u003eAlgorithm development\u003c/h3\u003e\n\u003cp\u003eThe model was designed to answer a simple clinical question: Given a patient\u0026rsquo;s current trajectory, at what age will their T-score cross into the osteoporotic range (\u0026le; \u0026minus;\u0026thinsp;2.5)?\u003c/p\u003e\n\u003cp\u003eFor analysis, only hip T-scores were used. At the proximal femur, measurements from the neck, trochanteric, and intertrochanteric regions were combined to yield a total hip value. Patient age and corresponding T-scores were entered into Microsoft Excel, where macros were developed to calculate slope and intercept values. Graphical outputs were generated via GraphPad Prism, and values were cross-checked against hand-written derivations to confirm accuracy.\u003c/p\u003e\n\u003ch3\u003eTwo-point model (TTOc, Fig.\u0026nbsp;)\u003c/h3\u003e\n\u003cp\u003eFor each pair of consecutive scans, a straight line was fitted between the two coordinates (x1,y1) and (x2,y2) where x\u0026thinsp;=\u0026thinsp;age and y\u0026thinsp;=\u0026thinsp;T-score. The slope was calculated as follows:\u003c/p\u003e\n\u003cdiv id=\"Equa\" class=\"Equation\"\u003e\n \u003cdiv class=\"mathdisplay\" id=\"FileID_Equa\" name=\"EquationSource\"\u003e$$\\:m=\\frac{{y}_{2}-{y}_{1}}{{x}_{2}-{x}_{1}}$$\u003c/div\u003e\n\u003c/div\u003e\n\u003cp\u003eThe line equation was then:\u003c/p\u003e\n\u003cdiv id=\"Equb\" class=\"Equation\"\u003e\n \u003cdiv class=\"mathdisplay\" id=\"FileID_Equb\" name=\"EquationSource\"\u003e\u003cimg src=\"data:image/png;base64,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\"\u003e\u003c/div\u003e\n\u003c/div\u003e\n\u003cp\u003eSolving for y\u0026thinsp;=\u0026thinsp;\u0026minus;\u0026thinsp;2.5 yielded the predicted age of entry into osteoporosis:\u003c/p\u003e\n\u003cdiv id=\"Equc\" class=\"Equation\"\u003e\n \u003cdiv class=\"mathdisplay\" id=\"FileID_Equc\" name=\"EquationSource\"\u003e$$\\:x=\\frac{-2.5-C}{m}$$\u003c/div\u003e\n\u003c/div\u003e\n\u003cp\u003eIf the patient was already osteoporotic (y\u0026thinsp;\u0026lt;\u0026thinsp;\u0026minus;\u0026thinsp;2.5) and the slope was positive, the same calculation instead estimated the age at which the trajectory would exit the osteoporotic range (\u003cem\u003etime to exit osteoporosis, TEO\u003c/em\u003e).\u003c/p\u003e\n\u003ch3\u003eNon-generating cases\u003c/h3\u003e\n\u003cp\u003eNo TTO or TEO values were reported when the slope pointed away from the osteoporotic threshold (i.e. improving T-scores in non-osteoporotic patients or worsening scores in already osteoporotic patients), as such predictions would not be clinically meaningful.\u003c/p\u003e\n\u003ch3\u003e\u003cstrong\u003eMultipoint regression model (TTOt, Figure 2a, 2b)\u003c/strong\u003e\u003c/h3\u003e\n\u003cp\u003eFor patients with three or more scans, a least-squares regression line was fitted through all available data points:\u003c/p\u003e\n\u003cp\u003e\u003cimg src=\"data:image/png;base64,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\"\u003e\u003c/p\u003e\n\u003cp\u003eThe cumulative age at which the regression line crossed the osteoporotic threshold was as follows:\u003c/p\u003e\n\u003cdiv id=\"Eque\" class=\"Equation\"\u003e\n \u003cdiv class=\"mathdisplay\" id=\"FileID_Eque\" name=\"EquationSource\"\u003e$$\\:x=\\frac{-2.5-C}{m}$$\u003c/div\u003e\n\u003c/div\u003e\n\u003cp\u003eThis regression approach provides a more stable trajectory by smoothing fluctuations that often occur in individual scan pairs.\u003c/p\u003e\n\u003cp\u003eFull pseudocode for TTOc and TTOt is provided in \u003cstrong\u003eSupplementary File 1\u003c/strong\u003e to facilitate reproducibility.\u003c/p\u003e"},{"header":"RESULTS","content":"\u003cp\u003eThe algorithms successfully generated a predicted age of entry into the osteoporotic range on the basis of serial DEXA trajectories (\u003cstrong\u003eTable 1\u003c/strong\u003e). Both the two-point model (TTOc) and the multipoint regression model (TTOt) produced estimates of \u003cem\u003etime to osteoporosis (TTO)\u003c/em\u003e, whereas the two-point model also identified cases of \u003cem\u003etime to exit osteoporosis (TEO)\u003c/em\u003e in patients showing recovery.\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003eTable 1\u003c/strong\u003e summarises outcomes from patients with two to five scans. In most cases, TTOc and TTOt produced consistent estimates; however, variability between consecutive scans occasionally led to divergence or non-generation of values, particularly when slopes were positive in non-osteoporotic patients or negative in already osteoporotic patients.\u003c/p\u003e\n\u003cp\u003eWorked examples illustrate these patterns. In one subject with five scans over 13 years (Fig. 2a), T-scores declined steadily from \u0026ndash;1.2 to \u0026ndash;3.2. TTOc predicted entry into osteoporosis at 76.9 years from the first two scans and at 71.8 years after the third scan. Once the patient entered the osteoporotic range, no further TTOc was generated; however, improvement at the fifth scan (T-score \u0026ndash;2.7 at age 79) allowed estimation of TEO at 82.5 years. The cumulative TTOt for this patient suggested overall entry into osteoporosis at 74.1 years, closely aligning with the two-point results.\u003c/p\u003e\n\u003cp\u003eIn contrast, a second patient (Fig. 2b) showed wide fluctuations in T-scores across four scans. Here, TTOc values varied markedly (57.4 to 94.0 years) or could not be generated, whereas the regression-based TTOt provided a more stable estimate of 73.6 years, smoothing the oscillations of individual measurements.\u003c/p\u003e"},{"header":"DISCUSSION","content":"\u003cp\u003eThis study showed that simple mathematical models can likely estimate the age at which patients enter or exit the osteoporotic T score range, and that serial DEXA data can be converted into interpretable, time-based indicators of bone health trajectory. The framework aims to bridge quantitative densitometry and clinical reasoning by reframing T scores into predicted ages of onset or recovery. These findings are consistent with the initial hypothesis that trajectory-based modelling may convert numerical bone density data into a more intuitive, patient centred metric.\u003c/p\u003e \u003cp\u003eAdherence to osteoporosis therapy remains a major challenge, and conventional reporting often provides limited guidance. Real world persistence with oral bisphosphonates varies widely, with medication possession ratios that are frequently low\u003csup\u003e\u003cspan citationid=\"CR6\" class=\"CitationRef\"\u003e6\u003c/span\u003e\u003c/sup\u003e. Systematic reviews suggest that multicomponent interventions involving education, counselling, and active patient involvement are more likely to improve adherence than single component approaches\u003csup\u003e\u003cspan citationid=\"CR7\" class=\"CitationRef\"\u003e7\u003c/span\u003e\u003c/sup\u003e. Evidence also indicates that personalised, numeric communication of osteoporosis risk may improve patient understanding and intent to treat when compared with qualitative descriptions\u003csup\u003e\u003cspan citationid=\"CR8\" class=\"CitationRef\"\u003e8\u003c/span\u003e\u003c/sup\u003e. In keeping with findings from the RICO study\u003csup\u003e\u003cspan citationid=\"CR8\" class=\"CitationRef\"\u003e8\u003c/span\u003e\u003c/sup\u003e, our model may help by translating abstract T scores into personalised, age-based predictions that patients can interpret more easily.\u003c/p\u003e \u003cp\u003eBeyond communication, the usability of any predictive method depends on its transparency. Unlike black box models, the deterministic structure of this framework is likely to maintain interpretability. Informatics literature has highlighted that opaque predictive systems may reduce users\u0026rsquo; trust, make validation more difficult, and obscure model failure modes. Rudin has argued that high stakes clinical decisions should rely on inherently interpretable models rather than post hoc methodological explanations, since transparency is essential for safety and auditability\u003csup\u003e\u003cspan citationid=\"CR9\" class=\"CitationRef\"\u003e9\u003c/span\u003e\u003c/sup\u003e. Lipton has similarly noted that parameter level interpretability allows end users to understand and challenge predictions when required\u003csup\u003e\u003cspan citationid=\"CR10\" class=\"CitationRef\"\u003e10\u003c/span\u003e\u003c/sup\u003e. Each of the proposed model parameters, including the slope, intercept, and the estimated threshold crossing age, is derived directly from measured DEXA T-scores, which are validated clinical indicators of bone mineral density.\u003c/p\u003e \u003cp\u003eThe framework also provides a transparent mathematical foundation for expansion. Hybrid systems could incorporate this deterministic layer within supervised learning pipelines. Gradient boosting regression or Gaussian process models may be trained on larger datasets using demographics, comorbidity profiles, and treatment histories as covariates, with TTO values used as interpretable targets. Recurrent neural networks may capture nonlinear BMD patterns while remaining anchored to the baseline logic of the slope-based trajectory. Such systems could enhance precision while remaining aligned with modern standards for explainable artificial intelligence.\u003c/p\u003e \u003cp\u003eIn clinical settings, age-based metrics may also offer practical support for communication and decision making. For example, a seventy-two-year-old individual with a TTO estimate of seventy-seven years could be informed that their current trajectory suggests a likely transition into the osteoporotic range within approximately five years. This type of framing may help clinicians time intervention and reinforce the rationale for follow up. Embedding TTO and TEO values directly into electronic DEXA reports may therefore support more effective shared decision making.\u003c/p\u003e \u003cp\u003eThis study also highlights methodological considerations. The two-point model is sensitive to fluctuations between consecutive scans, which may lead to divergent or non-generating values. The regression-based approach appears to provide a more stable long-term trajectory by smoothing measurement variability. These differences reflect the mathematical characteristics of the algorithms. TTOc derives slope and intercept from two serial measurements, whereas TTOt applies least squares fitting across all available data points. The predicted ages of entry into or exit from osteoporosis are obtained by solving these line equations at the diagnostic threshold of negative 2 point five. The worked examples support the internal consistency of these formulas and suggest that the overall method is valid at the proof-of-concept stage.\u003c/p\u003e\n\u003ch3\u003eAlgorithm Limitations\u003c/h3\u003e\n\u003cp\u003eThis work has several important limitations.\u003c/p\u003e \u003cp\u003e \u003cspan type=\"Underline\" class=\"Underline\" name=\"Emphasis\"\u003e1. Assumption of linearity\u003c/span\u003e \u003c/p\u003e \u003cp\u003eBone loss trajectories are often non-linear, with periods of accelerated decline (menopause) and periods of stabilisation (treatment). The model therefore approximates rather than represents true biology.\u003c/p\u003e \u003cp\u003e \u003cspan type=\"Underline\" class=\"Underline\" name=\"Emphasis\"\u003e2. Absence of confidence intervals\u003c/span\u003e \u003c/p\u003e \u003cp\u003ePredictions are point estimates, and the model does not generate measures of uncertainty.\u003c/p\u003e \u003cp\u003e \u003cspan type=\"Underline\" class=\"Underline\" name=\"Emphasis\"\u003e3. Effects of measurement error\u003c/span\u003e \u003c/p\u003e \u003cp\u003eDEXA reproducibility error (1\u0026ndash;2%) may influence slope estimation, particularly in two-point calculations.\u003c/p\u003e \u003cp\u003e \u003cspan type=\"Underline\" class=\"Underline\" name=\"Emphasis\"\u003e4. Treatment heterogeneity\u003c/span\u003e \u003c/p\u003e \u003cp\u003ePharmacologic therapy was not modelled and may modify trajectories.\u003c/p\u003e \u003cp\u003e \u003cspan type=\"Underline\" class=\"Underline\" name=\"Emphasis\"\u003e5. Non generating patterns\u003c/span\u003e \u003c/p\u003e \u003cp\u003eFor trajectories moving away from the osteoporotic threshold, TTO or TEO cannot be generated.\u003c/p\u003e"},{"header":"Conclusions","content":"\u003cp\u003eDespite these limitations, the proposed framework offers a novel and clinically meaningful method for transforming serial DEXA measurements into interpretable, age-based predictions. Its key strength lies in its transparency: each component of the model (slope, intercept, threshold-crossing age) corresponds directly to measurable properties of bone physiology. This aligns with the goals of explainable artificial intelligence (XAI), in which interpretability, reproducibility, and clarity are prioritised for safe clinical deployment.\u003c/p\u003e \u003cp\u003eThe deterministic structure of the model also provides a scalable foundation for future methodological extensions. As larger datasets become available, this framework can be expanded into hybrid informatics systems that retain interpretability while accommodating non-linear BMD changes and treatment effects. Candidates include Gaussian process regression, Bayesian hierarchical models, and interpretable neural ordinary differential equations, each of which could enhance predictive performance without compromising transparency.\u003c/p\u003e \u003cp\u003eBy converting abstract statistical information into patient-relevant metrics, the model has potential to improve clinical decision-making, strengthen communication around bone health, and support personalised risk assessment. Together, these findings suggest that time-based interpretation of serial DEXA data may offer a practical, transparent, and scalable foundation for future clinical decision support tools in osteoporosis care.\u003c/p\u003e"},{"header":"Abbreviations","content":"\u003cdiv class=\"DefinitionList\"\u003e \u003cdiv class=\"DefinitionListEntry\"\u003e \u003cdiv class=\"Term\"\u003e\u003cb\u003eAI\u003c/b\u003e\u003c/div\u003e \u003cdiv class=\"Description\"\u003e \u003cp\u003eArtificial Intelligence\u003c/p\u003e \u003c/div\u003e \u003c/div\u003e \u003cdiv class=\"DefinitionListEntry\"\u003e \u003cdiv class=\"Term\"\u003e\u003cb\u003eBMD\u003c/b\u003e\u003c/div\u003e \u003cdiv class=\"Description\"\u003e \u003cp\u003eBone Mineral Density\u003c/p\u003e \u003c/div\u003e \u003c/div\u003e \u003cdiv class=\"DefinitionListEntry\"\u003e \u003cdiv class=\"Term\"\u003e\u003cb\u003eC\u003c/b\u003e\u003c/div\u003e \u003cdiv class=\"Description\"\u003e \u003cp\u003eIntercept (linear model)\u003c/p\u003e \u003c/div\u003e \u003c/div\u003e \u003cdiv class=\"DefinitionListEntry\"\u003e \u003cdiv class=\"Term\"\u003e\u003cb\u003eDEXA\u003c/b\u003e\u003c/div\u003e \u003cdiv class=\"Description\"\u003e \u003cp\u003eDual-Energy X-ray Absorptiometry\u003c/p\u003e \u003c/div\u003e \u003c/div\u003e \u003cdiv class=\"DefinitionListEntry\"\u003e \u003cdiv class=\"Term\"\u003e\u003cb\u003eFRAX\u003c/b\u003e\u003c/div\u003e \u003cdiv class=\"Description\"\u003e \u003cp\u003eFracture Risk Assessment Tool\u003c/p\u003e \u003c/div\u003e \u003c/div\u003e \u003cdiv class=\"DefinitionListEntry\"\u003e \u003cdiv class=\"Term\"\u003e\u003cb\u003eFRDP\u003c/b\u003e\u003c/div\u003e \u003cdiv class=\"Description\"\u003e \u003cp\u003eFracture Risk Decision Point\u003c/p\u003e \u003c/div\u003e \u003c/div\u003e \u003cdiv class=\"DefinitionListEntry\"\u003e \u003cdiv class=\"Term\"\u003e\u003cb\u003eGE\u003c/b\u003e\u003c/div\u003e \u003cdiv class=\"Description\"\u003e \u003cp\u003eGeneral Electric Medical Systems\u003c/p\u003e \u003c/div\u003e \u003c/div\u003e \u003cdiv class=\"DefinitionListEntry\"\u003e \u003cdiv class=\"Term\"\u003e\u003cb\u003eHSE\u003c/b\u003e\u003c/div\u003e \u003cdiv class=\"Description\"\u003e \u003cp\u003eHealth Service Executive\u003c/p\u003e \u003c/div\u003e \u003c/div\u003e \u003cdiv class=\"DefinitionListEntry\"\u003e \u003cdiv class=\"Term\"\u003e\u003cb\u003eIOF\u003c/b\u003e\u003c/div\u003e \u003cdiv class=\"Description\"\u003e \u003cp\u003eInternational Osteoporosis Foundation\u003c/p\u003e \u003c/div\u003e \u003c/div\u003e \u003cdiv class=\"DefinitionListEntry\"\u003e \u003cdiv class=\"Term\"\u003e\u003cb\u003eISCD\u003c/b\u003e\u003c/div\u003e \u003cdiv class=\"Description\"\u003e \u003cp\u003eInternational Society for Clinical Densitometry\u003c/p\u003e \u003c/div\u003e \u003c/div\u003e \u003cdiv class=\"DefinitionListEntry\"\u003e \u003cdiv class=\"Term\"\u003e\u003cb\u003eLSTM\u003c/b\u003e\u003c/div\u003e \u003cdiv class=\"Description\"\u003e \u003cp\u003eLong Short-Term Memory\u003c/p\u003e \u003c/div\u003e \u003c/div\u003e \u003cdiv class=\"DefinitionListEntry\"\u003e \u003cdiv class=\"Term\"\u003e\u003cb\u003em\u003c/b\u003e\u003c/div\u003e \u003cdiv class=\"Description\"\u003e \u003cp\u003eSlope (linear model)\u003c/p\u003e \u003c/div\u003e \u003c/div\u003e \u003cdiv class=\"DefinitionListEntry\"\u003e \u003cdiv class=\"Term\"\u003e\u003cb\u003eNHANES\u003c/b\u003e\u003c/div\u003e \u003cdiv class=\"Description\"\u003e \u003cp\u003eNational Health and Nutrition Examination Survey\u003c/p\u003e \u003c/div\u003e \u003c/div\u003e \u003cdiv class=\"DefinitionListEntry\"\u003e \u003cdiv class=\"Term\"\u003e\u003cb\u003eOP\u003c/b\u003e\u003c/div\u003e \u003cdiv class=\"Description\"\u003e \u003cp\u003eOsteoporosis\u003c/p\u003e \u003c/div\u003e \u003c/div\u003e \u003cdiv class=\"DefinitionListEntry\"\u003e \u003cdiv class=\"Term\"\u003e\u003cb\u003eTEO\u003c/b\u003e\u003c/div\u003e \u003cdiv class=\"Description\"\u003e \u003cp\u003eTime to Exit Osteoporosis\u003c/p\u003e \u003c/div\u003e \u003c/div\u003e \u003cdiv class=\"DefinitionListEntry\"\u003e \u003cdiv class=\"Term\"\u003e\u003cb\u003eTEOc\u003c/b\u003e\u003c/div\u003e \u003cdiv class=\"Description\"\u003e \u003cp\u003eTime to Exit Osteoporosis (two-point calculation)\u003c/p\u003e \u003c/div\u003e \u003c/div\u003e \u003cdiv class=\"DefinitionListEntry\"\u003e \u003cdiv class=\"Term\"\u003e\u003cb\u003eT-score\u003c/b\u003e\u003c/div\u003e \u003cdiv class=\"Description\"\u003e \u003cp\u003eStandardised BMD score relative to young-adult reference\u003c/p\u003e \u003c/div\u003e \u003c/div\u003e \u003cdiv class=\"DefinitionListEntry\"\u003e \u003cdiv class=\"Term\"\u003e\u003cb\u003eTTO\u003c/b\u003e\u003c/div\u003e \u003cdiv class=\"Description\"\u003e \u003cp\u003eTime to Osteoporosis\u003c/p\u003e \u003c/div\u003e \u003c/div\u003e \u003cdiv class=\"DefinitionListEntry\"\u003e \u003cdiv class=\"Term\"\u003e\u003cb\u003eTTOc\u003c/b\u003e\u003c/div\u003e \u003cdiv class=\"Description\"\u003e \u003cp\u003eTime to Osteoporosis (two-point calculation)\u003c/p\u003e \u003c/div\u003e \u003c/div\u003e \u003cdiv class=\"DefinitionListEntry\"\u003e \u003cdiv class=\"Term\"\u003e\u003cb\u003eTTOt\u003c/b\u003e\u003c/div\u003e \u003cdiv class=\"Description\"\u003e \u003cp\u003eTime to Osteoporosis (regression-based calculation)\u003c/p\u003e \u003c/div\u003e \u003c/div\u003e \u003cdiv class=\"DefinitionListEntry\"\u003e \u003cdiv class=\"Term\"\u003e\u003cb\u003eUHL\u003c/b\u003e\u003c/div\u003e \u003cdiv class=\"Description\"\u003e \u003cp\u003eUniversity Hospital Limerick\u003c/p\u003e \u003c/div\u003e \u003c/div\u003e \u003cdiv class=\"DefinitionListEntry\"\u003e \u003cdiv class=\"Term\"\u003e\u003cb\u003eUSA\u003c/b\u003e\u003c/div\u003e \u003cdiv class=\"Description\"\u003e \u003cp\u003eUnited States of America\u003c/p\u003e \u003c/div\u003e \u003c/div\u003e \u003cdiv class=\"DefinitionListEntry\"\u003e \u003cdiv class=\"Term\"\u003e\u003cb\u003eWHO\u003c/b\u003e\u003c/div\u003e \u003cdiv class=\"Description\"\u003e \u003cp\u003eWorld Health Organization\u003c/p\u003e \u003c/div\u003e \u003c/div\u003e \u003cdiv class=\"DefinitionListEntry\"\u003e \u003cdiv class=\"Term\"\u003e\u003cb\u003eXAI\u003c/b\u003e\u003c/div\u003e \u003cdiv class=\"Description\"\u003e \u003cp\u003eExplainable Artificial Intelligence\u003c/p\u003e \u003c/div\u003e \u003c/div\u003e \u003c/div\u003e"},{"header":"Declarations","content":"\u003cp\u003e\u003cstrong\u003eEthics approval and consent to participate\u003c/strong\u003e\u003c/p\u003e\n\u003cp\u003eThe study was reviewed by the UL Hospital Group Research Ethics Committee, University Hospital Limerick. The Committee granted an ethics waiver on the basis that the study involved a retrospective analysis of anonymised clinical data and did not require full ethical review. This waiver was confirmed in writing by the Committee Chair, Prof. Colin Peirce, as documented in the approval letter (Supplementary File 2).\u003c/p\u003e\n\u003cp\u003eAll procedures were conducted in compliance with the Declaration of Helsinki, the General Data Protection Regulation (GDPR), the Data Protection Acts, and the Health Research Regulations, as outlined in the ethics correspondence.\u003c/p\u003e\n\u003cp\u003eBecause the dataset consisted solely of anonymised retrospective service data, the requirement for individual informed consent was waived by the UL Hospital Group Research Ethics Committee.\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003eConsent for publication\u003c/strong\u003e\u003c/p\u003e\n\u003cp\u003eNot applicable. The study uses anonymised retrospective data and contains no identifiable information. The UL Hospital Group Research Ethics Committee confirmed that consent for publication was not required.\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003eAvailability of data and materials\u003c/strong\u003e\u003c/p\u003e\n\u003cp\u003eThe clinical dataset analysed during this study is not publicly available owing to GDPR and HSE data-governance restrictions. De-identified data can be accessed upon reasonable request, contingent on approval from the UL Hospitals Group Data Controller and adherence to GDPR compliant safeguards.\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003eCompeting interests\u003c/strong\u003e\u003c/p\u003e\n\u003cp\u003eThe authors declare that they have no competing interests relevant to this work, including no commercial or proprietary involvement in the development or deployment of the algorithms described.\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003eFunding\u003c/strong\u003e\u003c/p\u003e\n\u003cp\u003eThis research received no specific grant from any funding agency in the public, commercial, or not-for-profit sectors.\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003eAuthors’ contribution\u003c/strong\u003e\u003c/p\u003e\n\u003cp\u003ePS conceived the study and designed the algorithm, AM wrote the manuscript and interpreted data, TS and AC involved in collecting and maintaining data, NA assisted with data curation and reference management, DL supervised and provided governance for the study.\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003eAcknowledgements\u003c/strong\u003e\u003c/p\u003e\n\u003cp\u003eThe authors gratefully acknowledge the staff and nurses of the Clinical Age Assessment Unit, University Hospital Limerick, whose meticulous collection and maintenance of the DEXA dataset made this work possible.\u003c/p\u003e"},{"header":"References","content":"\u003col\u003e\n \u003cli\u003eOsteoporosis prevention, diagnosis, and therapy. JAMA. 2001;285:785\u0026ndash;795.\u003c/li\u003e\n \u003cli\u003eLooker AC, Frenk SM. \u003cem\u003eOsteoporosis and low bone mass among adults aged 50 and over: United States, 2017\u0026ndash;2018.\u003c/em\u003e NCHS Data Brief, no. 405. Hyattsville, MD: National Center for Health Statistics; 2021.\u0026nbsp;\u003c/li\u003e\n \u003cli\u003eKrugh M, Langaker MD. \u003cem\u003eDual-Energy X-Ray Absorptiometry\u003c/em\u003e. In: StatPearls [Internet]. Treasure Island (FL): StatPearls Publishing; 2024 May 20.\u003c/li\u003e\n \u003cli\u003eKanis JA, Cooper C, Rizzoli R, Reginster JY; on behalf of ESCEO and IOF. European guidance for the diagnosis and management of osteoporosis in postmenopausal women. \u003cem\u003eOsteoporosis Int\u003c/em\u003e. 2019;30(1):3\u0026ndash;44.\u003c/li\u003e\n \u003cli\u003eInternational Society for Clinical Densitometry. \u003cem\u003eISCD Official Positions 2023\u003c/em\u003e. Middletown, CT: ISCD; 2023.\u003c/li\u003e\n \u003cli\u003eFatoye F, Garcia-Tormo G, Gebrye T, et al. Real-world persistence and adherence with oral bisphosphonates in patients treated for osteoporosis: a retrospective cohort study. \u003cem\u003eBMJ Open\u003c/em\u003e. 2019;9(4):e027049\u003c/li\u003e\n \u003cli\u003eCornelissen D, de Kunder S, Si L, et al. Interventions to improve adherence to anti-osteoporosis medications: an updated systematic review. \u003cem\u003eOsteoporosis Int\u003c/em\u003e. 2020;31(9):1645\u0026ndash;1669.\u003c/li\u003e\n \u003cli\u003eSharma M, Beaudart C, Clark P, et al. Clinical and demographic factors determining patient fracture‑risk decision point (FRDP): the improving risk communication in osteoporosis (RICO) project. \u003cem\u003eOsteoporosis Int\u003c/em\u003e. 2025;36:71\u0026ndash;80. doi:10.1007/s00198-024-07264-5\u003c/li\u003e\n \u003cli\u003eRudin C. Stop Explaining Black Box Machine Learning Models for High Stakes Decisions and Use Interpretable Models Instead. Nat Mach Intell. 2019 May;1(5):206-215. doi: 10.1038/s42256-019-0048-x. Epub 2019 May 13. PMID: 35603010; PMCID: PMC9122117.\u003c/li\u003e\n \u003cli\u003eLipton ZC. The mythos of model interpretability. Commun ACM. 2018;61(10):36\u0026ndash;43.\u003c/li\u003e\n\u003c/ol\u003e"},{"header":"Table","content":"\u003cp\u003eTable 1 is available in the supplementary files section\u003c/p\u003e"}],"fulltextSource":"","fullText":"","funders":[],"hasAdminPriorityOnWorkflow":false,"hasManuscriptDocX":true,"hasOptedInToPreprint":true,"hasPassedJournalQc":"","hasAnyPriority":false,"hideJournal":false,"highlight":"","institution":"","isAcceptedByJournal":true,"isAuthorSuppliedPdf":false,"isDeskRejected":"","isHiddenFromSearch":false,"isInQc":false,"isInWorkflow":false,"isPdf":false,"isPdfUpToDate":true,"isWithdrawnOrRetracted":false,"journal":{"display":true,"email":"[email protected]","identity":"bmc-medical-informatics-and-decision-making","isNatureJournal":false,"hasQc":true,"allowDirectSubmit":false,"externalIdentity":"midm","sideBox":"Learn more about [BMC Medical Informatics and Decision Making](http://bmcmedinformdecismak.biomedcentral.com/)","snPcode":"","submissionUrl":"https://www.editorialmanager.com/midm/default.aspx","title":"BMC Medical Informatics and Decision Making","twitterHandle":"BMC_series","acdcEnabled":true,"dfaEnabled":false,"editorialSystem":"em","reportingPortfolio":"BMC Series","inReviewEnabled":true,"inReviewRevisionsEnabled":true},"keywords":"DEXA, Bone Mineral Density, T-score, Predictive Modelling, Decision Support, Machine Learning","lastPublishedDoi":"10.21203/rs.3.rs-8028195/v1","lastPublishedDoiUrl":"https://doi.org/10.21203/rs.3.rs-8028195/v1","license":{"name":"CC BY 4.0","url":"https://creativecommons.org/licenses/by/4.0/"},"manuscriptAbstract":"\u003cp\u003e\u003cstrong\u003eBackground\u003c/strong\u003e\u003cbr\u003e\nDual energy X ray absorptiometry (DEXA) is the diagnostic standard for osteoporosis, yet its serial data remain underutilised in predictive analytics. To our knowledge, no published model provides explicit age based predictions of osteoporosis onset or recovery using serial DEXA T score trajectories. This study describes a mathematical framework for predicting time to osteoporosis (TTO), defined as the age at which a patient’s T score trajectory reaches negative 2 point five.\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003eMethods\u003c/strong\u003e\u003c/p\u003e\n\u003cp\u003eWe developed a mathematical framework that converts serial DEXA T-scores into \u003cem\u003etime-to-osteoporosis (TTO)\u003c/em\u003e and \u003cem\u003etime-to-exit osteoporosis (TEO)\u003c/em\u003e predictions. Using hip DEXA results from 50 consecutive patients with ≥2 scans, T-scores were plotted against age, and we applied two models: (a) a two-point slope algorithm (TTOc) and (b) a multipoint least-squares regression (TTOt). Both were designed to estimate the age at which the T-score trajectory would cross the diagnostic threshold of negative 2 point five.\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003eResults\u003c/strong\u003e\u003c/p\u003e\n\u003cp\u003eBoth algorithms successfully predict the age of entry into, or recovery from, osteoporosis. TTOc produced scan-pair–specific short-term projections, whereas TTOt provided smoothed cumulative trajectories. Worked examples demonstrated agreement between models in patients with monotonic decline and highlighted the stabilising effect of regression in fluctuating cases.\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003eDiscussion\u003c/strong\u003e\u003cbr\u003e\nThis framework transforms static DEXA outputs into patient specific, age-based predictions, enhancing clinical interpretability. In addition to their immediate clinical use, deterministic equations can serve as the foundation for a hybrid machine-learning model that uses slope and intercept values as interpretable features within ensemble or deep learning architectures to improve temporal prediction accuracy across larger datasets.\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003eConclusions\u003c/strong\u003e\u003c/p\u003e\n\u003cp\u003eThis proof-of-concept demonstrates the feasibility of trajectory-based modelling for osteoporosis risk prediction. It represents a step toward AI-assisted DEXA interpretation systems that are transparent, explainable, and directly usable at the point of care.\u003cbr\u003e\nIt also reframes DEXA outputs into an age-based measure that patients easily understand, offering clinicians a simplified parameter for monitoring therapy. The incorporation of these findings into DEXA reporting could strengthen patient engagement and adherence.\u003c/p\u003e","manuscriptTitle":"A mathematical and computational framework to predict the time to and recovery from osteoporosis via serial DEXA scans: a proof-of-concept model for digital decision support","msid":"","msnumber":"","nonDraftVersions":[{"code":1,"date":"2026-01-02 05:55:23","doi":"10.21203/rs.3.rs-8028195/v1","editorialEvents":[{"type":"communityComments","content":0},{"type":"decision","content":"Revision requested","date":"2026-04-17T06:34:04+00:00","index":"","fulltext":""},{"type":"editorInvitedReview","content":"","date":"2026-04-14T12:02:44+00:00","index":"hide","fulltext":""},{"type":"editorInvitedReview","content":"","date":"2026-04-12T01:44:59+00:00","index":"hide","fulltext":""},{"type":"reviewerAgreed","content":"112468950760462758768474275190070396645","date":"2026-04-09T17:54:29+00:00","index":"hide","fulltext":""},{"type":"reviewerAgreed","content":"52552632325999842538709371467153730308","date":"2026-04-09T11:56:26+00:00","index":"hide","fulltext":""},{"type":"reviewersInvited","content":"","date":"2025-12-30T09:36:26+00:00","index":"","fulltext":""},{"type":"editorAssigned","content":"","date":"2025-11-21T11:06:48+00:00","index":"","fulltext":""},{"type":"checksComplete","content":"","date":"2025-11-20T10:36:39+00:00","index":"","fulltext":""},{"type":"submitted","content":"BMC Medical Informatics and Decision Making","date":"2025-11-20T10:33:57+00:00","index":"","fulltext":""}],"status":"published","journal":{"display":true,"email":"[email protected]","identity":"bmc-medical-informatics-and-decision-making","isNatureJournal":false,"hasQc":true,"allowDirectSubmit":false,"externalIdentity":"midm","sideBox":"Learn more about [BMC Medical Informatics and Decision Making](http://bmcmedinformdecismak.biomedcentral.com/)","snPcode":"","submissionUrl":"https://www.editorialmanager.com/midm/default.aspx","title":"BMC Medical Informatics and Decision Making","twitterHandle":"BMC_series","acdcEnabled":true,"dfaEnabled":false,"editorialSystem":"em","reportingPortfolio":"BMC Series","inReviewEnabled":true,"inReviewRevisionsEnabled":true}}],"origin":"","ownerIdentity":"7a8931bf-2a52-4348-b530-0c96b2cca5fe","owner":[],"postedDate":"January 2nd, 2026","published":true,"recentEditorialEvents":[],"rejectedJournal":[],"revision":"","amendment":"","status":"under-review","subjectAreas":[],"tags":[],"updatedAt":"2026-05-08T09:39:42+00:00","versionOfRecord":[],"versionCreatedAt":"2026-01-02 05:55:23","video":"","vorDoi":"","vorDoiUrl":"","workflowStages":[]},"version":"v1","identity":"rs-8028195","journalConfig":"researchsquare"},"__N_SSP":true},"page":"/article/[identity]/[[...version]]","query":{"redirect":"/article/rs-8028195","identity":"rs-8028195","version":["v1"]},"buildId":"XKTyCvWXoU3ODBz1xrDgd","isFallback":false,"isExperimentalCompile":false,"dynamicIds":[84888],"gssp":true,"scriptLoader":[]}