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A poisson process model for hip fracture risk

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Abstract

The primary method for assessing fracture risk in osteoporosis relies primarily on measurement of bone mass. Estimation of fracture risk is most often evaluated using logistic or proportional hazards models. Notwithstanding the success of these models, there is still much uncertainty as to who will or will not suffer a fracture. This has led to a search for other components besides mass that affect bone strength. The purpose of this paper is to introduce a new mechanistic stochastic model that characterizes the risk of hip fracture in an individual. A Poisson process is used to model the occurrence of falls, which are assumed to occur at a rate, λ. The load induced by a fall is assumed to be a random variable that has a Weibull probability distribution. The combination of falls together with loads leads to a compound Poisson process. By retaining only those occurrences of the compound Poisson process that result in a hip fracture, a thinned Poisson process is defined that itself is a Poisson process. The fall rate is modeled as an affine function of age, and hip strength is modeled as a power law function of bone mineral density (BMD). The risk of hip fracture can then be computed as a function of age and BMD. By extending the analysis to a Bayesian framework, the conditional densities of BMD given a prior fracture and no prior fracture can be computed and shown to be consistent with clinical observations. In addition, the conditional probabilities of fracture given a prior fracture and no prior fracture can also be computed, and also demonstrate results similar to clinical data. The model elucidates the fact that the hip fracture process is inherently random and improvements in hip strength estimation over and above that provided by BMD operate in a highly “noisy” environment and may therefore have little ability to impact clinical practice.

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Notes

  1. Indeed age is a proxy for several physiologic factors such as vision, stability, impaired mental status, and frailty, which together with environmental factors (e.g., steps, carpets, cords, ice) should to as much an extent as possible be factored into the evaluation of an individual’s fall rate, λ.

  2. The remainder of this paper will refer only to the occurrence of no fracture or at least one fracture in a time interval (0,T], because of the relative simple forms of the expressions in (4) and (5). However, because of the relative rarity of a fracture event, the probability of at least one fracture will generally be close to the probability of exactly one fracture, in a given time interval. In any event, the exact expressions can easily be substituted if preferred.

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Authors and Affiliations

  1. CyberLogic, Inc., 611 Broadway, Suite 707, New York, NY, 10012, USA

    Zvi Schechner, Gangming Luo & Jonathan J. Kaufman

  2. VA New York Harbor HealthCare System, New York, NY, USA

    Gangming Luo

  3. Department of Rehabilitation Medicine, New York University School of Medicine, New York, NY, USA

    Gangming Luo

  4. Department of Orthopedics, The Mount Sinai School of Medicine, New York, NY, USA

    Jonathan J. Kaufman & Robert S. Siffert

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  1. Zvi Schechner
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Correspondence to Jonathan J. Kaufman.

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Schechner, Z., Luo, G., Kaufman, J.J. et al. A poisson process model for hip fracture risk. Med Biol Eng Comput 48, 799–810 (2010). https://doi.org/10.1007/s11517-010-0638-6

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  • DOI: https://doi.org/10.1007/s11517-010-0638-6

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