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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.

References

  1. Anonymous (2001) Osteoporosis prevention, diagnosis, and therapy. JAMA 285:785–95

    Google Scholar 

  2. Blake GM, Fogelman I (2007) Role of dual-energy X-ray absorptiometry in the diagnosis and treatment of osteoporosis. J Clin Densitom 10(1):102–110

    Article  PubMed  Google Scholar 

  3. Boehm HF, Horng A, Notohamiprodjo M, Eckstein F, Burklein D, Panteleon A, Lutz J, Reiser M (2008) Prediction of the fracture load of whole proximal femur specimens by topological analysis of the mineral distribution in DXA-scan images. Bone 43(5):826–831

    Article  PubMed  Google Scholar 

  4. Bonnick SL (2004) Bone densitometry in clinical practice. Humana Press, Totowa, NJ

    Google Scholar 

  5. Boonen S, Bischoff-Ferrari HA, Cooper C, Lips P, Ljunggren O, Meunier PJ, Reginster JY (2006) Addressing the musculoskeletal components of fracture risk with calcium and vitamin D: a review of the evidence. Calcif Tissue Int 78(5):257–270

    Article  CAS  PubMed  Google Scholar 

  6. Boutroy S, Bouxsein ML, Munoz F, Delmas PD (2005) In vivo assessment of trabecular bone microarchitecture by high-resolution peripheral quantitative computed tomography. J Clin Endocrinol Metab 90:6508–6515

    Article  CAS  PubMed  Google Scholar 

  7. Bouxsein ML, Coan BS, Lee SC (1999) Prediction of the strength of the elderly proximal femur by bone mineral density and quantitative ultrasound measurements of the heel and tibia. Bone 25(1):49–54

    Article  CAS  PubMed  Google Scholar 

  8. Bouxsein ML, Szule P, Munoz F, Thrall E, Sornay-Rendu E, Delmas PD (2007) Contribution of trochanteric soft tissues to force fall estimates, the factor of risk, and prediction of hip fracture risk. J Bone Miner Res 22(6):825–831

    Article  PubMed  Google Scholar 

  9. Breiman L (1973) Statistics with a view towards applications. Houghton Mifflin Company, Boston, MA

    Google Scholar 

  10. Carter DR, Hayes WC (1977) The compressive behavior of bone as a two-phase porous structure. J Bone Joint Surg Am 59:954–962

    CAS  PubMed  Google Scholar 

  11. Center JR, Bliue D, Nguyen TV, Eisman JA (2007) Risk of subsequent fracture after low-trauma fracture in men and women. JAMA 297(4):387–394

    Article  CAS  PubMed  Google Scholar 

  12. Centers for Disease Control and Prevention, U.S. Department Of Health and Human Services, NHANES - National Health and Nutrition Examination Survey Web site. Available at: http://www.cdc.gov/nchs/nhanes.htm. Accessed February 26, 2009

  13. Cheng XG, Lowet G, Boonen S, Nicholson PHF, Brys P, Nijs J, Dequeker J (1997) Assessment of the strength of proximal femur in vitro: relationship to femoral bone mineral density and femoral geometry. Bone 20(3):213–218

    Article  CAS  PubMed  Google Scholar 

  14. Cooper C, Aihie A (1995) Osteoporosis. Baillière’s Clin Rheu 9(3):555–564

  15. Cox DR, Isham V (1980) Point processes. CRC Press, Boca Raton, FL

    Google Scholar 

  16. Cummings SR, Nevitt MC, Browner WS, Stone K, Fox KM, Ensrud KE, Cauley J, Black D, Vogt TM (1995) Risk factors for hip fracture in white women. New Engl J Med 332(12):767–774

    Article  CAS  PubMed  Google Scholar 

  17. Cummings SR, Karpf DB, Harris F et al (2002) Improvement in spine bone density and reduction in risk of vertebral fractures during treatment with antiresorptive drugs. Am J Med 112:281–289

    Article  CAS  PubMed  Google Scholar 

  18. Dargent-Molina P, Favier F, Grandjean H, Baudoin C, Schott AM, Hausherr E, Meunier PJ, Breart G, for EPIDOS Group (1996) Fall-related factors and risk of hip fracture: the EPIDOS prospective study. Lancet 348:145–149

    Article  CAS  PubMed  Google Scholar 

  19. Eastell R, Barton I, Hannon RA, Chines A, Garnero P, Delmas PD (2003) Relationship of early changes in bone resorption to the reduction in fracture risk with risendronate. J Bone Miner Res 18:1051–1056

    Article  CAS  PubMed  Google Scholar 

  20. Gregg EW, Pereira MA, Caspersen CJ (2000) Physical activity, falls, and fractures among older adults: a review of the epidemiologic evidence. J Am Geriatr Soc 48(8):883–893

    CAS  PubMed  Google Scholar 

  21. Gullberg B, Johnell O, Kanis JA (1997) World-wide projections for hip fracture. Osteoporos Int 7:407

    Article  CAS  PubMed  Google Scholar 

  22. Hayes WC, Piazza SJ, Zysset PK (1991) Biomechanics of fracture risk prediction of the hip and spine by quantitative computed tomography. Radiol Clin North Am 29:1–18

    CAS  PubMed  Google Scholar 

  23. Hosmer DW Jr, Lemeshow S (1989) Applied logistic regression. John Wiley, New York

    Google Scholar 

  24. Hui SL, Slemenda CW, Johnston CC Jr (1988) Age and bone mass as predictors of fracture in a prospective study. J Clin Invest 81:1804–1809

    Article  CAS  PubMed  Google Scholar 

  25. International Osteoporosis Foundation. 2009 Facts and statistics about osteoporosis and its impact. International Osteoporosis Foundation Web site. Available at: http://www.iofbonehealth.org/facts-and-statistics.html. Accessed February 24, 2009

  26. Johnell O, Kanis JA, Oden A, Sernbo I, Redlund-Johnell I, Petterson C, De Laet C, Jonsson B (2004) Fracture risk following an osteoporotic fracture. Osteoporos Int 15:175–179

    Article  CAS  PubMed  Google Scholar 

  27. Johnell O, Kanis JA, Oden A, Johansson H, De Laet C, Delmas P, Eisman JA, Fujiwara S, Kroger H, Mellstrom D, Meunier PJ, Melton LJ III, O’Neill T, Pols H, Reeve J, Silman A, Tenenhouse A (2005) Predictive value of BMD for hip and other fractures. J Bone Miner Res 20:1185–1194

    Article  PubMed  Google Scholar 

  28. Kanis JA (2002) Diagnosis of osteoporosis and assessment of fracture risk. Lancet 359(9321):1929–1936

    Article  PubMed  Google Scholar 

  29. Kanis JA, Johnell O, De Laet C, Johansson H, Oden A, Delmas P et al (2004) A meta-analysis of previous fracture and subsequent fracture risk. Bone 35:375–382

    Article  CAS  PubMed  Google Scholar 

  30. Kanis JA, Johnell O, Oden A, Johansson H, McCloskey E (2008) FRAX™ and the assessment of fracture probability in men and women from the UK. Osteoporos Int 19:385–397

    Article  CAS  PubMed  Google Scholar 

  31. Kannus P, Sievanen H, Palvanen M, Jarvinen T, Parkkari J (2005) Prevention of falls and consequent injuries in elderly people. Lancet 366:1885–1893

    Article  PubMed  Google Scholar 

  32. Kaptoge S, Benevolenskaya LI, Bhalla AK, Cannata JB, Boonen S, Flach JA et al (2005) Low BMD is less predictive than reported falls for future limb fractures in women across Europe: results from the European Prospective Osteoporosis Study. Bone 36:387–398

    Article  CAS  PubMed  Google Scholar 

  33. Kaufman JJ, Siffert RS (2001) Non-invasive assessment of bone integrity. In: Cowin S (ed) Bone mechanics handbook. CRC Press, Boca Raton, FL, pp 34.1–34.25

    Google Scholar 

  34. Keaveny TM, Bouxsein ML (2008) Theoretical implications of the biomechanical fracture threshold (Perspective). J Bone Miner Res 23(10):1541–1547

    Article  PubMed  Google Scholar 

  35. Keaveny TM, Morgan EF, Niebur GL, Yeh OC (2001) Biomechanics of trabecular bone. Annu Rev Biomed Eng 3:307–333

    Article  CAS  PubMed  Google Scholar 

  36. Klotzbuecher CM, Ross PD, Landsman PB, Abbott TA III, Berger M (2000) Patients with prior fractures have an increased risk of future fractures: a summary of the literature and statistical synthesis. J Bone Miner Res 15(4):721–739

    Article  CAS  PubMed  Google Scholar 

  37. Lawless JF (1982) Statistical models and methods for lifetime data. Wiley, New York, NY

    Google Scholar 

  38. Lester G (2005) Bone quality: summary of NIH/ASBMR meeting. J Musculoskelet Neuronal Interact 5:309

    CAS  PubMed  Google Scholar 

  39. Lin JT, Lane JM (2004) Osteoporosis: a review. Clin Orthop Relat Res 425:126–134

    Article  PubMed  Google Scholar 

  40. Lochmuller EM, Miller P, Burklein D, Wehr U, Rambeck W, Eckstein F (2000) In situ femoral dual-energy X-ray absorptiometry related to ash weight, bone size and density, and its relationship with mechanical failure loads of the proximal femur. Osteoporos Int 11:361–367

    Article  CAS  PubMed  Google Scholar 

  41. Looker AC, Orwoll ES, Conrad Johnston C Jr, Lindsay RL, Wahner HW, Dunn WL, Calvo MS, Harris TB, Heyse SB (1997) Prevalence of low femoral bone density in older U.S. adults from NHANES III. J Bone Miner Res 12:1761–1768

    Article  CAS  PubMed  Google Scholar 

  42. Mc Donnell P, Mc Hugh PE, O’Mahoney D (2007) Vertebral osteoporosis and trabecular bone quality. Ann Biomed Eng 35(2):170–189

    Article  CAS  Google Scholar 

  43. McCreadie BR, Goldstein SA (2000) Biomechanics of fracture: is bone mineral density sufficient to assess risk? J Bone Miner Res 15(12):2305–2308

    Article  CAS  PubMed  Google Scholar 

  44. Melton LJ III (1988) Epidemiology of fractures. In: Riggs BL, Melton LJ III (eds) Osteoporosis: etiology, diagnosis, and management. Raven Press, New York, NY, pp 133–154

    Google Scholar 

  45. Miller CW (1978) Survival and ambulation following hip fracture. J Bone Joint Surg 60A:930–934

    Google Scholar 

  46. Mosteller F (1952) The world series competition. J Am Stat Assoc 47(259):355–380

    Article  Google Scholar 

  47. Orwoll ES, Marshall LM, Nielson CM, Cummings SR, Lapidus J, Cauley JA, Ensrud K, Lane N, Hoffman PR, Kopperdahl DL, Keaveny TM, for the Osteoporotic Fractures in Men Study Group (2009) Finite element analysis of the proximal femur and hip fracture risk in older men. J Bone Miner Res 24(3):475–483

    Article  PubMed  Google Scholar 

  48. Papoulis A, Pillai SU (2002) Probability, random variables and stochastic processes, 4th edn. McGraw Hill, New York, NY

    Google Scholar 

  49. Pinilla TP, Boardman KC, Bouxsein ML, Myers ER, Hayes WC (1996) Impact direction from a fall influences the failure load of the proximal femur as much as age-related bone loss. Calcif Tissue Int 58:231–235

    CAS  PubMed  Google Scholar 

  50. Reginster J-Y, Burlet N (2006) Osteoporosis: a still increasing prevalence. Bone 38(2 Supp1):4–9

    Article  Google Scholar 

  51. Rice JC, Cowin SC, Bowman JA (1988) On the dependence of the elasticity and strength of cancellous bone on apparent density. J Biomech 21(2):155–168

    Article  CAS  PubMed  Google Scholar 

  52. Riggs BL, Melton LJ III (2002) Bone turnover matters: the raloxifene treatment paradox of dramatic decreases in vertebral fractures without commensurate increases in bone density. J Bone Miner Res 17:11–14

    Article  PubMed  Google Scholar 

  53. Robbins JA, Schott AM, Garnero P, Delmas PD, Hans D, Meunier PJ (2005) Risk factors for hip fracture in women with high BMD: EPIDOS study. Osteoporos Int 16:149–154

    Article  CAS  PubMed  Google Scholar 

  54. Robinovich SN, Hayes WC, McMahon TA (1991) Prediction of femoral impact forces in falls on the hip. Trans ASME 113:366–374

    Google Scholar 

  55. Ross S (1995) Stochastic processes, 22nd edn. Wiley, New York

    Google Scholar 

  56. Ross SM (2006) Simulation, 4th edn. Elsevier Science and Technology Books, Amsterdam, The Netherlands

    Google Scholar 

  57. Roux C, Bischoff-Ferrari HA, Papapoulos SE, de Papp AE, West JA, Bouillon R (2008) New insights into the role of vitamin D and calcium in osteoporosis management: an expert roundtable discussion. Curr Med Res Opin 24(5):1363–1370

    Article  CAS  PubMed  Google Scholar 

  58. Schuit SC, van der Klift M, Weel AE, de Laet CE, Burger H, Seeman E, Hofman A, Uitterlinden AG, van Leeuwen JP, Pols HA (2004) Fracture incidence and association with bone mineral density in elderly men and women: the Rotterdam Study. Bone 34:195–202

    Article  CAS  PubMed  Google Scholar 

  59. Seeman E (2008) Bone quality: the material and structural basis of bone strength. J Bone Miner Metab 26(1):1–8

    Article  PubMed  Google Scholar 

  60. Siffert RS, Kaufman JJ (2007) Ultrasonic bone assessment: “the time has come”. Bone 40(1):5–8

    Article  PubMed  Google Scholar 

  61. Siffert RS, Luo GM, Cowin SC, Kaufman JJ (1996) Dynamic relationships of trabecular bone density, architecture, and strength in a computational model of osteopenia. Bone 18(2):197–206

    Article  CAS  PubMed  Google Scholar 

  62. Silva MJ (2007) Biomechanics of osteoporotic fractures. Injury 38(Suppl 3):S69–S76

    Article  PubMed  Google Scholar 

  63. Snyder DL, Miller MI (1975) Random point processes. Wiley, New York, NY

    Google Scholar 

  64. Turner C (2002) Biomechanics of bone: determinants of skeletal fragility and bone quality. Osteoporos Int 13(2):97–104

    Article  CAS  PubMed  Google Scholar 

  65. van den Kroonenberg A, Hayes W, McMahon TA (1996) Hip impact velocities and body configurations for experimental falls from standing height. J Biomech 29:807–811

    Article  PubMed  Google Scholar 

  66. Van Trees HL (1968) Detection, estimation, and modulation theory part I. Wiley, New York

    Google Scholar 

  67. Yang L, Peel N, Clowes JA, McCloskey EV, Eastell R (2009) Use of DXA-based structural engineering models of the proximal femur to discriminate hip fracture. J Bone Miner Res 24(1):33–42

    Article  PubMed  Google Scholar 

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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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