Medical & Biological Engineering & Computing

, Volume 48, Issue 8, pp 799–810 | Cite as

A poisson process model for hip fracture risk

  • Zvi Schechner
  • Gangming Luo
  • Jonathan J. Kaufman
  • Robert S. Siffert
Original Article

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.

Keywords

Fracture risk Poisson process Conditional probability Bayesian analysis Fall rate BMD DXA 

References

  1. 1.
    Anonymous (2001) Osteoporosis prevention, diagnosis, and therapy. JAMA 285:785–95Google Scholar
  2. 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):102–110CrossRefPubMedGoogle Scholar
  3. 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):826–831CrossRefPubMedGoogle Scholar
  4. 4.
    Bonnick SL (2004) Bone densitometry in clinical practice. Humana Press, Totowa, NJGoogle Scholar
  5. 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):257–270CrossRefPubMedGoogle Scholar
  6. 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 90:6508–6515CrossRefPubMedGoogle Scholar
  7. 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):49–54CrossRefPubMedGoogle Scholar
  8. 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):825–831CrossRefPubMedGoogle Scholar
  9. 9.
    Breiman L (1973) Statistics with a view towards applications. Houghton Mifflin Company, Boston, MAGoogle Scholar
  10. 10.
    Carter DR, Hayes WC (1977) The compressive behavior of bone as a two-phase porous structure. J Bone Joint Surg Am 59:954–962PubMedGoogle Scholar
  11. 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):387–394CrossRefPubMedGoogle Scholar
  12. 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. 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):213–218CrossRefPubMedGoogle Scholar
  14. 14.
    Cooper C, Aihie A (1995) Osteoporosis. Baillière’s Clin Rheu 9(3):555–564Google Scholar
  15. 15.
    Cox DR, Isham V (1980) Point processes. CRC Press, Boca Raton, FLGoogle Scholar
  16. 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):767–774CrossRefPubMedGoogle Scholar
  17. 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 112:281–289CrossRefPubMedGoogle Scholar
  18. 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 348:145–149CrossRefPubMedGoogle Scholar
  19. 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 18:1051–1056CrossRefPubMedGoogle Scholar
  20. 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):883–893PubMedGoogle Scholar
  21. 21.
    Gullberg B, Johnell O, Kanis JA (1997) World-wide projections for hip fracture. Osteoporos Int 7:407CrossRefPubMedGoogle Scholar
  22. 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 29:1–18PubMedGoogle Scholar
  23. 23.
    Hosmer DW Jr, Lemeshow S (1989) Applied logistic regression. John Wiley, New YorkGoogle Scholar
  24. 24.
    Hui SL, Slemenda CW, Johnston CC Jr (1988) Age and bone mass as predictors of fracture in a prospective study. J Clin Invest 81:1804–1809CrossRefPubMedGoogle Scholar
  25. 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. 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 15:175–179CrossRefPubMedGoogle Scholar
  27. 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 20:1185–1194CrossRefPubMedGoogle Scholar
  28. 28.
    Kanis JA (2002) Diagnosis of osteoporosis and assessment of fracture risk. Lancet 359(9321):1929–1936CrossRefPubMedGoogle Scholar
  29. 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 35:375–382CrossRefPubMedGoogle Scholar
  30. 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 19:385–397CrossRefPubMedGoogle Scholar
  31. 31.
    Kannus P, Sievanen H, Palvanen M, Jarvinen T, Parkkari J (2005) Prevention of falls and consequent injuries in elderly people. Lancet 366:1885–1893CrossRefPubMedGoogle Scholar
  32. 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 36:387–398CrossRefPubMedGoogle Scholar
  33. 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 34.1–34.25Google Scholar
  34. 34.
    Keaveny TM, Bouxsein ML (2008) Theoretical implications of the biomechanical fracture threshold (Perspective). J Bone Miner Res 23(10):1541–1547CrossRefPubMedGoogle Scholar
  35. 35.
    Keaveny TM, Morgan EF, Niebur GL, Yeh OC (2001) Biomechanics of trabecular bone. Annu Rev Biomed Eng 3:307–333CrossRefPubMedGoogle Scholar
  36. 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):721–739CrossRefPubMedGoogle Scholar
  37. 37.
    Lawless JF (1982) Statistical models and methods for lifetime data. Wiley, New York, NYGoogle Scholar
  38. 38.
    Lester G (2005) Bone quality: summary of NIH/ASBMR meeting. J Musculoskelet Neuronal Interact 5:309PubMedGoogle Scholar
  39. 39.
    Lin JT, Lane JM (2004) Osteoporosis: a review. Clin Orthop Relat Res 425:126–134CrossRefPubMedGoogle Scholar
  40. 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 11:361–367CrossRefPubMedGoogle Scholar
  41. 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 12:1761–1768CrossRefPubMedGoogle Scholar
  42. 42.
    Mc Donnell P, Mc Hugh PE, O’Mahoney D (2007) Vertebral osteoporosis and trabecular bone quality. Ann Biomed Eng 35(2):170–189CrossRefGoogle Scholar
  43. 43.
    McCreadie BR, Goldstein SA (2000) Biomechanics of fracture: is bone mineral density sufficient to assess risk? J Bone Miner Res 15(12):2305–2308CrossRefPubMedGoogle Scholar
  44. 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 133–154Google Scholar
  45. 45.
    Miller CW (1978) Survival and ambulation following hip fracture. J Bone Joint Surg 60A:930–934Google Scholar
  46. 46.
    Mosteller F (1952) The world series competition. J Am Stat Assoc 47(259):355–380CrossRefGoogle Scholar
  47. 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):475–483CrossRefPubMedGoogle Scholar
  48. 48.
    Papoulis A, Pillai SU (2002) Probability, random variables and stochastic processes, 4th edn. McGraw Hill, New York, NYGoogle Scholar
  49. 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 58:231–235PubMedGoogle Scholar
  50. 50.
    Reginster J-Y, Burlet N (2006) Osteoporosis: a still increasing prevalence. Bone 38(2 Supp1):4–9CrossRefGoogle Scholar
  51. 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):155–168CrossRefPubMedGoogle Scholar
  52. 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 17:11–14CrossRefPubMedGoogle Scholar
  53. 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 16:149–154CrossRefPubMedGoogle Scholar
  54. 54.
    Robinovich SN, Hayes WC, McMahon TA (1991) Prediction of femoral impact forces in falls on the hip. Trans ASME 113:366–374Google Scholar
  55. 55.
    Ross S (1995) Stochastic processes, 22nd edn. Wiley, New YorkGoogle Scholar
  56. 56.
    Ross SM (2006) Simulation, 4th edn. Elsevier Science and Technology Books, Amsterdam, The NetherlandsGoogle Scholar
  57. 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):1363–1370CrossRefPubMedGoogle Scholar
  58. 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 34:195–202CrossRefPubMedGoogle Scholar
  59. 59.
    Seeman E (2008) Bone quality: the material and structural basis of bone strength. J Bone Miner Metab 26(1):1–8CrossRefPubMedGoogle Scholar
  60. 60.
    Siffert RS, Kaufman JJ (2007) Ultrasonic bone assessment: “the time has come”. Bone 40(1):5–8CrossRefPubMedGoogle Scholar
  61. 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):197–206CrossRefPubMedGoogle Scholar
  62. 62.
    Silva MJ (2007) Biomechanics of osteoporotic fractures. Injury 38(Suppl 3):S69–S76CrossRefPubMedGoogle Scholar
  63. 63.
    Snyder DL, Miller MI (1975) Random point processes. Wiley, New York, NYGoogle Scholar
  64. 64.
    Turner C (2002) Biomechanics of bone: determinants of skeletal fragility and bone quality. Osteoporos Int 13(2):97–104CrossRefPubMedGoogle Scholar
  65. 65.
    van den Kroonenberg A, Hayes W, McMahon TA (1996) Hip impact velocities and body configurations for experimental falls from standing height. J Biomech 29:807–811CrossRefPubMedGoogle Scholar
  66. 66.
    Van Trees HL (1968) Detection, estimation, and modulation theory part I. Wiley, New YorkGoogle Scholar
  67. 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):33–42CrossRefPubMedGoogle Scholar

Copyright information

© International Federation for Medical and Biological Engineering 2010

Authors and Affiliations

  • Zvi Schechner
    • 1
  • Gangming Luo
    • 1
    • 2
    • 3
  • Jonathan J. Kaufman
    • 1
    • 4
  • Robert S. Siffert
    • 4
  1. 1.CyberLogic, Inc.New YorkUSA
  2. 2.VA New York Harbor HealthCare SystemNew YorkUSA
  3. 3.Department of Rehabilitation MedicineNew York University School of MedicineNew YorkUSA
  4. 4.Department of OrthopedicsThe Mount Sinai School of MedicineNew YorkUSA

Personalised recommendations