Computer simulations on the mechanical behaviors of biphasic calcium phosphates

  • Xingtao Ma
  • Li ZhangEmail author
  • Hong Wu
  • Xingdong Zhang
  • Mingli YangEmail author
Original Paper


Biphasic calcium phosphate (BCP) bioceramics, the mixture of hydroxyapatite (HA) and beta- tricalcium phosphate (β-TCP), are widely used as bone repair materials. Optimization of its composition and mixing pattern is crucial for its design and preparation. A series of BCP structures with a HA/β-TCP mass ratio of 0/100, 30/70, 50/50, 70/30, and 100/0 were constructed and studied with a simulated annealing molecular dynamics method. On the basis of equilibrated BCP structures, their elastic constants and moduli were computed and analyzed. With increasing HA content, the elastic moduli of BCP increase. Under the same mass ratio (50/50), the elastic moduli have no distinct changes for different mixing patterns. Calculations on the uniaxial extension of BCP reveal a sophisticated nonlinear and loading-path dependent behavior. The maximum stress decreases with increasing β-TCP content and mixing level.


Biphasic calcium phosphate Elastic constant Elastic moduli Molecular dynamics Simulated annealing Uniaxial extension 



The authors thank financial support from National High Technology Research and Development Program of China (No. 2015AA034202) and National Natural Science Foundation of China (No. 21373140). Part of the calculations were carried out at High-Performance Computing Center of Sichuan University.

Supplementary material

894_2017_3316_MOESM1_ESM.docx (354 kb)
ESM 1 (DOCX 354 kb)


  1. 1.
    Palmer I, Nelson J, Schatton W, Dunne NJ, Buchanan F, Clarke SA (2016) Biocompatibility of calcium phosphate bone cement with optimised mechanical properties: an in vivo study. J Mater Sci Mater Med 27:191. doi: 10.1007/s10856-016-5806-2 CrossRefGoogle Scholar
  2. 2.
    Woodard JR, Hilldore AJ, Lan SK, Park CJ, Morgan AW, Eurell JAC, Clark SG, Wheeler MB, Jamison RD, Johnson AJW (2007) The mechanical properties and osteoconductivity of hydroxyapatite bone scaffolds with multi-scale porosity. Biomaterials 28:45–54. doi: 10.1016/j.biomaterials.2006.08.021 CrossRefGoogle Scholar
  3. 3.
    Habibovic P, van der Valk CM, van Blitterswijk CA, de Groot K (2004) Influence of octacalcium phosphate coating on osteoinductive properties of biomaterials. J Mater Sci Mater Med 15:373–380. doi: 10.1023/B:JMSM.0000021104.42685.9f CrossRefGoogle Scholar
  4. 4.
    Kokubo T, Kim HM, Kawashita M (2003) Novel bioactive materials with different mechanical properties. Biomaterials 24:2161–2175. doi: 10.1016/S0142-9612(03)00044-9 CrossRefGoogle Scholar
  5. 5.
    Ducheyne P, Radin S, King L (1993) The effect of calcium phosphate ceramic composition andstructure on in vitro behavior. I. Dissolution. J Biomed Mater Res 27:25–34CrossRefGoogle Scholar
  6. 6.
    Kotani S, Fujita Y, Kitsugi T, Nakamura T, Yamamuro T, Ohtsuki C, Kokubo T (1991) Bone bonding mechanism of β-Tricalcium phosphate. J Biomed Mater Res 25:1303–1315CrossRefGoogle Scholar
  7. 7.
    Yokozeki H, Hayashi T, Nakagawa T, Kurosawa H, Shibuya K, Ioku K (1998) Influence of surface microstructure on the reaction of the active ceramics in vivo. J Mater Sci Mater Med 9:381–384CrossRefGoogle Scholar
  8. 8.
    Ducheyne P (1987) Bioceramics: material characteristics versus in vivo behavior. J Biomed Mater Res 21:219–236Google Scholar
  9. 9.
    Yaszemski MJ, Payne RF, Hayes WC, Lander R, Mikos AG (1996) Evolution of bone transplantation: molecular, cellular and tissue strategies to engineer human bone. Biomaterials 17:175–185CrossRefGoogle Scholar
  10. 10.
    Daculsi G, Laboux O, Malard O, Weiss P (2003) Current state of the art of biphasic calcium phosphate bioceramics. J Mater Sci Mater Med 14:195–200CrossRefGoogle Scholar
  11. 11.
    Ripamonti U, Richter PW, Nilen RW, Renton L (2008) The induction of bone formation by smart biphasic hydroxyapatite tricalcium phosphate biomimetic matrices in the non-human primate Papio ursinus. J Cell Mol Med 12:2609–2621. doi: 10.1111/j.1582-4934.2008.00312.x CrossRefGoogle Scholar
  12. 12.
    Cheng L, Ye F, Yang R, Lu X, Shi Y, Li L, Fan H, Bu H (2010) Osteoinduction of hydroxyapatite/β-tricalcium phosphate bioceramics in mice with a fractured fibula. Acta Biomater 6:1569–1574. doi: 10.1016/j.actbio.2009.10.050 CrossRefGoogle Scholar
  13. 13.
    Lindgren C, Mordenfeld A, Hallman M (2012) A prospective 1-year clinical and radiographic study of implants placed after maxillary sinus floor augmentation with synthetic biphasic calcium phosphate or deproteinized bovine bone. Clin Implant Dent Relat Res 14:41–50. doi: 10.1111/j.1708-8208.2010.00224.x CrossRefGoogle Scholar
  14. 14.
    Hallman M, Thor A (2008) Bone substitutes and growth factors as an alternative/complement to autogenous bone for grafting in implant dentistry. Periodontol 47:172–192CrossRefGoogle Scholar
  15. 15.
    LeGeros RZ, Lin S, Rohanizadeh R, Mijares D, LeGeros JP (2003) Biphasic calcium phosphate bioceramics: preparation, properties and applications. J Mater Sci Mater Med 14:201–209CrossRefGoogle Scholar
  16. 16.
    Daculsi G, LeGeros RZ, Heughebaert M, Barbieux I (1990) Formation of carbonate-apatite crystals after implantation of calcium phosphate ceramics. Calcif Tissue Int 46:20–27CrossRefGoogle Scholar
  17. 17.
    Nery EB, LeGeros RZ, Lynch KL, Lee K (1992) Tissue response to biphasic calcium phosphate ceramic with different ratios of HA/β-TCP in periodontal osseous defects. J Periodontol 63:729–735. doi: 10.1902/jop.1992.63.9.729 CrossRefGoogle Scholar
  18. 18.
    Jensen SS, Broggini N, Hjørting-Hansen E, Sclenk R, Buser D (2006) Bone healing and graft resorption of autograft, anorganic bovine bone and β-tricalcium phosphate. A histologic and histomorphometric study in the mandibles of minipigs. Clin Oral Implants Res 17:237–243. doi: 10.1111/j.1600-0501.2005.01257.x CrossRefGoogle Scholar
  19. 19.
    Raynaud S, Champion E, Lafon JP, Bernache-Assolant D (2002) Calcium phosphate apatites with variable Ca/P atomic ratio III. Mechanical properties and degradation in solution of hot pressed ceramics. Biomaterials 23:1081–1089CrossRefGoogle Scholar
  20. 20.
    Bouler JM, Trécant M, Delécrin J, Royer J, Passuti N, Daculsi G (1996) Macroporous biphasic calcium phosphate ceramics: Influence of five synthesis parameters on compressive strength. J Biomed Mater Res 32:603–609CrossRefGoogle Scholar
  21. 21.
    Ramay HRR, Zhang M (2004) Biphasic calcium phosphate nanocomposite porous scaffolds for load-bearing bone tissue engineering. Biomaterials 25:5171–5180. doi: 10.1016/j.biomaterials.2003.12.023 CrossRefGoogle Scholar
  22. 22.
    Yang Y, He F, Ye J (2016) Preparation, mechanical property and cytocompatibility of freeze-cast porous calcium phosphate ceramics reinforced by phosphate-based glass. Mater Sci Eng C 69:1004–1009. doi: 10.1016/j.msec.2016.08.008 CrossRefGoogle Scholar
  23. 23.
    Yetmez M (2014) Sintering behavior and mechanical properties of biphasic calcium phosphate ceramics. Adv Mater Sci Eng 2014:871749. doi: 10.1155/2014/871749 CrossRefGoogle Scholar
  24. 24.
    Kim DH, Kim KL, Chun HH, Kim TW, Park HC, Yoon SY (2014) In vitro biodegradable and mechanical performance of biphasic calcium phosphate porous scaffolds with unidirectional macro-pore structure. Ceram Int 40:8293–8300. doi: 10.1016/j.ceramint.2014.01.031 CrossRefGoogle Scholar
  25. 25.
    Hochrein O, Zahn D (2011) On the molecular mechanisms of the acid-induced dissociation of hydroxy-apatite in water. J Mol Model 17:1525–1528. doi: 10.1007/s00894-010-0855-9 CrossRefGoogle Scholar
  26. 26.
    Zahn D, Hochrein (2008) On the composition and atomic arrangement of calcium-deficient hydroxyapatite: an ab-initio analysis. J Solid State Chem 181:1712–1716. doi: 10.1016/j.jssc.2008.03.035 CrossRefGoogle Scholar
  27. 27.
    Calderín L, Stott MJ, Rubio A (2003) Electronic and crystallographic structure of apatites. Phys Rev B 67:134106. doi: 10.1103/PhysRevB.67.134106 CrossRefGoogle Scholar
  28. 28.
    Misra A, Ching WY (2013) Theoretical nonlinear response of complex single crystal under multi-axial tensile loading. Sci Rep 3:1488. doi: 10.1038/srep01488 CrossRefGoogle Scholar
  29. 29.
    Ou X, Han Q (2014) Molecular dynamics simulations of the mechanical properties of monoclinic hydroxyapatite. J Mol Model 20:2505. doi: 10.1007/s00894-014-2505-0 CrossRefGoogle Scholar
  30. 30.
    de Leeuw NH (2004) A computer modelling study of the uptake and segregation of fluoride ions at the hydrated hydroxyapatite (0001) surface: introducing a Ca10(PO4)6(OH)2 potential model. Phys Chem Chem Phys 6:1860–1866. doi: 10.1039/b313242k CrossRefGoogle Scholar
  31. 31.
    Yin X, Calderin L, Stott MJ, Sayer M (2002) Density functional study of structural, electronic and vibrational properties of Mg- and Zn-doped tricalcium phosphate biomaterials. Biomaterials 23:4155–4163CrossRefGoogle Scholar
  32. 32.
    Matsunaga K, Kubota T, Toyoura K, Nakamura A (2015) First-principles calculations of divalent substitution of Ca2+ in tricalcium phosphates. Acta Biomater 23:329–337. doi: 10.1016/j.actbio.2015.05.014 CrossRefGoogle Scholar
  33. 33.
    Liang L, Rulis P, Ching WY (2010) Mechanical properties, electronic structure and bonding of alpha- and beta-tricalcium phosphate with surface characterization. Acta Biomater 6:3763–3771. doi: 10.1016/j.actbio.2010.03.033 CrossRefGoogle Scholar
  34. 34.
    Kay MI, Young RA, Posner AS (1964) Crystal structure of hydroxyapatite. Nature 204:1050–1052CrossRefGoogle Scholar
  35. 35.
    Yashima M, Sakai A, Kamiyama T, Hoshikawa A (2003) Crystal structure analysis of β-tricalcium phosphate Ca3(PO4)2 by neutron powder diffraction. J Solid State Chem 175:272–277. doi: 10.1016/S0022-4596(03)00279-2 CrossRefGoogle Scholar
  36. 36.
    Menéndez-Proupin E, Cervantes-Rodríguez S, Osorio-pulgar R, Franco-Cisterna M, Camacho-Montes H, Fuentes ME (2011) Computer simulation of elastic constants of hydroxyapatite and fluorapatite. J Mech Behav Biomed Mater 4:1011–1020. doi: 10.1016/j.jmbbm.2011.03.001 CrossRefGoogle Scholar
  37. 37.
    Hughes JM, Rakovan (2002) The crystal structure of apatite, Ca5(PO4)3(F, OH, Cl). Rev Mineral Geochem doi: 10.2138/rmg.2002.48.1 Google Scholar
  38. 38.
    Kirkpatrick S, Gelatt CD, Vecchi MP (1983) Optimization by simulated annealing. Science 220:671–680CrossRefGoogle Scholar
  39. 39.
    Plimpton S (1995) Fast parallel algorithms for short-range molecular dynamics. J Comput Phys 117:1–19CrossRefGoogle Scholar
  40. 40.
    Lin TJ, Heinz H (2016) Accurate force field parameters and pH resolved surface models for hydroxyapatite to understand structure, mechanics, hydration, and biological interfaces. J Phys Chem C 120:4975–4992. doi: 10.1021/acs.jpcc.5b12504 CrossRefGoogle Scholar
  41. 41.
    Voigt W (1928) Lehrbuch der Kristallpysik. Springer Vieweg, Berlin. doi: 10.1007/978-3-663-15884-4Google Scholar
  42. 42.
    Reuss A, Angnew Z (1929) A calculation of the bulk modulus of polycrystalline materials. Math Meth 9:55. doi: 10.1007/BF00544497 Google Scholar
  43. 43.
    Hill R (1952) The elastic behavior of a crystalline aggregate. Proc Phys Soc Lond Sect A 65:349–355. doi: 10.1088/0370-1298/65/5/307 CrossRefGoogle Scholar
  44. 44.
    Lee WT, Dove MT, Salje EKH (2000) Surface relaxations in hydroxyapatite. J Phys Condens Matter 12:9829–9841CrossRefGoogle Scholar
  45. 45.
    Grenoble DE, Katz JL, Dunn KL, Gilmore RS, Murty KL (1972) The elastic properties of hard tissues and apatites. J Biomed Mater Res 6:221. doi: 10.1002/jbm.820060311 CrossRefGoogle Scholar
  46. 46.
    Schroeder LW, Dickens B, Brown WE (1977) Crystallographic studies of role of Mg as a stabilizing impurity in beta-Ca3(PO4)2.2. refinement of Mg-containing beta-Ca3(PO4)2. J Solid State Chem 22:253–262. doi: 10.1016/0022-4596(77)90002-0 CrossRefGoogle Scholar
  47. 47.
    Katz JL, Ukrainck K (1971) Anisotropic elastic properties of hydroxyapatite. J Biomech 4:221–227CrossRefGoogle Scholar
  48. 48.
    Akao M, Aoki H, Kato K (1982) Dense polycrystalline β-tricalcium phosphate for prosthetic applications. J Mater Sci 17:343–346CrossRefGoogle Scholar
  49. 49.
    Lopes MA, Silva RF, Monteiro FJ, Santos JD (2000) Microstructural dependence of Young’s and shear moduli of P2O5 glass reinforced hydroxyapatite for biomedical applications. Biomaterials 21:749–754CrossRefGoogle Scholar
  50. 50.
    Gilmore RS, Katz JL (1982) Elastic properties of apaties. J Mater Sci 17:1131–1141CrossRefGoogle Scholar
  51. 51.
    Ching WY, Rulis P, Misra A (2009) Ab initio elastic properties and tensile strength of crystalline hydroxyapatite. Acta Biomater 5:3067–3075. doi: 10.1016/j.actbio.2009.04.030 CrossRefGoogle Scholar
  52. 52.
    Lees SRF (1972) Anisotropy in hard dental tissues. J Biomech 5:557–566. doi: 10.1016/0021-9290(72)90027-9 CrossRefGoogle Scholar
  53. 53.
    Snyders R, Music D, Sigumonrong D, Schelnberger B, Jensen J, Schneider JM (2007) Experimental and ab initio study of the mechanical properties of hydroxyapatite. Appl Phys Lett 90:193902. doi: 10.1063/1.2738386 CrossRefGoogle Scholar
  54. 54.
    de Leeuw NH, Bowe JR, Rabone JAL (2007) A computational investigation of stoichiometric and calcium-deficient oxy- and hydroxy-apatites. Faraday Discuss 134:195–214. doi: 10.1039/b602012g CrossRefGoogle Scholar
  55. 55.
    Sun JP, Song Y, Wen GW, Wang Y, Yang R (2013) Softening of hydroxyapatite by vacancies: a first principles investigation. Mater Sci Eng C 33:1109–1115. doi: 10.1016/j.msec.2012.12.001 CrossRefGoogle Scholar
  56. 56.
    Marković S, Lukić MJ, Škapin SD, Stojanović B, Uskoković D (2015) Designing, fabrication and characterization of nanostructured functionally graded HAp/BCP ceramics. Ceram Int 41:2654–2667. doi: 10.1016/j.ceramint.2014.10.079 CrossRefGoogle Scholar
  57. 57.
    Impens S, Schelstraete R, Luyten J, Mullens S, Thijs I, van Humbeeck J, Schrooten J (2009) Production and characterisation of porous calcium phosphate structures with controllable hydroxyapatite/β-tricalcium phosphate ratios. Adv Appl Ceram 108:494–500. doi: 10.1179/174367609x422243 CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2017

Authors and Affiliations

  1. 1.Institute of Atomic and Molecular PhysicsSichuan UniversityChengduChina
  2. 2.National Engineering Research Center for BiomaterialsSichuan UniversityChengduChina

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