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Journal of Mathematical Biology

, Volume 69, Issue 6–7, pp 1383–1429 | Cite as

A coupled mechano-biochemical model for bone adaptation

  • Václav KlikaEmail author
  • Maria Angelés Pérez
  • José Manuel García-Aznar
  • František Maršík
  • Manuel Doblaré
Article

Abstract

Bone remodelling is a fundamental biological process that controls bone microrepair, adaptation to environmental loads and calcium regulation among other important processes. It is not surprising that bone remodelling has been subject of intensive both experimental and theoretical research. In particular, many mathematical models have been developed in the last decades focusing in particular aspects of this complicated phenomenon where mechanics, biochemistry and cell processes strongly interact. In this paper, we present a new model that combines most of these essential aspects in bone remodelling with especial focus on the effect of the mechanical environment into the biochemical control of bone adaptation mainly associated to the well known RANKL-RANK-OPG pathway. The predicted results show a good correspondence with experimental and clinical findings. For example, our results indicate that trabecular bone is more severely affected both in disuse and disease than cortical bone what has been observed in osteoporotic bones. In future, the methodology proposed would help to new therapeutic strategies following the evolution of bone tissue distribution in osteoporotic patients.

Keywords

Mechano-biochemical model Bone remodelling BMU  Damage Mineralization RANKL-RANK-OPG pathway 

List of symbols

Used notation of quantities:

\(v_b\)

Bone volume fraction [1]

\(v_m\)

Mineral volume fraction of bone [1]

\(v_o\)

Organic volume fraction of bone [1]

\(v_v\)

Void volume fraction [1]

\(dam\)

Damage concentration [1]

\(r_{\alpha }\)

Rate of \(\alpha \)-th interaction \(\left[ \text{ kmol }\cdot \text{ m }^{-3}\cdot \text{ s }^{-1}\right] \)

\(\mathcal {A}_{\alpha }\)

Affinity of \(\alpha \)-th reaction \(\left[ \text{ J }\cdot \text{ kmol }^{-1}\right] \)

Open image in new window

The deformation rate tensor Open image in new window and its j-\(th\) invariant; \(j=1\) and \(j=2\) represent rate of volume dilatation and shear rate, respectively [\(s^{-1}\)]

\({\varvec{\varepsilon }},~\varepsilon ^{(j)}\)

The deformation tensor \(\varepsilon \) and its j-\(th\) invariant [1]

\(l_{v\alpha },~l_{\alpha \alpha }\)

Phenomenological constants in classical irreversible thermodynamics (CIT)

\(C_j, [\mathrm{C_j}]\)

j-\(th\) substance and its molar concentration \(\left[ \text{ kmol }\cdot \text{ m }^{-3}\right] \)

\(\nu _{j\alpha },~\nu '_{j\alpha }\)

Stoichiometric coefficients of substance \(C_j\) entering \(\alpha \)-th reaction or being produced in it (denoted with a prime), respectively [1]

\(k_{+\alpha },~k_{-\alpha }\)

Forward and backward reaction rate constant of \(\alpha \)-th reaction

\(\alpha \)

Ash fraction [1]

\(\nu \)

Poisson ratio [1]

\(E\)

Young’s modulus \(\left[ \text{ M } \text{ kg }\cdot \text{ m }^{-1}\cdot \text{ s }^{-1}\right] \)

\(\xi \)

Daily strain history [1]

\(L\)

Number of loading cases [1]

\(N_i\)

Number of cycles of i-\(th\) loading case [1]

\(\bar{\varepsilon }\)

Strain level [1]

\(\bar{d}\)

Strain rate level [\(s^{-1}\)]

\(U\)

Strain energy density \(\left[ \text{ kg }\cdot \text{ m }^{-1}\cdot \text{ s }^{-2}\right] \)

\(f_i\)

Frequency of a considered loading case [\(s^{-1}\)]

\(T_i\)

Period of a considered loading case [s]

\(\mathcal {D}_{\alpha }\)

Parameters describing the influence of mechanical stimulus on \(\alpha -th\) interaction [1]

\(A\)

Parameter used for relating \(\mathcal {D}_{\alpha }\) to \(\xi \) [1]

\(\mathcal {D}_{\alpha ,ref},~\xi _{ref}\)

Reference values of mechanical stimulus [1]

\(\varepsilon _{crit}\)

Critical strain value of bone tissue [1]

\(N_f\)

Fatigue life expectation (number of cycles) [1]

\(K_i\)

(\(i=t,c\)) Constant of proportionality in fatigue life expectation [1]

\(\delta _i\)

(\(i=t,c\)) Exponent in fatigue life expectation [1]

\(dam_i\)

(\(i=t,c\)) Damage in trabecular or cortical bone [1]

\(C_{c,t_1,t_2},~\gamma _{c,t}\)

Fitted parameters in evolution laws for damage growth [1]

\(\varepsilon _u\)

Ultimate tensile strain, function of calcium content [1]

\(v_{b,res}\)

Amount of resorbed bone volume fraction [1]

\(\rho _{res}\)

Difference in bone tissue density caused by resorption [kg\(\cdot \text{ m }^{-3}\)]

\(\rho _{max}\)

Maximal amount of bone density (true bone density) [kg\(\cdot \text{ m }^{-3}\)]

\(J_{Old\_B}\)

Normalised turnover rate in equilibrium [1]

\(n_{C_j}\)

Normalised concentration of \(j\)-th substance [1]

\({n_{Old\_B}}_{,max}\)

Maximal possible value of \(n_{Old\_B}\) corresponding to intact bone tissue under the maximal mechanical stimulus \(\xi (\varepsilon _{crit})\) [1]

\(m_m,~m_o\)

Mineral/organic mass [kg\(\cdot \text{ m }^{-3}\)]

\(\rho _m,~\rho _o\)

Mineral/organic true densities [kg\(\cdot \text{ m }^{-3}\)]

\(v_m^{\mathrm{secondary}}\)

Exponential law for secondary mineralisation [1]

\(v_m^{prim}\)

Mineral volume fraction value at the end of the primary phase [1]

\(v_j^0\)

Initial value of \(v_j,~j=m,b,o\) [1]

\(\tau _{BR}\)

Time corresponding to the end of the primary phase [1]

\(v_m^{max}\)

\(v_m\) corresponding to the maximal mineral content found in bone [1]

\(\kappa \)

Exponent in secondary mineralisation phase [1]

\(v_m^{\mathrm{OldB}}\)

Average mineral content of old bone \(Old\_B\) [1]

\(v_m^{\mathrm{NewB}}\)

Average mineral content of new bone \(New\_B\) [1]

\(n_{prim}\)

Fraction of bone undergoing primary mineralisation [1]

\(X^0\)

Initial value of \(X\) (e.g. \(v_b^0,\alpha ^0, [\mathrm{PTH}]_0\))

\(BT_{ind}\)

Bone tissue index (a measure of total bone mass) [\(\text{ m }^3\)]

\(BMD_{ind}\)

Bone mineral index (a measure of total mineral mass) [kg]

\(BMD(i)\)

Bone mineral density at \(i\)-th element [kg\(\cdot \text{ m }^{-3}\)]

\(V(i)\)

Volume of the \(i\)-th element [\(\text{ m }^3\)]

Used notation in Appendices:

\(n_{C_{j,0}}\)

Initial normalised concentration of j-\(th\) substance [1]

\([\mathrm{C_j}]_{st}\)

Standard concentration of j-\(th\) substance [1]

\(n_{C_j,st}\)

Normalised equivalent standard concentration of j-\(th\) substance [1]

\(n_{i_0}^{RRO,C_j}\)

\(C_j \in \{PTH,~estr,~NO\}\), \(i \in \{RKL,~OPG\}\); predicted initial value of \(n_i\) (from a submodel for \(C_j\) influence on RANKL-RANK-OPG pathway) used as input in the RANKL-RANK-OPG model [1]

Mathematics Subject Classification (2000)

74F25 92C45 92E20 74A15 74B99 

Notes

Acknowledgments

This research was supported by the Instituto Aragones de Ciencias de la Salud through the research project (PIPAMER10/015) and the Spanish Ministry of Science and Technology through the Research Project DPI2011-22413. Further support was received from the CENTEM project, reg. no. CZ.1.05/2.1.00/03.0088, that is partially supported by the ERDF within the OP RDI Programme of the Ministry of Education, Youth and Sports and from the institutional support RVO:61388998.

Supplementary material

285_2013_736_MOESM1_ESM.f (34 kb)
Supplementary material 1 (F 35 kb)
285_2013_736_MOESM2_ESM.for (3 kb)
Supplementary material 2 (FOR 3 kb)

References

  1. Aubin JE, Bonnelye E (2000) Osteoprotegerin and its ligand: a new paradigm for regulation of osteoclastogenesis and bone resorption. Medscape Womens Heal 5(2):5Google Scholar
  2. Beaupre GS, Orr TE, Carter DR (1990) An approach for time-dependent bone modeling and remodeling: theoretical development. J Orthop Res 8:651–661CrossRefGoogle Scholar
  3. Bergmann G, Graichen F, Rohlmann A (1993) Hip joint loading during walking and running, measured in two patients. J Biomech 26(8):969–990CrossRefGoogle Scholar
  4. Bougherara H, Klika V, Maršík F, Mařík I, Yahia L (2010) New predictive model for monitoring bone remodeling. J Biomed Mater Res Part A 95(1):9–24CrossRefGoogle Scholar
  5. Charopoulos I, Tournis S, Trovas G, Raptou P, Kaldrymides P, Skarandavos G, Katsalira K, Lyritis GP (2006) Effect of primary hyperparathyroidism on volumetric bone mineral density and bone geometry assessed by peripheral quantitative computed tomography in postmenopausal women. J Clin Endocr Metab 91(5):1748–1753CrossRefGoogle Scholar
  6. Cohen M Jr (2006) The new bone biology: pathologic, molecular, and clinical correlates. Am J Med Genet A 140(23):2646–2706CrossRefGoogle Scholar
  7. Collin-Osdoby P (2004) Regulation of vascular calcification by osteoclast regulatory factors RANKL and osteoprotegerin. Circ Res 95:1046–1057CrossRefGoogle Scholar
  8. Cowin SC, Hegedus DH (1976) Bone remodelling i: theory of adaptaive elasticity. J Elast 6:313–326CrossRefzbMATHMathSciNetGoogle Scholar
  9. Currey JD (2004) Tensile yield in compact bone is determined by strain, post-yield behaviour by mineral content. J Biomech 37(4):549–556CrossRefGoogle Scholar
  10. Doblaré M, García J (2002) Anisotropic bone remodelling model based on a continuum damage-repair theory. J Biomech 35(1):1–17CrossRefGoogle Scholar
  11. Doblaré M, García J, Gómez M (2004) Modelling bone tissue fracture and healing: a review. Eng Fract Mech 71:1809–1840CrossRefGoogle Scholar
  12. Ettinger B, Pressman A, Sklarin P, Bauer DC, Cauley JA, Cummings SR (1998) Associations between low levels of serum estradiol, bone density, and fractures among elderly women: The study of osteoporotic fractures. J Clin Endocr Metab 83(7):2239–2243Google Scholar
  13. Fang G, Ji B, Liu XS, Guo XE (2010) Quantification of trabecular bone microdamage using the virtual internal bond model and the individual trabeculae segmentation technique. Comput Method Biomech Biomed Eng 13(5):605–615CrossRefGoogle Scholar
  14. Frost HM (1963) Bone remodelling dynamics. C C Thomas, SpringfieldGoogle Scholar
  15. Frost HM (2000) The utah paradigm of skeletal physiology: an overview of its insights for bone, cartilage and collagenous tissue organs. J Bone Miner Metab 18:305–316CrossRefGoogle Scholar
  16. Fukunaga T, Kurata K, Matsuda J, Higaki H (2008) Effects of strain magnitude on mechanical responses of three-dimensional gel-embedded osteocytes studied with a novel 10-well elastic chamber. J Biomech Sci Eng 3(1):13–24CrossRefGoogle Scholar
  17. Fyhrie DP, Schaffler MB (1995) The adaptation of bone apparent density to applied load. J Biomech 28:135–146CrossRefGoogle Scholar
  18. García-Aznar JM, Rueberg T, Doblaré M (2005) A bone remodelling model coupling microdamage growth and repair by 3D BMU-activity. Biomech Model Mechanobiol 4:147–167CrossRefGoogle Scholar
  19. Goemaere S, Van Laere M, De Neve P, Kaufman J (1994) Bone mineral status in paraplegic patients who do or do not perform standing. Osteoporosis Int 4(3):138–143CrossRefGoogle Scholar
  20. Gong J, Arnold J, Cohn S (1964) Composition of trabecular and cortical bone. Anat Rec 149(3):325– 331CrossRefGoogle Scholar
  21. Hazelwood SJ, Martin RB, Rashid MM, Rodrigo JJ (2001) A mechanistic model for internal bone remodeling exhibits different dynamic responses in disuse and overload. J Biomech 34:299–308CrossRefGoogle Scholar
  22. Hazenberg JG, Taylor D, Lee TC (2007) The role of osteocytes and bone microstructure in preventing osteoporotic fractures. Osteoporosis Int 18:1–8CrossRefGoogle Scholar
  23. Hernandez C (2001) Simulation of bone remodeling during the development and treatment of osteoporosis. Phd thesis, Stanford University, StanfordGoogle Scholar
  24. Hernández-Gil I, Gracia M, Pingarrón M, Jerez L (2006) Physiological bases of bone regeneration ii. the remodeling process. Med Oral Patol Oral Cir Bucal 11:E151–215Google Scholar
  25. Hill PA (1998) Bone remodelling. Brit J Orthod 25:101–107CrossRefGoogle Scholar
  26. Hjelmstad KD (2005) Fundamentals of structural mechanics. Springer, BerlinGoogle Scholar
  27. Huiskes R, Ruimerman R, van Lenthe GH, Janssen JD (2000) Effects of mechanical forces on maintenance and adaptation of form in trabecular bone. Nature 405:704–706CrossRefGoogle Scholar
  28. Ingber DE (2008) Tensegrity-based mechanosensing from macro to micro. Prog Biophys Mol Bio 97: 163179CrossRefGoogle Scholar
  29. Jacobs CR (1994) Numerical simulation of bone adaptation to mechanical loading. Phd thesis, Stanford University, StanfordGoogle Scholar
  30. Jacobs CR, Simo JC, Beaupré GS, Carter DR (1997) Adaptive bone remodeling incorporating simultaneous density and anisotropy considerations. J Biomech 30(6):603–613CrossRefGoogle Scholar
  31. Jimi E, Akiyama S, Tsurukai T, Okahashi N, Kobayashi K, Udagawa N, Nishihara T, Takahashi N, Suda T (1999) Osteoclast differentiation factor acts as a multifunctional regulator in murine osteoclast differentiation and function. J Immunol 163(1):434–442Google Scholar
  32. Kim C, You L, Yellowley C, Jacobs C (2006) Oscillatory fluid flow-induced shear stress decreases osteoclastogenesis through rankl and opg signaling. Bone 39(5):1043–1047CrossRefGoogle Scholar
  33. Klika V (2010) Comparison of the effects of possible mechanical stimuli on the rate of biochemical reactions. J Phys Chem B 114(32):10,567–10,572CrossRefGoogle Scholar
  34. Klika V, Maršík F (2009) Coupling effect between mechanical loading and chemical reactions. J Phys Chem B 113:14,689–14,697CrossRefGoogle Scholar
  35. Klika V, Maršík F (2010) A thermodynamic model of bone remodelling: The influence of dynamic loading together with biochemical control. J Musculoskelet Neuron Interact 10(3):220–230Google Scholar
  36. Klika V, Maršík F (2011a) Biomechanics, vol 1, INTECH, Vienna, chap Feasible simulation of diseases related to bone remodelling and of their treatment. ISBN 978-953-307-312-5, [online] http://www.sciyo.com
  37. Klika V, Maršík F (2011b) Feasible predictions of bone remodelling using modelling techniques. Locomot Appar 1+2:26–41, [online] http://www.pojivo.cz
  38. Klika V, Maršík F, Mařík I (2010) Dynamic Modelling, INTECH, Vienna, chap Influencing the Effect of Treatment of Disease Related to Bone Remodelling by Dynamic Loading. ISBN 978-953-7619-68-8, [online] http://www.sciyo.com
  39. Klika V, Grmela M (2013) Coupling between chemical kinetics and mechanics that is both nonlinear and compatible with thermodynamics. Phys rev E 87(1–1):012,141–012,141CrossRefGoogle Scholar
  40. Kobayashi Y, Udagawa N, Takahashi N (2009) Action of rankl and opg for osteoclastogenesis. Crit Rev Eukaryot Gene Expr 19(1):61CrossRefGoogle Scholar
  41. Komarova SV, Smith RJ, Dixon SJ, Sims SM, Wahl LM (2003) Mathematical model predicts a critical role for osteoclast autocrine regulation in the control of bone remodeling. Bone 33(2):206–215CrossRefGoogle Scholar
  42. Kroll MH (2000) Parathyroid hormone temporal effects on bone formation and resorption. Bull Math Biol 61(1):163–188CrossRefGoogle Scholar
  43. Kurata K, Heino TJ, Higaki H, Vaananen HK (2006) Bone marrow cell differentiation induced by mechanically damaged osteocytes in 3D gel-embedded culture. J Bone Miner Res 21(4):616–625CrossRefGoogle Scholar
  44. Lai Y, Qin L, Hung V, Chan K (2005) Regional differences in cortical bone mineral density in the weight-bearing long bone shafta pqct study. Bone 36(3):465–471CrossRefGoogle Scholar
  45. Lemaire V, Tobin FL, Greller LD, Cho CR, Suva LJ (2004) Modeling the interactions between osteoblast and osteoclast activities in bone remodeling. J Theor Biol 229(3):293–309CrossRefMathSciNetGoogle Scholar
  46. Manolagas SC (1999) Cell number versus cell vigor-what really matters to a regeneration skeleton? Endocrinology 140(10):4377–4381Google Scholar
  47. Martin RB (1995) A mathematical model for fatigue damage repair and stress fracture in osteonal bone. J Orthop Res 13:309–316Google Scholar
  48. Martin RB (2007) Targeted bone remodeling involves BMU steering as well as activation. Bone 40(6):1574–1580CrossRefGoogle Scholar
  49. Martin T (2004) Paracrine regulation of osteoclast formation and activity: Milestones in discovery. J Musculoskelet Neuron Interact 4:243–253Google Scholar
  50. Martínez-Reina J, García-Aznar JM, Dominguez J, Doblaré M (2008) On the role of bone damage in calcium homeostasis. J Theor Biol 254:704–712CrossRefGoogle Scholar
  51. Nakashima T, Hayashi M, Fukunaga T, Kurata K, Oh-hora M, Feng J, Bonewald L, Kodama T, Wutz A, Wagner E, et al. (2011) Evidence for osteocyte regulation of bone homeostasis through rankl expression. Nat med 11:1231-1234Google Scholar
  52. van Oers RF, Ruimerman R, Tanck E, Hilbers PA, Huiskes R (2008) A unified theory for osteonal and hemiosteonal remodeling. Bone 42(2):250–259CrossRefGoogle Scholar
  53. Parfitt AM (2002) Life history of osteocytes: relationship to bone age, bone remodeling and bone fragility. J Musculoskelet Neuron Interact 2(6):499–500Google Scholar
  54. Pattin CA, Caler WE, Carter DR (1996) Cyclic mechanical property degradation during fatigue loading of cortical bone. J Biomech 29(1):69–79CrossRefGoogle Scholar
  55. Pivonka P, Zimak J, Smith D, Gardiner B, Dunstan C, Sims N, John Martin T, Mundy G (2008) Model structure and control of bone remodeling: a theoretical study. Bone 43(2):249–263CrossRefGoogle Scholar
  56. Pivonka P, Zimak J, Smith D, Gardiner B, Dunstan C, Sims N, John Martin T, Mundy G (2010) Theoretical investigation of the role of the rank-rankl-opg system in bone remodeling. J Theor Biol 262(2):306– 316CrossRefGoogle Scholar
  57. Prendergast PJ, Taylor D (1994) Prediction of bone adaptation using damage accumulation. J Biomech 27(8):1067–1076CrossRefGoogle Scholar
  58. Proff P, Römer P (2009) The molecular mechanism behind bone remodeling: a review. Clin Oral Invest 13:355–362CrossRefGoogle Scholar
  59. Qu C, Qin QH, Kang Y (2006) A hypothetical mechanism of bone remodeling and modeling under electromagnetic loads. Biomaterials 27:4050–4057CrossRefGoogle Scholar
  60. Ramtani S, Zidi M (2001) A theoretical model of the effect of continuum damage on a bone adaptation model. J Biomech 34(4):471–479CrossRefGoogle Scholar
  61. Rattanakul C, Lenbury Y, Krishnamara N, Wollkind DJ (2003) Modeling of bone formation and resorption mediated by parathyroid hormone: response to estrogen/PTH therapy. Biosystems 70:55–72CrossRefGoogle Scholar
  62. Robling AG, Castillo AB, Turner CH (2006) Biomechanical and molecular regulation of bone remodeling. Annu Rev Biomed Eng 8:455–498CrossRefGoogle Scholar
  63. Rodan G (1998) Bone homeostasis. Proc Natl Acad Sci USA 95(23):13,361–13,362CrossRefGoogle Scholar
  64. Rodan GA, Martin TJ (1981) Role of osteoblasts in hormonal control of bone resorption- a hypothesis. Calcif Tissue Int 33(4):349–351CrossRefGoogle Scholar
  65. Roux W (1881) Der zuchtende Kampf der teile, oder die Teilauslee im Organismus (Theorie der funktionellen anpassung). Wukgekn Ebgeknabb, LeipzigGoogle Scholar
  66. Rubin J, Murphy T, Nanes MS, Fan X (2000) Mechanical strain inhibits expression of osteoclast differentiation factor by murine stromal cells. Am J Physiol-Cell Ph 278:C1126–C1132Google Scholar
  67. Ryser M, Nigam N, Komarova S (2008) Mathematical modeling of spatio-temporal dynamics of a single bone multicellular unit. J Bone Miner Res 24(5):860–870CrossRefGoogle Scholar
  68. Sikavitsas VI, Temenoff JS, Mikos AG (2001) Biomaterials and bone mechanotransduction. Biomaterials 22:2581–2593CrossRefGoogle Scholar
  69. Skerry TM (1998) Methods in bone biology, 1st edn, chap 6, Chapman & Hall, London, UK, pp 149–176. ISBN 0 412 75770 2Google Scholar
  70. Stacey E, Korkia P, Hukkanen M, Polak J, Rutherford O (1998) Decreased nitric oxide levels and bone turnover in amenorrheic athletes with spinal osteopenia. J Clin Endocr Metab 83(9):3056–3061Google Scholar
  71. Tan S, de Vries T, Kuijpers-Jagtman A, Semeins C, Everts V, Klein-Nulend J (2007) Osteocytes subjected to fluid flow inhibit osteoclast formation and bone resorption. Bone 41(5):745–751CrossRefGoogle Scholar
  72. Tudor-Locke C, Bassett J (2004) How many steps/day are enough?: Preliminary pedometer indices for public health. Sports Med 34(1):1–8CrossRefGoogle Scholar
  73. Vaira S, Alhawagri M, Anwisye I, Kitaura H, Faccio R, Novack DV (2008) RelA/p65 promotes osteoclast differentiation by blocking a RANKL-induced apoptotic JNK pathway in mice. J Clin Invest 118(6):2088–2097Google Scholar
  74. Virtama P, Telkkä A (1962) Cortical thickness as an estimate of mineral content of human humerus and femur. Brit J Radiol 35(417):632–633CrossRefGoogle Scholar
  75. Wang H, Ji B, Liu XS, Guo XE, Huang Y, Hwang KC (2012) Analysis of microstructural and mechanical alterations of trabecular bone in a simulated three-dimensional remodeling process. J biomech 45:2417-2425Google Scholar
  76. Wang H, Ji B, Liu X, Oers R, Guo X, Huang Y, Hwang KC (2013) Osteocyte-viability-based simulations of trabecular bone loss and recovery in disuse and reloading. Biomech Model Mechanobiol, pp 1–14. doi: 10.1007/s10237-013-0492-1
  77. Whalen RT, Carter DR, Steele CR (1988) Influence of physical activity on the regulation of bone density. J Biomech 21(10):825–837CrossRefGoogle Scholar
  78. Wimalawansa SJ (2007) Rationale for using nitric oxide donor therapy for prevention of bone loss and treatment of osteoporosis in humans. Ann NY Acad Sci 1117:283–297CrossRefGoogle Scholar
  79. Wolff J (1892) Das Gesetz der transformation der knochen. Hirchwild, BerlinGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  • Václav Klika
    • 1
    • 2
    Email author
  • Maria Angelés Pérez
    • 3
  • José Manuel García-Aznar
    • 3
  • František Maršík
    • 4
  • Manuel Doblaré
    • 3
  1. 1.Institute of ThermomechanicsAcademy of Sciences of the Czech RepublicPrague 8Czech Republic
  2. 2.Department of MathematicsFNSPE, Czech Technical University in PraguePrague 2Czech Republic
  3. 3.Mechanical Engineering Department, Aragon Institute of Engineering Research (I3A)Universidad de ZaragozaZaragozaSpain
  4. 4.New Technologies-Research CenterUniversity of West Bohemia in PilsenPilsenCzech Republic

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