This paper deals with the torsional moment depending on the angle of torsion of the
compact bone in laboratory animals and humans. Based on the data from laboratory
animals, obtained by measurement, the data on dependence of the torsional moment
and the angle of torsion were assumed for humans. Measurements were carried out on
four groups of compact bone in laboratory animals. One was the control group, and
three other groups were treated by various vitamin D3 metabolites. Equal measurements
were performed in only one group of compact bone in humans, due to the impossibility
to treat humans with vitamin D3 metabolites. Functional relations between the
angle of torsion and the torsional moment for all groups of animal body tissue were determined
by measurements, and the results were used to assume the reaction of human
compact bone tissue if treated by vitamin D3 metabolites

Andrija Bošnjak

Diana Milčić

Jadranka Keros

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Coll. Antropol. 27 Suppl. 2 (2003) 9–15UDC 572.781:612.753Original scientific paperAnalysis of Bone TissueMechanical PropertiesDiana Mil~i}1, Jadranka Keros2 and Andrija Bo{njak31 Department of Applied Mechanics, Faculty of Mechanical Engineering and NavalArchitecture, University of Zagreb, Zagreb, Croatia2 Department of Dental Anthropology, School of Dental Medicine, University of Zagreb,Zagreb, Croatia3 Department of Periodontology, School of Dental Medicine, University of Zagreb, Zagreb,CroatiaA B S T R A C TThis paper deals with the torsional moment depending on the angle of torsion of thecompact bone in laboratory animals and humans. Based on the data from laboratoryanimals, obtained by measurement, the data on dependence of the torsional momentand the angle of torsion were assumed for humans. Measurements were carried out onfour groups of compact bone in laboratory animals. One was the control group, andthree other groups were treated by various vitamin D3 metabolites. Equal measure-ments were performed in only one group of compact bone in humans, due to the impossi-bility to treat humans with vitamin D3 metabolites. Functional relations between theangle of torsion and the torsional moment for all groups of animal body tissue were de-termined by measurements, and the results were used to assume the reaction of humancompact bone tissue if treated by vitamin D3 metabolites.Key words: vitamin D3, biomechanics, torsional movement, bone, osseous tissueIntroductionMicromeasurement relationship be-tween tension and deformation of bonetissue has already been extensivelyinvestigated1–4. Bone is considered to be aphysical body, behaving according to thelaws of physics and mechanics. Mechani-cal behavior of the bone in standard phys-ical situations equals the mechanical be-havior of an elastic body. This is a conclu-sion based on primarily mechanical char-acteristics of the skeleton, in which bonediseases are firstly manifested as me-chanical shortcomings of the construc-tion. The bond between structure andfunction of the bone is obvious1–5.Precise mechanical analysis of the os-seous tissue from the mechanical stand-9Received for publication April 28, 2003.point is, therefore, a complex issue. Theanalysis is based primarily on the hard-ness of the material, the relationship be-tween deformation and tension of thebone, etc. These are considered as the re-lationships between molecules and atoms,which in turn means that every deforma-tion study and osseous tissue tension stu-dy surmises substantial knowledge of thephysical characteristics of the bone. Ex-actly this is a problem, since it is widelyknown that the composition and the shapeof the bone are changing throughout thelife3,6–9.Therefore, the choice of an appropri-ate bone model is of greatest importancein the determination of relationships be-tween deformation and tension, as wellas of the so-called threshold status whichcharacterizes the bone as a material ofthreshold constructions. The most impor-tant feature of a threshold constructionmaterial is that the minimum of masshas the maximum of hardness.The elasticity of the compact bone, aswell as the other physical characteristicsof the bone, are changing according to themineral content. This notion was the im-pulse to investigate the changes in thehardness of the osseous tissue during aprogrammed flow of vitamin D3 and itsmetabolic products on laboratory animalsthat were treated by the metabolites indifferent combinations and doses prior tosacrifice. The investigation was inspiredby the fact that vitamin D3 and its majormetabolic product, 1,25-dyhydrooxycho-lecalciferol (1,25-(OH)2D3), as well as itssecond metabolic product, 24,25-dihydro-cholecalciferol (24,25-(OH)2D3), are im-portant for the metabolism of the bone,and its regeneration after fracturing.The aim of the investigation was to de-termine the influence of the mineral com-position of the bone on total rigidity oflong bones. In this investigation, we usedlong bones of laboratory animals whosecomposition and shape are similar to thehuman long bones.Material and MethodsPhysical and chemical properties ofthe osseous tissue are changing. They arein concordance with loading circum-stances – strength and direction of theforce applied, etc. These properties de-pend on the part of the observed bone, inwhich the mode of the loading plays avery important role. Therefore, any at-tempt to simplify the procedure can sub-stantially and completely alter the ob-served features.Based on the hypothesis that osseoustissue of the long bones, whose cross-sec-tion is much smaller than their length, istransversely isotropic, we chose torsionalloading for the experiment. The theoreti-cal solution was aimed at isotropic tor-sion.The other important problem thatmakes this investigation difficult is pro-ducing the artificial bone samples in de-fined dimensions, and making sure thatthe axes of samples are parallel to the fi-ber direction. With that purpose we in-vestigated the dependency of the staticmoment of torsion, torsion angle, andgeometrical significance for the bone oflaboratory animals in the control groupas well as in three experimental groups,treated by metabolic products of vitaminD3. Equal investigations were to be per-formed on human osseous tissue, and af-terwards, based on the determined rela-tionship between torsion angle andtorsion moment for all groups of osseoustissue of animals. It was projected to pre-dict the properties of the human osseoustissue in the given situations of vitaminD3 metabolites implementation.The tibial bones of compared animalgroup were labeled I to IV, the group la-beled V to VIII was treated by 1,25-di-hydrocholecalciferol, the group labeled IX10D. Mil~i} et al.: Bone Tissue Mechanical Properties, Coll. Antropol. 27 Suppl. 2 (2003) 9–15to XII with 24,25-dihydrocholecalciferol,while the group labeled XIII to XVI wastreated by a combination of the two meta-bolic products of vitamin D3. The dura-tion of the experiment and time of thesacrifice of animals were equal in allgroups.The measurements were performed onfive human tibial bones in order to estab-lish the values of torsion moment and an-gles of torsion, as well as changes of tor-sional inertia moment, depending on thelocation of the studied cross-section.Since all human tibiae were macerated, itwas only possible to compare dimensionalproperties of human and animal bones.Human bones were studied at room tem-perature (23°C). Bone temperature dur-ing the experiment was 40.2°C.Since the industrial torsional deviceswith great measurement precision pre-sented excessive financial demands, weconstructed much simpler torsional de-vices. These have proved to be acceptablesince expected changes of dependence be-tween the torsional moment and the an-gle of torsion are acceptably significant inorder to calculate the effect of vitamin D3metabolites on the toughness of the com-pact bone. The accuracy was between 0.5and –0.5 degrees, which cannot be consid-ered as an outstanding error due to thedifferences in the mechanical hardnessduring the change of the mineral compo-sition of the bone.We constructed two different mea-surement units – one small and one large– since we examined two different sam-ples the dimensions of which did not al-low the use of only one unit. We measuredhuman tibiae (length 25 cm), and labora-tory animals tibiae (length 6 cm).It was not possible to perform the ex-periment on the human sample treatedby different metabolites of vitamin D3.Therefore, the experiment on humansamples was performed only once, bymeans of the larger measurement unit.The results of measurements on hu-man and animal samples enabled us todetermine the functional relationshipsbetween them. Based on this relation-ship, we simulated the behavior of thehuman tissue as if it were treated by thesame metabolites as the osseous tissue oflaboratory animals.An original program was set up inMicrosoft Visual Basic (Microsoft Corp.,Seattle, WA, USA). All data from the ex-periment, as well as the other data, wereinterpolated by means of the programpackage Mathematica and stored in MSAccess (Microsoft Corp., USA) databasethat was linked to the Visual Basic pro-gram, so every change was refreshed in-stantly.ResultsHardness measurements were per-formed on five samples of macerated hu-man tibiae that were bent until they frac-tured. The total diagram of the obtainedresults is shown in Figure 1, and it is onlya comparative depiction of mechanicalbehavior of human tibiae in the condi-tions of torsional tension. The data sug-gest a relationship between the torsionalmoment and the angle of torsion.It has been confirmed that the tensionup to about 400 Nm can be considered tobe linear, whereby when the tension ex-ceeds 400 Nm, it becomes alinear. Thefracture of the tibiae was observed in therange between 550 and 600 Nm, whilethe highest value of angle of torsion wasbetween 11 and 14°.Figure 2 shows the data obtained bymeasuring the relationships between dif-ferent samples over time. It can be ob-served that the angle of torsion is a verychangeable value that differs betweensamples. For the control group (I–IV) it11D. Mil~i} et al.: Bone Tissue Mechanical Properties, Coll. Antropol. 27 Suppl. 2 (2003) 9–15was 22.08°2.245, for the first experi-mental group (V–VIII) 17.83°3.116, forthe second experimental group (IX–XII)21.25°6.669, while for the last experi-mental group (XIII–XVI) it was 23.083°5.928.The measurement of torsional momentin relationship to time can be seen in fig-ure 3. It is obvious that the group treatedby 1,25-(OH)2D3 and 24,25-(OH)2D3 canwithstand greater torsional momentwhen compared to other experimentalgroups.The results in figure 4 represent therelationship of torsional hardness of thestudied groups. The groups treated by1,25-(OH)2D3 and 24,25-(OH)2D3 havehigher hardness when compared to the12D. Mil~i} et al.: Bone Tissue Mechanical Properties, Coll. Antropol. 27 Suppl. 2 (2003) 9–1501002003004005006000 2 3 3,9 4,9 5,1 6,9 8,8 12Torsional angle (degrees)torsionalmomenttorsional momentpoly. (torsional moment)Fig. 1. Dependence of torsional moment and torsional angle on human compact bone.051015202530355 10 15 20 25 30Days of treatmenttorsional angle for group I-IVtorsional angle for group V-VIIItorsional angle for group IX-XIItorsional angle for group XIII-XVItorsionalmomentFig. 2. Effect of time on torsional angle changes.control groups, but also when comparedto other experimental groups.Discussion and ConclusionThe relationship between bone struc-ture and its function has not yet beenestablished2,3. The osseous tissue is anentity that changes throughout life. Thisis especially related to module of elastic-ity changes of the compact bone,depending on the mineral composition ofthe bone. These changes are sudden andleap-like, so a mineral change of about 2to 3% can result in module of elasticitychanges to 60 to 100%. Changes of themodule of elasticity result in changes ofother physical values, and other physical13D. Mil~i} et al.: Bone Tissue Mechanical Properties, Coll. Antropol. 27 Suppl. 2 (2003) 9–1505101520253035405 10 15 20 25 30Days of treatmenttorsionalhardness(x10Pa)–2torsional hardness of group I-IVtorsional hardness of group V-VIIItorsional hardness of group IX-XIItorsional hardness of group XIII-XVIFig. 4. Effect of time on torsional hardness.01002003004005006007005 10 15 20 25 30Days of treatmenttorsionalmomenttorsional moment for group I-IVtorsional moment for group V-VIIItorsional moment for group IX-XIItorsional moment for group XIII-XVIFig. 3. Effect of time on torsional moment changes.properties as well. Therefore, in solvingand determining the physical propertiesof biological materials one must includepresumptions and simplifying proceduresthat cause substantial differences in com-puted and measured values9–13.Structural difference of osseous tissuecompared to crystalline materials, whosedeformation mechanism is explained bymultiplication of dislocatory fields, is notacceptable in fibrous materials like bone.In order to discover a suitable model forexplaning the mechanism of deformationdevelopment in osseous tissue Mufti}5,11had considered an assumption thatosteon fibers (cylindrical osseous units),connected one to another by osseous ma-trix, could be exhibiting two differenttypes of deformations: elastic and plasticdeformation phases. In the elastic phasethere is an ongoing elongation of osteonsand a relative returning movement whichis allowed by the matrix. If the tissue re-turns to its initial size, the total move-ment is considered to be an »elastic« de-formation. The phase of plastic defor-mation is characterized by the so-calledripping of the osteons out of the matrix,which results in the inability of the tissueto return to its initial size once the forcehas been removed. Therefore, permanentdeformation develops10–14.It can be concluded that 400 Nm is theborderline for linear deformation, and thatthe deformation becomes nonlinear withthe increase in force. The fracture thresh-old is 550 to 600 Nm, while the greatestangle of torsion is 11 to 14°.The torsional angle for the studiedgroups differs greatly from that of thecontrol group which is 20.56°. This couldmean that the group treated by 1,25-(OH)2D3 and 24,25-(OH)2D3 can sustain agreater torsional moment, when com-pared to other experimental groups. Itdoes not mean, however, that this groupexhibits greater hardness than othergroups, since this measurement has nottaken into account the geometrical char-acteristics of the cross-section of thebones.It can be, however, concluded, that thetorsional angle of the groups treated byvitamin D3 metabolic products signifi-cantly differs from the mean value of thecontrol group.The group treated by the combinationof both metabolic products has greaterhardness than the control group, as wellas the other studied groups.This experiment has undoubtedlyproved that the torsional angle depend-ing on the torsion moment can, withsubstanitial accuracy, determine the in-fluence of vitamin D3 on physical proper-ties of the osseous tissue. This should en-courage further studies of the influence ofthe mineral composition of the osseoustissue on physical properties of the saidtissue. Such studies might point to cru-cial data for the human and veterinarymedicine, but biomechanics as well.Due to the small sample sizes it wasnot possible to establish the criteria forstatistical significance, a fact that speaksin favor of extending the research in thisfield.14D. Mil~i} et al.: Bone Tissue Mechanical Properties, Coll. Antropol. 27 Suppl. 2 (2003) 9–15R E F E R E N C E S1. COWIN, S. C.: Bone mechanics. (CSC Press,Boca Raton, New York, 1991). — 2. CURREY, J.D., K.BREAR, P. ZIOUPOS, J., Biomechanics, 29 (1996)257. — 3. MUFTI], O., Strojarstvo, 15 (1973) 74. —4.FUNG, Y.C.: Stress-strain history relations of softtissues in simple elongation. In: Biomechanics. Itsfoundations and objectives (Prentice-Hall, EnglewoodCliffs, New York 1972). — 5. MUFTI], O., D.J. KerosDepartment of Dental Anthropology, School of Dental Medicine, Gunduli}eva 5,10000 Zagreb, CroatiaANALIZA MEHANI^KIH SVOJSTAVA KO[TANOG TKIVAS A @ E T A KU radu je utvr|en suodnos momenta svijanja i kuta svijanja tkiva zbite kosti pokus-nih `ivotinja i ~ovjeka. Temeljem mjerenja utvr|enih podataka za pokusne `ivotinjepretpostavljena je i zavisnost momenta svijanja za kost u ~ovjeka. Mjerenje je prove-deno na ~etiri skupine tkiva zbite kosti pokusnih `ivotinja. Pri tome su tri skupine `i-votinja dobivale razli~ite metabolite vitamina D3, a ~etvrta je skupina bila kontrolna.Zbog nemogu}nosti obra|ivanja ~ovjeka metabolicima vitamina D3, istovjetno je mjere-nje provedeno samo na jednoj skupini tkiva zbite kosti ~ovjeka. Pri mjerenju su za sveskupine ko{tanig tkiva `ivotinja utvr|ene funkcionalne veze izme|u kuta svijanja imomenta svijanja. Temeljem dobivenih rezultata predmijeva se kako bi se pona{alazbita kost u ~ovjeka tretiranog metaboliticima vitamina D3.Key words: vitamina D3, biomehanika, moment svijanja, kost, ko{tano tkivo15D. Mil~i} et al.: Bone Tissue Mechanical Properties, Coll. Antropol. 27 Suppl. 2 (2003) 9–15MIL^I], Sigurnost, 2 (1997) 67. — 6. SASSKI, N., N.MATSUSHIMA, T. IKAWA, A. FUKUDA, J.Biomechanics, 22 (1989) 157. — 7. VUKI^EVI], S.,^. BAGI, G. VUJI^I], B. KREMPIE, A. STAVLJE-NI], M. HERAK, Bone and Mineral, 1 (1986) 383. —8. KEROS, J., I. BAGI], @. VERZAK, D. BUKOVI]Jr., O. LULI]-DUKI], Coll. Antropol., 22 (1998) 195.— 9. MIL^I] D., Analysis of mechanical properties ofbone tissue based on its mucrostructure. M.Sc. Thesis(Faculty of Mechanical Engineering and Naval Archi-tecture, University of Zagreb, Zagreb 1997). — 10.MUFTI] O., D. MIL^I], J. SAUCHA, V. CAREK,Coll. Antropol. 24 Suppl. (2000) 97. — 11. JUR^E-VI], T., O. MUFTI], Coll. Antropol. 22 (1998) 585. —12. MIL^I] D., J. KEROS, J. SAUCHA, Z. RAJI], R.PEZEROVI]-PANIJAN, Coll. Antropol. 24 Suppl(2000) 15. — 13. SHIGLY, J.E.: Mechanical Engineer-ing Design (McGraw-Hill, New York, 1986). — 14.LOVASI], I., T. [KARI]-JURI], B. BUDISELI], L.SZIROVICZA, Coll. Antropol, 22 (1998) 307.

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