Journal of Biomechanics 31 (1998) 741745

Representative assessment of long bone shaft biomechanicalproperties: an optimized testing method

Jos A.M. Bramer!,*, Robbert H Barentsen", Maarten vd Elst!, Elly S.M. de Lange#,Peter Patka!, Henk J.Th.M. Haarman!

! Department of Traumatology, University Hospital Vrije Universiteit, Amsterdam, The Netherlands" Department of Clinical Physics and Engineering, University Hospital Vrije Universiteit, Amsterdam, The Netherlands# Department of Epidemiology and Biostatistics, University Hospital Vrije Universiteit, Amsterdam, The Netherlands

Received in final form 17 June 1998

Abstract

Whole bone bending tests are commonly used in mechanical evaluation of long bones. Reliable information about the midshaft canonly be obtained if the bending moment is uniformly distributed along the shaft, and if the distribution of the bending stress is notadversely influenced by rigid clamping of the bone ends. A testing device was developed to determine bending stiffness of long bones in24 directions, perpendicular to the bone axis. For optimal distribution of bending moment and stress, four-point bending wasperformed, and bone ends were simply supported, not rigidly clamped. The method was validated by repeated testing of a stainlesssteel rod, and a sheep femur. Leftright ratios were assessed twice in 2 groups of 5 sheep: one control group, and one group in whichthe left femur was stabilized with a stainless steel interlocking nail for 2.5 yr, after a midshaft osteotomy. Test results obtained with thesteel rod reproducibly were close to predicted values. Measurements with the sheep femurs were reproducible and precise for 3 of the4 parameters of the bending test. Stiffness parameters were significantly higher in the operated sheep than in the control group. Weconclude that the method described here provides accurate and reproducible information, which is representative for the long boneshaft. ( 1998 Elsevier Science Ltd. All rights reserved.

Keywords: Animal experiments; Long bone shaft; Mechanics; Four-point bending; Stress distribution

1. Introduction

The rigidity of long bones reveals differences if bendingtests are done in several directions of the transverse plane(Ruff and Hayes, 1983; Lovejoy et al., 1976). These differ-ences are even more pronounced after fracture fixation(Foux et al., 1993). Foux developed a method to assessthe distribution of the bone rigidity by performing three-point bending tests in 24 directions perpendicular to thebone axis (Foux et al., 1990). A disadvantage of three-point bending tests is the local deformation of the bone atthe site where the force is applied, resulting in an under-estimation of the Youngs modulus (Turner, 1993). More-over, the bending moment is maximal at this site, whichwill have a major effect on the test results. If a four-point

* Corresponding author. Tel.: 31-20-6151301; fax: 31-20-6151301.

bending test is used, the bending moment will be uniformbetween the applied forces (Fig. 1) and the weakest partof the shaft will determine the outcome of the test (Tim-oshenko and Goudier, 1970). Moreover, the influence ofshearing force will be reduced (Torzilli et al., 1981).

Furthermore, test results will be influenced by themethod of fixating the bone ends in the testing device.The theory of elasticity shows that rigid clamping ofa beam in a bending device, will result in maximal stressnear the points of fixation, and minimal stress in themiddle. More appropriate distribution of the stress, withthe maximum in the middle, will be achieved if the endsare allowed translation in the plane perpendicular tothe plane of bending (Fig. 2) (Timoshenko and Goudier,1970; Griffel, 1966).

In this report we describe a device for the assessment ofthe rigidity of long bones in a four-point bending test in24 planes, in which the bone ends are not rigidly fixed butsimply supported. This results in a uniform distribution

0021-9290/98/$19.00 ( 1998 Elsevier Science Ltd. All rights reserved.P I I S 0 0 2 1 - 9 2 9 0 ( 9 8 ) 0 0 1 0 1 - 8

Fig. 1. Bending moment distribution along the length of a beam in three- and four-point bending.

Fig. 2. Bending stress distribution along the length of a beam in case of fixated and simply supported ends.

of the bending moment, and a more appropriate distribu-tion of the bending stress along the shaft.

2. Device design

The long bone was placed with the ends in cylindricalmetal cups. The axis was centred. Fixation took place byfilling the cups with a low melting point Bismuth alloy (A301, Degussa, Wolfgang, Germany; melting point 47C)in the liquid state. The outside of the cups consisted of 24small facets, corresponding with the 24 facets of the ringsin which they were placed (Fig. 3). These rings had a lugon two sides which was placed on a saddle.

The device was positioned on a bending machine(Hounsfield H5000M, Hounsfield Test Equipment, Eng-land) which recorded load and deflection. A four-pointbending test was performed by applying two equal forcesat the edges of both cups. The cups were allowed transla-tion in the horizontal plane perpendicular to the plane ofbending. Bending was performed with a constant cross-head speed of 1 mm/minute. Deflection was measured atthe probe where the bending machine applied the force(F in Fig. 3). The test was nondestructive, no plastic

deformation was introduced. After each test the cupswere taken out of the rings, turned 15 and placed in thedevice again for the next test. This was repeated 24 times,untill a full revolution was made.

3. Validation of the method

The method was validated in four experimental set-tings:(A) A stainless-steel rod (asi no. 316), with a diameter of

8 mm, was tested 5 times in the device.(B) The entire procedure, including fixation of the bone

ends in the cups, and mounting of the device, wasrepeated 4 times with one sheep femur.

(C) Both femurs of five healthy adult sheep (controls)were tested to assess leftright differences. The entireprocedure was repeated for a second time to assessreproducibility of the method (duplo tests).

(D) Both femurs of five operated sheep were tested. Thesesheep received a stainless-steel interlocking nail 2.5 yrearlier stabilizing a midshaft osteotomy of the leftfemur. Before testing, the nail was removed. Fourfemurs of two sheep were tested twice (duplo tests).

742 J.A.M. Bramer et al. / Journal of Biomechanics 31 (1998) 741745

d"12Fx2y(EI)bone

# 12Fx33(EI)cups

(EI)bone

" [F/d]1@2x2y

C1!CF

dDx3

6(EI)cup3D

Fig. 3. Device for bending test in 24 directions: (1) tested bone;(2) metal cups and (3) corresponding rings; (4) lug at the side of therings; (5) saddle; (6) testprobes for applying force.

All tests were performed 8 weeks after the sheep weresacrificed. The femora were stored in 70% alcohol, andkept wet during testing (Linde and Sorensen, 1993). Allexperiments were approved by the DEC (Animal Experi-ment Committee) and carried out in accordance with theDutch regulations of Animal Welfare.

4. Processing of the test results

After linear regression on the loaddeflection curves,24 loaddeflection quotients were obtained for each fe-mur. Because the main contents of the cups consistedof the bismuth alloy, the elasticity of the alloy(4]109 Nm~2) was assumed for this part of the device.The flexural rigidity (EI) of the bone in every directiontested could be determined by the following equation,derived from beam theory.

Equation for calculating flexural stiffness (EI):

where x is the distance from the center of the lug to thepoint where the force was applied, y half the length of thelong bone shaft, F the applied bending force, d the dis-placement of the probe and I the moment of inertia.

For each femur the 24 EI-values were plotted in polarcoordinates and elliptical regression analyses was per-formed, resulting in an ellipse. The ellipses of both femursof one sheep were plotted in the same coordinate set,centralized and the ellipse of the right femur was mir-rored in the -axis.

With the use of semimajor axis (a), semiminor axis (b),and angle of inclination (a) of both ellipses (Fig. 4), thefollowing parameters were calculated:

Stiffness index:

SI"CEItestbone

EIcontralateralDminimal .This is the ratio of the stiffness of the testbone versus thecontralateral bone in the direction in which this ratio isminimal.

Area ratio:

AR" (ab)testbone(ab)contralateral

.

This ratio represents the total stiffness of the testbone, ascompared to the contralateral bone.

Flatness ratio:

FR" (b/a)testbone(b/a)contralateral

.

This represents the relative distribution of the stiffness ofthe bone in different directions.

Inclination difference:

ID"atestbone!acontralateral .This is the difference between the angles the semiminoraxis make with the anteriorposterior plane.

5. Results

(A) In repeated testing of the steel rod in the device,the mean value of flexural stiffness was

J.A.M. Bramer et al. / Journal of Biomechanics 31 (1998) 741745 743

Fig. 4. Ellipses of both femurs of operated sheep no. 4, plotted in the same set of coordinates. a"semi-major axis (e.g. 93.5 Nm2 for the left, and71.4 Nm2 for the right femur) b"semi-minor axis (e.g. 86.4 N m2 for the left, and 68.4 Nm2 for the right femur). a"angle of the semi-minor axis withAP-plane (inclination) (e.g. 145.62 for the left, and 122.7 for the right femur).

31.98 Nm2 (standard deviation 2.02), whereasthe calculated theoretical value was 35.39 Nm2.

(B) Repeated performance of the test with one sheepfemur resulted in mean values of 108.30 Nm2 forthe semimajor axis (S.D. 4.04), and 101.25 Nm2for the semiminor axis (S.D. 4.15) of the ellipse.The mean area was 109.72 N2m4 (S.D. 7.39), andthe mean flatness was 0.94 (S.D. 0.04). Inclinationshowed a mean value of 123.00 (SD 49.99).

(C)/(D) In the bending tests with the control sheep, theStiffness Index, Area Ratio, and Flatness Ratioall approached 1, as expected (Table 1). In theoperated sheep, the stiffness index and the ariaratio were significantly higher (p"0.009, MannWhitney test). The flatness Ratio in the operatedsheep approached 1, not differing from this valuein the control group (p"0.917). The Inclinationdifference appeared to be highly variable withinthe two test groups, and the difference betweenthese groups was not significant (p"0.754).

From the seven duplo tests of five control and twooperated sheep, the standard deviation of the differenceof the duplo, and the repeatability coefficient were cal-culated for each parameter (Bland and Altman, 1986).

For the Stiffness Index, Area Ratio, and Flatness Ratiothe standard deviations were acceptable (0.07, 0.15, and0.04, respectively) as were the repeatability coefficients(0.14, 0.30, and 0.08). However, the standard deviationand repeatability coefficient of the inclination differencewere extremely high (64.27 and 128.54).

6. Discussion

In vivo, long bones are mostly exposed to torsion andbending forces (Ruff and Hayes, 1983; Raftoupoulos andQassem, 1983; Bertram and Biewener, 1988), represent-ing the most common cause of long bone fractures(Lovejoy et al., 1976; Evans et al., 1951; Alms, 1961). Themajority of fractures occur in the middle third of the shaft(Dencker, 1965). Biomechanical evaluation in animal ex-periments should produce accurate information aboutthe whole shaft, especially the middle third.

In the presented method this was achieved by usinga four-point bending test without rigid fixation of thebone ends. This way the bending moment was uniformalong the whole length of the shaft, and the bendingstress was maximal in the shaft between the appliedforces, and minimal at the proximal and distal ends. No

744 J.A.M. Bramer et al. / Journal of Biomechanics 31 (1998) 741745

local deformation of the midshaft was introduced, be-cause the loads were not applied directly to the bone.

The stiffness index, area ratio, and flatness ratio pro-ved to be useful in comparing leftright differences indifferent groups of animals. These ratios approached 1 inthe control group, indicating no difference between leftand right femur. This symmetry of mechanical propertieswas demonstrated before (Kersey et al., 1994; Mather,1967; Sumner et al., 1988). It results in the contralateralfemur being the ideal control. The stiffness index andarea ratio were significantly higher in the operated sheep.The flatness ratio approached 1 in both the controls andthe operated sheep, meaning that there was no preferen-tial direction of rigidity. This might be explained by theintramedullary fixation of the fractures used here, mak-ing stress shielding and other possible effects similar in alldirections. In the duplo tests, the stiffness index, the arearatio, and the flatness ratio appeared to be very repro-ducible. The inclination difference showed a largevariation and a poor reproducibility, which might beexplained by the fact that the flatness ratio always ap-proached 1. This means that the ellipse of each boneapproached a circle, making inaccuracy in assessment ofthe direction of the semiminor axis more likely. If there isno preferential direction of the stiffness, the inclinationdifference seems not a very relevant parameter. However,if flatness ratios would not equal 1, this parameter mightproduce more reproducible and useful results.

We conclude that the described method accurately andreproducibly determines mechanical properties of thelong bone shaft. By optimal distribution of bending mo-ment and bending stress, appropriate information can beobtained about the complete shaft.

Acknowledgements

The authors wish to acknowledge the BiomaterialsGroup of the University of Leiden, the Netherlands,Klaas Boshuizen of the Dept. of Clinical Physics andEngineering, and Ger Vink and the other workers ofthe Clinical Animal Experimental Laboratory of theVrije Universiteit Amsterdam for their support in thisstudy.

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