PII: SOO21-9290(97)00007-9

TECHNICAL NOTE

J. Bmnrchnnu, Vol. 30, No. 1, 737 142 pp. 1YY7 (’ 1997 Elsevrrr Saence Ltd. All rqhts rescrvcd

Prmed in Great Britain OO?l 924w7 5 I7 00 + .a0

DEVELOPMENT AND VALIDATION OF A SERIES OF THREE-DIMENSIONAL FINITE ELEMENT MODELS OF THE EQUINE METACARPUS

C. M. Les,* J. H. Keyak,? S. M. Stover* and K. T. Taylor*

* Veterinary Orthopedic Research Laboratory, Department of Anatomy, Physiology, and Cell Biology. School of Veterinary Medicine, University of California, Davis, CA 95616-8732, U.S.A.; and t Biomechanics

Research Laboratory. Department of Orthopaedic Surgery, UC1 Medical Center, 101 The City Drive, Bldg. 29A, Orange, CA, U.S.A.

Abstract-Three-dimensional finite element (FE) models of the left metacarpi of five adult horses were developed from quantitative computed tomography data, using the algorithms of Keyak et al. (1990, J Biomed. Enyny 12, 389-397). The metacarpi were then equipped with 12 rosette strain gauges and loaded non-destructively in a mechanical testing machine. The bones and the models were loaded in axial compression, with the load evenly distributed across the distal row of carpal bones, and with a point load placed mediad to the sagittal midline, to a load equivalent to three times body weight ( - 15 kN); and in sagittal four-point bending to -2 kN. Maximum and minimum principal strains from the models were compared with those at the strain gauge rosettes.

There were significant (p < 0.001) and strong (0.69 < r < 0.90) correlations between predicted and observed surface principal strains, most often resolving as second- or third-order polynomial relationships. In most cases, particularly at extreme strain magnitudes, the models tended to overestimate the observed strain magnitudes.

These data suggest that the models are robust and accurate predictors of surface strains. Validation of these models lends further support for the use of this method of automated three-dimensional FE modeling, with its emphasis on accurate, individualized portrayal of structural geometry and material distribution, in research applications, and specifically for the use of these models in the study of the biology and mechanics of the equine metacarpus. #!“I 1997 Elsevier Science Ltd

Kegwords: Equine; Metacarpus; Finite element; Bone.

INTRODUCTION

The equine metacarpus is subjected to a harsh biomechanical environ- ment (Barnes and Pinder, 1974; Davies et al., 1993; Gross et al., 1992; Nunamaker et al., 1990; Rybicki et uI., 1977; Turner et al., 1975). with frequent failure in athletic animals (Estberg et al., 1996; Johnson et al., 1994; Nunamaker 1987; Peloso et (I/., 1994; Stashak 1987; Stover et ai., 1988b). It has been hypothesized that this extreme mechanical milieu may be responsible for some of the individual variation in metacarpal morphology (Les et al., 1995~; Nunamaker et ul., 1989), although certain aspects of this heteromorphism may be developmental in origin (Norwood and Haynes, 1982; Stover et al., 1988a).

One approach to understanding the implications of this environment is through the use of finite element (FE) models (Huiskes and Hollister, 1993). This technique is particularly powerful when the models accu- rately represent the geometry, material property distribution, and load- ing conditions encountered in tlitio. Before using such analytic models, however, it is important to compare their predictions with measure- ments taken from the structure itself, under defined loading conditions.

Properly validated, these models can be confidently used to estimate mechanical parameters at inaccessible sites, to study the effects of changes in geometry, structure, or material distribution, and to evaluate the mechanical implications of various fracture fixation strategies. The objective of this study was to develop a set of three-dimensional, individualized FE models representing the metacarpi of five adult horses, using an automated model-generation algorithm previously described (Keyak et al., 1990), and to compare the surface strain predictions of these models with actual surface strains measured in a variety of ex viuo loading conditions.

Received in final form 17 December 1996. Address correspondence to: C. M. Les, Department of Anatomy,

Physiology, and Cell Biology, School of Veterinary Medicine, Univer- sity of California, Davis. CA 95616-8732, U.S.A.

MATERIALS AND METHODS

Under animal use review committee approval, the left metacarpus (Second, third, and fourth metacarpal bones; MC2, MC3, MC4) and distal row of carpal bones were harvested within 5 min of euthanasia from 5 Thoroughbred horses (2 I-yr-old females, and 3-, 4-, and 5-yr- old geldings), and stored in 0.9% saline solution at -2O’C.

Each specimen was thawed and suspended in water, and 96-103 contiguous transverse computed tomographic (CT) images were takenf: (70 mA, 140 kVp, 3 s, 3 mm-thick slices, 1.08 x 1.08 mm pixels, 320 x 320 pixel image). A K2HP04 calibration phantom (Cann and Genant. 1980) was included in each image, and data were converted to K2HP04 equivalent densities (pppE). Specimens were then re-frozen.

Using an algorithm previously described (Keyak et al., 1990) and validated (Keyak et al., 1993) for the human femur, a three-dimensional FE model of each specimen (14,479-19,338 3 x 3 x 3 mm linear isoparametric isotropic elements, 18,034-24,140 nodes) was generated from the CT images (Figs 1 and 2). The elastic modulus for each element was calculated from the pppE of the ~0x1s in the element (Les et ul.. 1994). A 0.4 Poisson’s ratio was assumed.

Specimens were thawed and soft tissues removed. Distal MC3 was potted5 to the level of the palmaroproximal aspect of the metacar- pophalangeal joint capsule. Twelve strain gauge rosettesl’ were at- tached with cyanoacrylateli to the subperiosteal surface at the proximal diaphvsis (dorsal, palmar MC3; abaxial aspects of MC2. MC4). mid- diaphysis idorsal, palmar, medial, lateral kC3), and distal diaphysis (dorsal, palmar, medial, lateral MC3). A 0.8 mm Tungsten carbide bead** was glued with cyanoacrylate at the convergence of each

$ GE 9800 Research Scanner. General Electric, Milwaukee, WI. 5 Coe Tray Plastic. GC America Inc., Chicago, IL 60658. 7 Models CEA-06-062UR-350 and CEA-060125UR-350, Measure-

ments Group Inc., Raleigh, NC 27611. I/ M-Bond 200, Measurements Group, Inc.. Raleigh. NC 27611. **Industrial Techtonics, Inc. Dexter, MI 48130.

131

Technical Note

Fig. 1. Palmarolateral view of an FE model of the left metacarpus and distal aspect of the carpus from a 5-year-old Thoroughbred gelding. This model (including the steel plate, not shown) consists of 19,338 3 x 3 x 3 mm elements. and 24.140 nodes, and is fully constrained at the

distal end.

Fig. 2. Transverse mid-diaphyseal(50% metacarpal length) sections of each model, demonstrating individual variation in bone geometry and modulus. Bone material was divided, using data from quantitative CT scans, into 81 moduli; the scale indicates the minimum modulus for a given grey shade. There is a substantial amount of individual vari-

ation in both metacarpal geometry and material distribution.

Fig. 3. Mechanical testing assembly for XI F P loading cclnciitlon. iicsr- sal view. Proximal aspect is to the top. medial to the left. The lod 1s applied via a self-levelling platen to the proximal aspect of the <ted

plate.

rosette’s gauges. The assembly was radiographed (2 vIewA. 300 mA, 70 kV, 0.03 s) in an acrylic reference object. The images were digitized??, and three-dimensional coordinates of the beads were deter- mined using a direct linear transform (Abdel-Aziz and Karara. 1971: Marzan and Karara, 1975). From these data, the surface node closest to each rosette was determined.

The proximal 1 cm of the distal row of carpal bones was removed m a cutting jig to provide a flat surface, to which a 10 x IO x I cm steel plate was attached.11 The plate was represented in tbe FE models by 4624 3 x 3 x 3 mm cubic linear isotropic elements (modulus 200 GPa. Poisson’s ratio 0.4), in a 34 x 34 x 4 element array.

The assembly was tested under load control (1 kN s- ‘) in a servohyd- raulic mechanical testing machine,& using load conditions (Les C? ul., 1995b, 1996) that reproduced the in rim trot surface strain distribution reported at mid-MC3 diaphysis (Gross et a/., 1992; Rybicki et (11.. 1977: Turner et al., 1975). ‘Full-platen’ (FP) compressive loads of ~ 15 kN (approximately three times body weight) were applied via a self-levell- ing platen77 to the entire surface of the plate (Fig. 3). ‘Point-load medial’ (PLM) compressive loads of - 15 kN were applied to a point 0.5 cm mediad to the center of the plate with a 20 mm stainless steel bearing (Fig. 4). In the FP models. a - I5 kN load was divided evenly between the 1225 proximal plate surface nodes. PLM loads were distributed between five plate surface nodes at the contact site. The models were fully constrained at the potting level.

tt Digi-Pad 21A71D4, GTCO Corporation, Columbia, MD 21046. 11 Coe Tray Plastic, GC America Inc., Chicago, IL 60658. $4 Model 809.10: MTS, Eden Prairie, MN 55424. 77 Model 643.41C, MTS, Eden Prairie, MN.

Technical Note 139

Fig. 4. Mechanical testing assembly for a PLM loading condition, dorsal view. Proximal aspect is to the top, medial to the left. The load is applied via a 20 mm-diameter stainless-steel bearing to the proximal

aspect of the steel plate. 0.5 cm mediad to the plate center.

To evaluate model robusticity, constructs were tested in sagittal four-point bending (4PB). Specimens were removed from the compres- sion fixture and placed in a 4PB fixture (Fig. 5), with inner supports (15 mm long x 12 mm diameter steel cylinders) on the palmar aspect, and outer supports (160 x 12 mm steel cylinders) on the dorsal aspect. Distance between supports was adjusted to accomodate the extent of parallel dorsal and palmar surfaces. Preliminary testing suggested that failure could be expected at - 14 to -20 kN. Therefore, maximum 4PB loads were held to -2 kN. Mid-diaphyseal bending moments averaged 17.85 Nm (range 1620Nm). For FE models, each load application site was represented by 10 surface nodes enclosing an area of 12 x 3 mm. Models were fully constrained at the proximal dorsal support, and constrained in the dorsopalmar direction at the distal dorsal support. Dorsally-directed I kN loads were applied at each palmar support.

Twelve rosettes were monitored during compressive (FP, PLM) tests (One rosette in the distal arcade of one metacarpus failed during these tests.); only the four mid-diaphyseal rosettes were monitored during 4PB procedures. Observed minimum and maximum principal strains (s,mbsn, E~C) were calculated for each rosette at max- imum load.

Models were analyzed with commercial software. 1111 Maximum and minimum principal strains (EF, E?“) were calculated at the node closest to each rosette. Dorsal plane strains were used to calculate sur- face strains at dorsal and palmar locations. and sagittal plane strains were used for lateral and medial locations. Predicted maximum

11 I/ Abaqus 5.3, Hibbitt, Karlsson, & Sorensen, Inc, Pawtucket, RI.

Fig. 5. Mechanical testing assembly for a 4PB loading condition. me- dial view. Proximal aspect is to the right, palmar is to the top.

and minimum principal strains (sr, s;l,l” were adjusted to the actual load recorded during mechanical testing (e.g. c;“,“’ = actual load x EF/ 15 kN).

Maximum and minimum principal strains were pooled for analysis. A Pearson partial correlation,*** with horse, rosette site. and loading condition partialled as appropriate, determined the sfe-sobs correlation. An incremental polynomial regressionttt (IPR) for orders s-3 quanti- fied the relationships (Kleinbaum et al., 1988). Alpha levels of 0.05 were used to establish significance.

RESULTS

There were significant, strong, positive efe-sobs correlations (Table I). The sre-sDbs relationship appeared to be non-linear (Figs. 6 and 7). For data from all rosettes, IPR yielded third-order polynomials (monotonic where -5663 PE < sre < 3075 ~.a; maximum slope < 1). Data from the four mid-diaphyseal rosettes for all, compressive, and FP tests followed second-order polynomials (monotonic, and with maximum slopes i 1 throughout the data range). Data from the mid-diaphyseal rosettes for PLM tests followed a linear relationship (slope significantly < 1); and data for 4PB tests were best described by a second-order

polynomial (monotonic throughout the data range. slope < 1 when Q. < 180 p).

*** SAS CORR procedure, SAS Institute Inc., Cary, NC. ttt Sigmastat 1.02, Jandel Scientific, San Rafael, CA.

740 Technical Note

Table 1. Relationships between observed and predicted principal strains

Tests

Pearson partial

correlation i

Incremental polynomial regression, orders 0 ~3 e ohs = M, + M,cr, + M&e + M&,

MO Ml MZ M3 $

AU Rosettes (12lmetacarpus)

All tests* (n = 276) Compressive tests (n = 236) FP tests (n = 118) PLM tests (n = 118)

0.700 - 204.5 0.630 - 4.681E-5 - 1.206E-8 0.647 0.700 - 260.2 0.624 - 3.979E-5 - l.l30E-8 0.645 0.708 - 265.2 0.619 - 4.071E-5 - l.O50E-8 0.691 0.696 - 243.0 0.641 - 4.707E-5 - 1.395E-8 0.624

Mid-diaphyseal rosettes (4/metacarpus)

All tests* (n = 120) 0.845 Compressive tests (n = 80) 0.858 FP tests (n = 40) 0.895 PLM tests (n = 40) 0.840 4PB tests* (n = 40) 0.765

- 163.1 0.624 3.154E-5 - 0.724 - 287.7 0.634 5.409E-5 0.754 - 333.5 0.648 7.266E-5 - 0.816 - 170.2 0.524 0.705

- 39.1 0.829 4.757E-4 0.636

Note. Pearson partial correlations, corrected for pseudoreplication, and incremental polynomial regressions between cobs and sre, pooled data from five horses. Regressions marked with an asterisk are plotted in Figs 6 and 7. P values for all correlations and regressions < 0.001.

The models tended to overestimate the observed principal strain magnitudes, particularly at higher strain magnitudes (both compressive and tensile). There were significant, strong, and positive correlations between sot,* and sre, and in most cases, the relationship was best characterized by a second- or third-order polynomial, whose maximum slope was significantly less than one. For the PLM mid-diaphyseal data, however, the relationship was best characterized by a simple linear model.

3

2

.E 1

I^0

ES .c iz -, EB -0

:g

c -2

8 -3

-4

-5 T

AH locations P

Observed=Pmdlcted

-10 -0 -6 -4 -2 0 2 4 Predicted strain, microstrain

(Thousands)

Fig. 6. Comparison between observed and predicted principal strains. Circles represent data from FP tests, open squares represent data from PLM tests, and triangles represent data from 4PB tests. The dashed line (‘Observed = Predicted’) represents what would be a perfect linear correlation between s,as and sr,. The heavy solid line represents the best-fit third-order polynomial regression for all horses, all locations, all loading conditions; the dotted line represents the best-fit second-order polynomial regression for all horses, diaphyseal locations, all loading conditions; and the light solid line represents the best-fit second-order polynomial regression for all horses, diaphyseal locations, 4PB load condition (equations in Table 1). The models tended to be conservative, in that they consistently overestimated observed strain magnitudes,

particularly at extreme strains.

DISCUSSION dieted and observed values. Quantitatively, the relationship between predicted and observed values was best described by multi-order stat- istical constructs.

Our goal was to develop and validate a series of equine metacarpal The FE models tended to overestimate IE,~~/. particularly at extreme FE models, representing the geometry and modulus distribution found 1~~~) values. In effect, these metacarpi were more stiff at extreme condi- in five animals. Qualitatively, there was good agreement between pre- tions than was predicted by linear isotropic models. This phenomenon

Technical Note

Diaphyseal locations

All locations P

/’ /’

,I’

Obswved=Prediied I !

741

-10 -0 -6 4 -2 0 2 4 Predicted strain, microstrain

(Thousands)

Fig. 7. Fig. 6, regressions only.

was not due to local yield, which would result in lsfel values that would have underestimated 1~~~~1 at extreme conditions. Moreover, maximum observed strain magnitude ( -4263 WE) was less than half the compres- sive ayield of this material (Les et al., 1994).

Underestimation of the modulus under a rosette would tend to cause the FE models to overestimate laobsl. Such an underestimation could result from CT and/or model generation volume averaging. In addition, our CT calibration consistently predicts lower moduli for a given density than do relationships established for other bones (Carter and Hayes, 1977; Keller et al., 1990, 1993; Schaffler and Burr, 1988; Snyder and Schneider, 1991).

at the University of California, Davis, with funds provided by the Oak Tree Racing Association, the State of California satellite wagering fund, and contributions by private donors. Computer facilities were provided by a gift from Miss Lorna Talbot to the VORL, and by the donation of computer time on the facilities of the Department of Orthopaedics, University of California, San Francisco.

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742 Technical Note

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