Transactions on Biomedicine and Health vol 1, © 1993 WIT Press, www.witpress.com, ISSN 1743-3525
Validation offiniteelement models of
canine forelimbs before and after ulnar
osteotomy using experimentally determined
physiological loading
Z.M. Oden", R.T. Hart", D.B. Buif
& M.R. Forwood&
"Department of Biomedical Engineering,
Tulane University, New
Orleans LA,
USA
^Department of Anatomy, Indiana University,
Indianapolis IN, USA
As a portion of on-going research into the relationship between a bone's form and function,
finite element models of five canine forelimbs were built. The models were created using
computed tomography scans to determine both the bone geometry and the structural density of
each element (Hart et al [1]). The density of each element was then used to assign orthotropic
material properties to each of the elements (Oden et al [2]). A convergence test was performed
on one of the models to ensure accuracy. Validity of die models is demonstrated with * benchtop* loading and boundary conditions as well as physiological boundary conditions before and
after ulnar osteotomy. Future uses of these valid and accurate finite element models of the
canine radius and ulna include prediction of functional adaptation of the radius following ulnar
osteotomy.
INTRODUCTION
Bone, which provides structural and metabolic support for the body, alters its architecture in
response to a changing mechanical environment. These changes are referred to as functional
adaptation of bone. Through functional adaptation, bone adjusts itself to withstand the
increased demands of a professional athlete or to minimize its size when not used as in
immobilization or space flight (Jones et al [3], Uhthoff et al [4]). It has been experimentally
shown that adult bone will react to elevated mechanical strain with bone formation in the
marrow cavity and on the cortical-endosteal surfaces (Burr et al [5], Burr et al [6], Lanyon et
al [7], Rubin et al [8], Woo et al [9]). Preliminary computational studies have demonstrated
the capability to describe changes that occur in the functional adaptation of cortical bone
(Cowin et al [10], Hart [11], Parrish [12]). In order to apply the computational techniques to
bone, valid and accurate models representing the structure must be created. The theories also
require knowledge equilibrium strain environment of the bone under normal conditions.
Presented here is the development of finite element models of five canine forelimbs,
verification of their accuracy and validity, as well as the determination of physiological
loading parameters that will provide the appropriate equilibrium strain environment.
The radius and ulna has been a useful experimental model for studying functional adaptation in
bone (Burr et al [5], Burr et al [6], Lanyon et al [13]). A sample finite element model is
shown in Figure 1 with relevant anatomy and coordinate systems identified. An ulnar
osteotomy removes a porion of the ulna causing an overload of the radius. In response, due to
the altered strain environment, the radius will alter its shape. Therefore, this is a convenient
procedure to induce bone adaptation in vivo. Finite element models are used tofirstobtain
initial strain data. Then, using a remodeling finite element program, the various theories
which have been postulated can then be tested.
Transactions on Biomedicine and Health vol 1, © 1993 WIT Press, www.witpress.com, ISSN 1743-3525
248
Computational Biomedicine
METHODS
BUILDING THE FINITE ELEMENT MODELS
The computed tomography (CT) scan is a convenient tool to obtain the correct geometry of
bone. The excised bone is stored in a frozen saline bath while a CT scan is made. The ice
surrounding the bone serves the dual purpose of holding the bone rigidly in place while
reducing the 'beam hardening* effects associated with abrupt changes in radiopacity. Between
forty-seven and fifty-five slices, each several millimeters apart, along the long axis of the bone
were used to build the radius/ulna models.
The resultant image from a CT scan is an array of integers representing the Houndsfield units
of the scanned tissues or objects. The Houndsfleld numbers (H) are calibrated so that water (or
ice) has a CT number of 0 H and air is -1000 H. Soft tissues range from -100 H to 100 H and
cortical bone can range from 500 H to more than 2000 H (Ruff ef al [14]). The use of CT
can also be beneficial because the Houndsfield units yield information about the structural
density of the specimen.
A user interactive FORTRAN program, IMGRID was written for use on a Raster
Technologies RT Model One/10 terminal running on a VAX computer system (Digital
Equipment Corporation, Maynard, MA). IMGRID is used to extract the geometry of each
slice. The inner and outer edges of the bones can be identified, as well as the border between
cortical and cancellous bone if sufficient resolution exists in the image. Points along a
boundary, called grid points, are selected to represent the edge. This process must be repeated
for every CT slice in order to build a complete model of the bone. PATRAN, (PDA
Engineering, Costa Mesa, CA) a finite element pre- and post-processing program, is used to
generate the rest of the model. A file containing the grid points at each slice are read into
PATRAN and 'fit* with cubic curves. These curves are connected to form a two dimensional
surface of the bone called a 'patch/ Once patches have been created for every slice, they are
joined in the third dimension by layering the slices on top of each other, creating volumes
called 'hyperpatches.' The completed geometrical shape is the resulting set of hyperpatches.
These hyperpatches can then be paved with a desired density of elements, with each element
being assigned an appropriate material property. * Shape intrinsic* orthotropic material
properties (Cowin [15]) were assigned using a quadratic relationship between density and
modulus proposed by Rice et al for densities below 1.0g/cm3, and a power relationship
proposed by Snyder etal for densities greater than 1.0g/cm3.
E (GPa) = .06 + .9 p % for 0.0 < p al.O (Rice et al [16])
E (GPa) = 2.875 p * for 1.0 * p < 2.0 (Snyder et al [17]).
The above relationships were chosen following a study of the effect of different density to
modulus rules on finite element analysis results (Oden ef al [2]). Figure 1 contains a sample
finite element model.
DETERMINING ACCURACY OF THE FINITE ELEMENT MODEL
In order to ensure that thefiniteelement models are able to solve the mathematical problem
accurately, convergence tests are performed. The convergence test consists of progressively
increasing the number of elements in the model, and, while applying the same loading and
boundary conditions, comparing the finite element analysis results at specific points in the
structure. Theoretically, the results for displacement will converge to the exact solution
monotonically from below (Cook [18]). At greater and greater element densities, the change in
the result from one refinement to the next will become smaller and smaller. A final mesh is
selected as a balance of the best solution for the available computer resources. When the
difference between the results from one mesh density to the next are less than 5%, the mesh
can usually be considered sufficiently refined. A convergence test was performed on a statically
loaded model of dog 1 before any other testing was performed.
Transactions on Biomedicine and Health vol 1, © 1993 WIT Press, www.witpress.com, ISSN 1743-3525
Computational Biomedicine
249
i PROXIMAL END
PHYSIOLOGICAL LOADING
CANINE FORELIMB
FINITE ELEMENT MODEL, DOC 1
RADIUS
NODE H'
DISTAL END
ZERO DISPLACEMENT
BOUNDARY CONDITIONS
Figure 1: Finite element model of canine radius and ulna. Example
physiological loading and boundary conditions are marked.
DETERMINING VALIDITY OF THE FINITE ELEMENT MODELS
Next, the validity of the models, or how well they represented the real-world structure of the
canine radius and ulna, was determined for all of the models. The validity test consisted of
using an MTS 810 (MTS Systems Corporation, Minneapolis, MN) machine to apply a point
load to the bone at a reproducible location and direction and then experimentally measuring the
strain at six locations around the circumference of the radius and ulna. The load is applied by
'potting* the bone with Cerro Bend (Cerro Metal Products, Bellafonte, Pa), a low melting
temperature bismouth alloy, into a container that subsequently mounted onto an angled fixture
on the MTS machine. The angled apparatus can hold the bone at either 20 or 30 degrees from
horizontal. Six strain rosettes from Micromeasurements Group (Measurements Group, Inc.
Raleigh, NC), model SA-06-015RC-120 were applied to each forelimb. In addition to the
rosettes, two axial gauges, model CEA-06-24OUZ-120 (Measurements Group), were applied.
In all cases, three rosettes were placed around the circumference of both the radius and the ulna.
An attempt was made to place all six rosettes at the same distance from the distal end of the
bones. Although this was difficult on the ulna, due to the small size of the bone, complete
strain information for the whole circumference is only possible with three rosettes around the
circumference. The axial gauges were applied at various locations on either bone. Identical
loading and boundary conditions were assigned to the finite element models and the computed
strain at the same points around the circumference of the radius and ulna were compared to the
experimentally determined strains.
For two of the forelimbs, the testing was performed on the intact forelimb bones. For the
other two specimens, in order to ascertain the relative load sharing between the ulna and
radius, an initial test was performed on the intact structure, followed by an * osteotomy' of the
ulna. The identical loading test was then repeated. The rosettes on these bones then could
measure the before and after effect of an osteotomy on the radius and quantify the load sharing
Transactions on Biomedicine and Health vol 1, © 1993 WIT Press, www.witpress.com, ISSN 1743-3525
Computational Biomedicine
250
contributions of the two bones. Table 1 lists all of the validity tests conducted.
PHYSIOLOGICAL LOADING DETERMINATION
Prior to sacrifice, each of the canine subjects was outfitted with five strain rosettes on the
bones of the forelimb under the guidance of Dr. D. B. Burr at Indiana University. Three
rosettes were places around the circumference of the radius and two rosettes around the ulna.
All five were at the same cross sectional level. The animals were then walked across a force
plate where the magnitudes of the force in the global X, medio-lateral, Y, anterior-posterior,
and Z, proximal-distal, directions were measured. The magnitudes and directions of the loads
measured from the force plate can then be used as input to determine the approximate
physiological loading for the finite element models of the forelimb. The chosen loading is
assumed appropriate if the strains calculated in the finite element model are the same as the
experimentally measured strains.
It should be noted that this problem is not closed, there can be an infinite number of loading
combinations with the same resultant loads and boundary conditions that will yield the same
strains at the rosette locations. However, the main goal is to accurately reflect the strain
atmosphere throughout the bone for functional adaptation, particularly at the level of the
rosette application, and this can be achieved without the exact physiological loading. Points of
application of the load are chosen to imitate anatomical structures such as the humerus, upper
arm bone, which articulates on the surfaces of the radius and ulna, or muscle or tendon
attachments.
Using a trial-and-error approach, three different loading schemes were tested on all bones. In all
three cases the nodes at the distal end of the radius and ulna were assigned zero displacement
boundary conditions. First, the humerus was assumed to be at a 30 degree angle to the
forelimb, and all the forces were assumed to be transmitted through the humerus. The second
iteration used the same loading scheme, however, the modulus of elasticity assigned to the
interosseous membrane elements was more realistic, 1.4 X l(r N/cm^, rather than the
previously assigned bone material properties (Reuben et al [19]). The third attempt assumed
that the angle between the radius/ulna and the humerus was 43 degrees during stance as shown
by Charteris et al ([20]). The quality of the trial was determined by comparing the in vivo
measured strains to the strains calculated for the same points in the finite element model.
Table 1: Validity Tests Conducted
TEST NUMBER
MODEL
LOAD
1
DOC1-NORMAL
111.2 N
2
DOG1-NORMAL
177. 9N
3
DOG2-NORMAL
224. 6N
4
DOC2-OSTEOTOMY
224. 6N
5
DOG3-NORMAL
55. 6N
6
DOG3-NORMAL
66. 7N
7
DOG4-NORMAL
133.4N
3
DOG4-OSTEOTOMY
133- 4N
LOADING POINT
LOCATION
(prox. Up of
bone)
Anterior Edge of
Radius
Anterior Edge of
Radius
Pt. Load on A/L
Radius
Pt. Load on A/L
Radius
A/M Edge of
Radius
A/M Edge of
Radius
Anterior Edge of
Radius
Anterior Edge of
Radius
BOUNDARY
CONDITIONS
(zero dlsp.)
Distal 29mm of
radius
Distal 29mm of
radius
Distal 18mm of
radius
Distal 13mm of
radius
Distal 15mm of
radius
Distal 15mm of
radius
Distal 18mm of
radius
Distal 18mm of
radius
Transactions on Biomedicine and Health vol 1, © 1993 WIT Press, www.witpress.com, ISSN 1743-3525
Computational Biomedicine
251
RESULTS
Currently,fivefiniteelement models of the canine radius and ulna have been built However,
only four were available for mechanical testing. Therefore, the validity testing could only be
performed on the first four models. However, force plate data, without the corresponding strain
data was available for thefifthcanine. The physiological loading was applied to this model as
it was to the others.
CONVERGENCE TEST
The convergence test to determine accuracy of thefiniteelement models was applied to the
first canine model. Six continually more refined meshes, using 20-noded quadrilateral brick
elements, were used in the analysis until the convergence was achieved. Total displacement
and axial displacement at ten different nodes were measured. Figure 2 shows a plot of the total
displacement versus the degrees of freedom of the model at the point H, as shown in figure 1.
The difference between the total displacement in the second most refined model which contains
42,783 degrees of freedom (d.o.f.) and thefinalmodel containing 55,671 d.o.f, was 4.09%. At
all 10 nodes, for both axial displacement and total displacement the maximum percentage
difference between thefinaltwo mesh refinements was 6.08%. The same mesh density was
used for allfivesubsequent models. The final models contained an average of 3500 elements,
corresponding to approximately 58,000 degrees of freedom.
7
X
% 6-
t
Q
o0
10000
20000 30000 40000
Degrees of Freedom
50000
60000
Figure 2: Convergence test of the finite element model for DOG
point H.
1, at node
VALIDITY TESTING
For three of the four bones, all experimentally measured strains and computationally measured
strains agreed with respect to the sign of the strain, (compressive or tensile). The magnitudes
of the strain also matched very well in all but a few cases, with differences generally less than
12%, but with a few locations the differences approached as high as 30%. The difficulty in
choosing appropriate nodes for comparison may have contributed to the differences since the
strain gages average the strain over a small area and the nodal results are for only one point.
The gradients of the strain are very high, as can be seen infigure2. As a consequence, a node
chosen at a slightly different location may yield largely different results. However, these
models conclusively showed the proper trends. Table 2 contains the results for validity
testing of the finite element model of dog 1. Validity testing is currently being completed on
the final models before they will be used in further study of functional adaptation.
Interesting results were obtained from the validity studies which included in vitro
'osteotomy.' When a load was applied to the radius after osteotomy, the strain on radius was,
on average, one- fourth to one-third greater that the strain caused by the identical load on the
intact forelimb. This result was reflected in the finite element analysis.
Transactions on Biomedicine and Health vol 1, © 1993 WIT Press, www.witpress.com, ISSN 1743-3525
252
Computational Biomedicine
Table 2: Experimental versus finite element results for validity study on
intact forelimb of dog 1.
Location. Distance Experimental
Finite Element
Percent Differencefrom Distal End of Results. Axial e
Results. Axial E
(Exp e-FE e)/ Exp e
Porellmb
(pstr)
(MC)
A/L Radius. 106 mm
220
185
15.9%
A/M Radius. 106 mm
1087
1319
-21.3%
Post. Radius. 106 mm
-1880
-1894
-.01%
Ant. Ulna. 95 mm
-158
-170
-7.5%
P/L Ulna. 95 mm
-422
-513
-21.5%
P/M Ulna. 95 mm
89"
723
712.3%**
Ant. Radius. 50 mm
2546
2850
-12%
Ant Radius. 130 mm
1090
1171
-7.4%
The strain rosette which measured this value broke shortly after this reading was obtained.
The strain value measured is believed to be inacuraie.
PHYSIOLOGICAL LOADING
Strain data was available from Indiana University for four of the canine models. There were
five rosettes placed on the radius or ulna in all animals. However, due to the complication of
in vivo strain gaging and a necessary delay post-operatively before the force plate testing
could begin, not all of the rosettes functioned perfectly. Results from the rosettes that did
function properly were compared to the finite element results obtained at the same location.
This comparison was done for all three loading physiological schemes on all four radius/ulna
models. In all cases the third loading scheme, which distributed the resultant load onto the
radius and the side of the ulna, where the humerus articulates resulted in the best strain
comparison. This loading also accounted for the interosseous membrane in a more realistic
manner. For example, in the dog 1 forelimb model, a 95.7N load was distributed over 122
nodes on the ulna, at an angle of 43 degrees from horizontal. Another 230N load was applied,
spread over 33 nodes at the proximal end of the radius. Figure 1 shows the physiological
loading described above. In order to computationally simulate an ulnar osteotomy, a portion of
the ulna was removed. Figure 3 shows the axial strain of the forelimb before and after the
simulated osteotomy. Table 3 contains the finite element and in vivo results for a similar
loading scheme for dog 4. One noticable result shown in Table 3 is that the applied
physiological loading does, with finite element analysis, yield a strain pattern similar to that
measured in vivo. In most cases the calculated strains are within the range seen in the strain
gaging of the forelimb.
Table 3: Finite Element and in vivo results of physiological loading in
dog 4.
Location
Value
RA - Antero/Lateral
Surface of Radius
Maximum e
Minimum e
Axial e
Circum. e
Maximum e
Minimum e
Axial e
Circum. e
Maximum e
Minimum e
Axial G
Circum. e
RB - Antero/Mediai
Surface of Radius
RC - Posterior
Surface of Radius
In Vivo Strain
Range (pstr)
180-260
(-450M560)
150-200
(-420X540)
400-480
(-360H-890)
(-410)4700)
120-200
390-410
(-900M-1500)
(-550H750)
250-300
Finite Element
Results (pstr)
286.8
-611
202
-519
1147
-568.8
-545.4
112.3
484.8
-719.1
-581.0
346.6
Transactions on Biomedicine and Health vol 1, © 1993 WIT Press, www.witpress.com, ISSN 1743-3525
Computational Biomedicine
253
DISCUSSION AND CONCLUSION
Accurate and valid modeling is essential before finite element analysis results can be believed
(Huiskes et al [21]). The accuracy of the models was determined using an extensive
convergence test This project establishes the validity of finite element models of canine
forelimbs, both in vivo and in vitro. The unique opportunity to obtain force plate loads as
well as corresponding strain data in vivo, has allowed for determination of equilibrium, or
'normal* loading of the canine forelimb. Additionally, mechanical testing, in vitro further
validated the finite element models.
Now that these models have been validated, they provide a tool in the further study of bone
adaptation using both animal and computational models. The accurate and valid models will
now be used to describe and predict the functional adaptation of the canine radius.
DOG 1- INTACT ULNA
PHYSIOLOGICAL LOADING
AXIAL STRAIN
DOG 1- OSTEOTOMY MODEL
PHYSIOLOGICAL LOADING
AXIAL STRAIN
-.00500 I
-.00357 I
-.00214
-.000714 I
.000714
.00214 I
.00357
.00500J
Figure 3: Axial strain results of the physiological loading on DOG
before and after osteotomy.
1,
Transactions on Biomedicine and Health vol 1, © 1993 WIT Press, www.witpress.com, ISSN 1743-3525
254
Computational Biomedicine
ACKNOWUEDGEMENTS
This work has been supported, in part, by a grant from the Pittsburgh Super-computing
Center, MSM860007P; and the National Institutes of Health, AR40655.
REFERENCES
1. Hart, R.T., ZM. Oden, and S.W. Parrish, Computational methods for bone mechanics
studies. Int. Journal of Supercomputer Applications, 6(2): p. 164-174. 1992.
2. Oden, Z.M., R.T. Hart, and D.B. Burr. The Influence of Assumed Relationships
Between Bone's Mechanical Properties and CT Density on Results of Finite Element
Analyses.PTQceedings of the Twelfth Southern Biomedical Engineering
Conference, New Orleans, LA.p.57-59. 1993.
3. Jones, H.H., J.D. Priest, W C. Hates, C.C. Tichenor, and D.A. Nagel, Humeral
hypertrophy In response to exercise. J. Bone and Joint Surg, 59-A: p. 204-208. 1977.
4. Uhthoff, H.K. and Z.F.G. Jaworski, Bone loss in response to long term immobilization.
Journal of Bone and Joint Surgery, 60B: p. 420-429. 1978.
5. Burr, D.B., M.B. Schaffler, K.H. Yang, M. Lukoschek, N. Sivaneri, J.D. Blaha, and
E.L. Radin, Skeletal change In response to altered strain environments: Is woven bone a
response to elevated strain? Bone, 10: p. 223-233. 1989.
6. Burr, D.B., M.B. Schaffler, K.H. Yang, D.D. Wu, M. Lukoschek, D. Kandzari, N.
Sivaneri, J.D. Blaha, and E.L. Radin, The effects of altered strain environments on bone
tissue kinetics. Bone, 10: p. 215-222. 1989.
7. Lanyon, L.E., A.E. Goodship, C.J. Pye, and J.H. MacFie, Mechanically adaptive bone
remodeling. Journal of Biomechanics, 15(3): p. 141-154. 1982.
8. Rubin, C.T. and L.E. Lanyon, Regulation of bone mass by mechanical strain
magnitude. Calcified Tissue International,37: p. 411-417. 1985.
9. Woo, S.L.Y., S.C. Kuei, W.A. Dillon, D. Amiet, F.C White, and W.H. Akeson, The
effect ofprolonged physical training on the properties of long bone —a study of Wolffs Law.
Journal of Bone and Joint Surgery, 63-A: p. 780-787. 1981.
10. Co win, S.C., R.T. Hart, J.R. Balser, and D.H. Kohn, Functional Adaptation in long
bones: Establishing in vivo values for surface remodeling rate coefficients. Journal of
Biomechanics, 18(9): p. 665-684. 1985.
11. Hart, R.T., Computational Techniques for Bone Remodeling, in Bone Mechanics.
S.C.Cowin, Editor. CRC Press: Boca Raton, p. 279-304. 1989.
12. Parrish, S.W., Computational Prediction of in vivo Bone Remodeling., Senior Honor's
Thesis, Department of Biomedical Engineering, Tulane University, New Orleans, LA. 1988.
13. Lanyon, L.E., P.T. Magee, and D.G. Baggott, The relationship of functional stress and
strain to the process of bone remodeling. An experimental study on the sheep radius.
Journal of Biomechanics, 12: p. 593-600. 1979.
14. Ruff, C.B. and F.P. Leo, Use of computed tomography hi skeletal structure research.
Yearbook of Physical Anthropology, 29: p. 181-196. 1986.
15. Cowin, S.C., Mechanics of Materials, in Bone Mechanics. S.C. Cowin, Editor. CRC
Press: Boca Raton, FL. p. 15-42. 1989.
16. Rice, J.C., S.C. Cowin, and J.A. Bowman, On the dependence of the elasticity and
strength of cancellous bone on apparent density. Journal of Biomechanics, 21(2): p.
155-168. 1988.
17. Snyder, S.M. and E. Schneider, Estimation of mechanical properties of cortical bone by
computed tomography. Journal of Orthopaedic Research, 9: p. 422-431. 1991.
18. Cook, R.D., Concepts and Applications of Finite Element Analysis. Second
Edition.New York: John Wiley and Sons. 1981.
19. Reuben, J.D., J.E. Akin, and F. Hou, The effect of the fibula and interosseous
membrane complex on proximal tibial stress distribution. Trans. Orthopaedic
Research Society/14) p. 201. 1989.
20. Charteris, J., D. Leach, and C. Taves, Comparative kinematic analysis of bipedal and
quadrupedal locomotion: a cyclographic technique. Journal of Anatomy, 128(4): p. 803819. 1979.
21. Huiskes, R., and E. Y. S. Chao, A survey of finite element analysis in orthopaedic
biomechanics: the first decade. Journal of Biomechanics, 16(6) p. 385-409. 1983.