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Bone stress in the horse forelimb during locomotion at different gaits: A comparison of two experimental methods
1. Domechomcs Vol
16. No
8. pp 565-576.
OOZI-9290 83 13.00 +
1983 g
Prmted m Great Bntam
00
1983 Pergamon Press Ltd.
BONE STRESS IN THE HORSE FORELIMB DURING LOCOMOTION AT DIFFERENT GAITS: A COMPARISON TWO EXPERIMENTAL METHODS
OF
A. A. BIEWENER Museum of Comparative Zoology, Harvard University, Cambridge, MA, U.S.A. J. THOMASON Royal Ontario Museum, University of Toronto, Ontario, Canada A. GOODSHIP Department of Veterinary Science, University of Bristol, U.K. and L. E. LANYON Department of Anatomy, Tufts University, Boston. MA, U.S.A. Abstract-Longitudinal stresses acting in the cranial and caudal cortices of the radius and the dorsal and palmar cortices of the metacarpus in the horse were determined using two independent methods simultaneously. One approach involved the use of rosette strain gauges to record in uiuobone strain; the other involved filming the position of the horse’s forelimb as it passed over a force plate. Agreement between the two analyses was better for the radius than for the metacarpus. Both methods showed the radius to be loaded primarily in sagittal bending acting to place the caudal cortex in compression and the cranial cortex in tension. At each gait the magnitude of peak stress in each cortex based on the film/force analysis was 1.5-2 times higher than that determined from the bone strain recordings. In the metacarpus, the magnitude of stress in each cortex calculated from the film/force method was 2-3 times greater at each gait than that shown by the bone strain recordings. However, whereas the film/force analysis indicated that the metacarpus was loaded in sagittal bending (acting to place the pahnar cortex in compression and the dorsal cortex in tension), the bone strain recordings showed the metacarpus to be loaded primarily in axial compression at each gait. Because the film/force method depends on an accurate measure of limb segment orientation relative to the direction of ground reaction force, comparatively small errors in calculations of bending moments may lead to a significant difference in the level and distribution of stress determined to act in the bone’s cortices. The discrepancy in metacarpal loading obtained by the two methods may be explained in part by the simplicity of the biomechanical model which, for instance, neglected the force exerted by the sesamoids on the distal end of the metacarpus. The records of stress determined from the in oiuobone strain recordings showed that each bone was subjected to a consistent loading regime despite changes of gait. Such a consistent strain distribution should allow these bones to maximize economy in the use of tissue required to support the dynamic loads applied. Peak stresses measured from the bone strain recordings in the radius during locomotion at constant speed ( - 40.8 k 4.1 MN m-‘) were significantly larger than those in the metacarpus ( - 25.1 f 2.8 MN m- *), regardless of speed and gait. During acceleration and deceleration, however, peak stress rose dramatically in the metacarpus ( -40.6+ 3.4 MNm-*) but remained constant in the radius ( - 37.8 f 5.8 MN me2). This suggests that when thecommonlyencountered loadingconditions likely to cause the highest strains are taken into account. both bones have similar safety factors to failure.
NOMENCLATURE F, (N) RI, (N)
R (NJ R, (N) R, (N) M, (N m) M,, (N m)
PI (N) P, (N)
vertical component of ground reaction force horizontal component of ground reaction force resultant ground reaction force acting on distal end of bone axial component of ground force acting on bone transverse component of ground force acting on bone moment exerted by ground force about carpal joint moment exerted by ground force about metacarpo-phalangeal joint
Receiced 28 Augur
1981: in recisrdform
18 January
x1 (m) x2 (m) 4 (m’) 1 (m4) c (m) u, (MN m-‘) a, (MN m-‘) 0 (degrees)
1983. 565
force exerted by extensors about carpal joint force exerted by extensors about metacarpophalangeal joint moment arm of extensors at carpal joint moment arm of extensors at metacarpophalangeal joint cross-sectional area of bone at its midshaft area moment of inertia for bending in the antero-posterior plane at the midshaft of the bone maximum distance from neutral plane of bending to the periosteal surface of the cortex at midshaft stress due to compression stress due to bending angle of maximum principal stress to the longitudinal axis of the bone
566
A. A. BIEWENER, J. THOMASON, A. G~~DSHIPand L. E. LANYON INTRODUCTlON
At present two methods are available for assessing the forces transmitted through bones and across joints. One technique involves determining the animal’s limb position from high speed tine film and using a force plate to measure the components of force exerted by each animal’s foot while it is in contact with the ground. The external forces and moments acting at each joint and the resulting bone stresses can then be calculated using the geometry of the limb and the material properties of the bone. The second approach is to measure directly in uiuo strain developed within the bone using rosette strain gauges attached to the bone’s surfaces. From these strain measurements, the stresses and hence the forces within the bone can be calculated using the same procedure as in the previous analysis, only in reverse. Both approaches have their limitations. Strain gauges can only respond to the deformation of the bone’s surface to which they are directly attached. The calculation of local stresses from local strains is based on experimental values for the physical properties of bone tissue. By using published values of longitudinal and transverse elastic moduli, shear modulus and Poisson’s ratio, the anisotropy of the bone can be taken into account and reasonable confidence placed in the stress values obtained. Extrapolation to the loading of the whole bone involves assumptions not only of its material and structural properties but also simplification of its manner of loading. If gauges are attached to the midshaft of the bone it is reasonable to assume that there are no stress concentrations such as could arise from local muscle insertions. To calculate forces from stresses and vice versa, in both the strain gauge and the film/force method it is assumed that the bone is homogeneous throughout its cross-section and along its length. The film/force method also assumes that the point of application of the ground force acting on the foot is known. While it is evident that details of the loading regime and the resulting stresses that act in a bone may be more accurately determined from rosette strain gauges, the film/force analysis has the advantage of being non-invasive and applicable to animals which are small or unsuitable for surgery. This technique has been used by a variety of investigators interested in the forces and stresses developed in limb muscles, tendons and bones in different animals such as: kangaroo rats, dogs, kangaroos and humans (Biewener et al., 1981; Alexander, 1974; Alexander and Vernon, 1975; Morrison, 1970) and in the forces acting across the knee and hip joints of humans (Paul, 1966, 1967; Morrison, 1968), as well as experimental hip prostheses in humans (Paul, 1973). The rosette strain gauge approach has been used to measure in vivo bone strain in the radius and tibia of sheep (Lanyon et 01., 1979; Lanyon and Bourn, 1979), the radius of pigs (Goodship et al., 1979), the tibia and radius of horses and dogs (Lanyon and Rubin, 1980), the tibia of man
(Lanyon et al., 1975). the femur of dogs (Carter et al., 1981), as well as the mandible of primates (Hylander, 1979). A knowledge of the distribution and magnitude of forces and stresses within the skeleton is a necessary prerequisite to an understanding of how bones, or joints, are designed to resist the loads applied. Moreover, a knowledge of the manner in which forces are transmitted across joints, and the stresses which are developed in the articulating bones, is relevant to an understanding of the mechanical factors that may influence the deterioration and ‘ultimate failure of articular cartilage or bone, as well as to the design of replacement prostheses. In this study we addressed two questions relevant to the structural design of the radius and metacarpus in the horse: (1) Are the manner of loading and amplitude of stress similar for these two bones at a given speed and gait and (2) does the stress distribution within each bone change with gait and speed? In attempting to answer these questions, we employed both the strain gauge method and the film/force method simultaneously. This study provides the only opportunity to date for comparing these two methods. Since the anatomy of the horse forelimb is less complex than in other species, the comparison between the film/force analysis and direct strain measurement we employed for the radius and metacarpus should provide the best chance for agreement between the two techniques. In more proximal locations and for more complex limbs, such as the hip and knee in humans, the assumptions required by the film/force analysis are more extensive and the possibility of error greater.
MATERIALS AND METHODS
Data collection
Three small horses weighing 268, 281 and 291 kg were trained to be led at a constant speed over a Kistler force plate mounted in a rubber covered, concrete track 45 m in length. The force plate was designed to withstand a peak force in the vertical direction of 49 kN. Passes over the force plate were made at three gaits and at four different speeds: walk, slow trot, fast trot and canter. Velocity was determined from an electric clock triggered by two photocells mounted 10 m apart, before and after the force plate. The horses were filmed in lateral view using light cinematography at 64 f s- i as they moved over the plate. The animals’ hooves were whitened for contrast against the black rubber matting and a white background was used for contrast against the animals’ limbs. The vertical, horizontal, and lateral components of force, as well as the raw strain data, were recorded using an FM tape recorder and were synchronized with the film using a clock placed in the camera’s field of view which marked pulses on the tape at one second intervals.
567
Bone stress in the horse forelimb during locomotion Once a series of pre-operative
force recordings
had
been obtained, rosette strain gauges were attached to the cranial and caudal cortices of the right radius and to the dorsal and palmar cortices of the right metacarpus of each animal. The gauge preparation and operative procedure was similar to that previously described by Lanyon (1976). Anaesthesia was induced with thiopentone sodium and maintained, after intubation. with a halothane/oxygen/nitrous oxide mixture. The areas to which gauges were attached were approached from the medial side of the limb. Surgical interference of the soft tissues caused some postoperative lameness. especially in the case of the metacarpus where the suspensory ligament had to be retracted to attach the palmar gauge. Phenylbutazone was included in the animals’ diet as a post-operative analgesic. A measure of the relative degree of lameness suffered by the animals due to surgery was that the post-operative records of vertical force were 75-80 “/, of the pre-operative forces recorded at each gait. Following the surgery, two or three days were allowed for recovery, after which simultaneous bone strain, film and force recordings were made on each of four days. The lengths of the bones, the position and orientation of the strain gauges relative to the bone’s longitudinal axis, the moment arms of tendons about joints, the moment arms due to the curvature of the bones, and at the bone’s midshaft: the cross-sectional area, area moment of inertia, and maximum distance from the neutral plane of bending to the periosteal surface were determined from radiographs or from post-mortem dissection.
The film records were analyzed using a Vanguard motion analyzer which digitized the coordinates of the joints for each frame. The corresponding vertical and horizontal components of the ground force for each frame were also determined (the lateral component of the ground force was neglected because its magnitude was small compared to that of the vertical and horizontal components of the ground force). Figure 1 shows a drawing of the forelimb position and corresponding ground force vector acting at the hoof for three frames of film of a representative stride over the force plate at a trot (4.3 m s ‘). The vertical (F,) and horizontal (Fh) components of the ground force are shown below. Both sets of data were entered into a PDP-11 microprocessor to calculate the moments and forces acting on each bone. The resulting stresses developed at the midshaft were then calculated based on the bones’ physical dimensions and an assumed
c Rodlus
1 1
b
Metocorpus
L
Fig. 1. A drawing of the lateral aspect of the radius articulating with the metacarpus. The transverse (R,) and axial (R,) components of the resultant ground force (R) acting on the distal end of each bone are shown. The forces exerted by the extensor muscles acting about the carpal (PI ) and metacarpophalangeal (P2) joints that are necessary to counteract the moments exerted by the ground force about each of these joints (M, and kf,,,, respectively) are also shown along with their correspondmg moment arms (x1 and x,). The orientations of P, and P2, determined from radiographs and dissection, were found to act parallel to the longitudinal axis of each bone. The position of the limb corresponds to the second frame (no. 5) drawn in Fig. 1.
longitudinal elastic modulus of 18.2 GN m-2. This value represents the mean ( _t S.D. = 0.20) for equine cortical bone determined by Schryver (1978).
frame number 5 shown in Fig. 1. The forces exerted by the extensor muscles acting about the carpal (PI) and metacarpo-phalangeal (P2) joints to counteract the moments exerted by the ground reaction force about each of these joints (M, and Mm,,, respectively) are also shown along with their moment arms (x1 and xz). No activity was assumed in antagonistic muscles during the support phase. Moreover, inertial forces due to angular accelerations of these limb segments were not calculated and were considered to be relatively insignificant. The stress due to axial compression in the case of radius is then given by
A drawing of the radius and metacarpus showing the transverse (R,) and axial (R,) components of the ground force (R) acting at the distal end of each bone is presented in Fig. 2. The orientation of the bones relative to the ground force vector corresponds to
where A is the cross-sectional area of the radius at its midshaft; and the stress due to sagittal bending is
A. A. BIEWENER,J. THOMSON, A. G~~DSHIP and L. E. LANYON
568 Fast
(4 3 ms-I)
trot
Frame
I 0
1
I
01
02
Time
1
03
(s)
Fig. 2. A drawing of the forelimb position of a horse for three frames of film during a stride over the force plate at a trot of 4.3 m s- ‘. Recordings of the vertical (F,) and horizontal (Fk) components of the ground reaction force acting at the hoof arc shown below. The dashed lines indicate the values of F, and F, corresponding to each frame shown.
sectional geometry at the midshaft to determine A and I rather than assuming an ideal geometrical shape. The measurement of r was made directly from radiographs and represents the perpendicular distance taken from the chord, drawn from the proximal to the distal end of the bone, to the center of the bone at its midshaft. The measurement of A and I was made from photographs of bone cross-sections at the midshaft which were magnified and traced along the endosteal and periosteal surfaces, using a digitizing table to enter the geometric data into the microprocessor. The bone strain data were entered into the microprocessor from the magnetic tape using an A/D converter. The magnitude and orientation of each principal strain at the gauge site was then calculated using standard formulae (Dally and Riley, 1965). The corresponding stresses in the longitudinal and transverse material directions of each bone were determined based on the anisotropic analysis of principal bone strain data described by Carter (1978). This procedure considers bone to be an orthotropic material that is transversely isotropic and assigns values for the transverse elastic modulus (12.2 GN m-‘) and shear modulus (3.4 GN m-‘) in proportion to the longitudinal elastic modulus of bone according to the results of Reilly and Burstein (1975). The values of the transverse elastic and shear moduli were calculated from the values of Poisson’s ratio determined for compact bone by these workers (0.46 f 0.14 when maximum principal stress is in the longitudinal direction and 0.31 f0.09 when it is in the transverse direction), using a value of 18.2 GN m-* for the longitudinal elastic modulus of compact bone (Schryver, 1978). The components of bending and compressive force acting on the radius and metacarpus were then calculated and compared with the values obtained from the film/force plate method.
given by d = b
[RJ+(Ra+P,)rlc I
(2)
where 1is the moment arm of the ground force about the midshaft of the radius, r is the moment arm of the force exerted by the extensors about the midshaft due to bone curvature, c is the distance from the neutral plane of bending to the surface of the cortex and I is the area moment of inertia (for bending in the anteroposterior direction) at the midshaft. The line of action of the extensor muscles and their tendons was found to be closely parallel to the longitudinal axis of both the radius and metacarpus. Consequently the force exerted by them was not considered to contribute to the net transverse component of force acting on each bone (see Appendix). However, in the case of the radius the axial component of the ground force (R,) and the force exerted by the extensor muscles (PI) do, in fact, induce a bending moment about the bone’s midshaft due to its curvature. The analysis described here is similar to that of Alexander (1974), except that it accounts for the curvature of each bone and uses the actual cross-
RESULTS
Figure 1 shows the anatomical relations of the radius and metacarpus in the forelimb. The ulna, which is fused to the caudal aspect of the radius, is greatly reduced so that in the middle forearm the radius provides the only resistance to axial force between the elbow and carpal joints. The third metacarpal (or ‘cannon’ bone) is similarly the single weight bearing bone between the carpal and metacarpo-phalangeal joints. The first and fifth metacarpals are not present and the second and fourth are small and attached along the sides of the third metacarpal to which the gauges were attached. From a lateral view the radius shows a slight caudally concave curvature whereas the metacarpus is relatively straight. Representative records of longitudinal stress determined to occur in the dorsal (D) and palmar (P) cortices of the metacarpus at each gait based on the bone strain recordings are presented in Fig. 3 and compared with equivalent records (dashed curves)
Bone stress in the horse forelimb
Metacarpus Walk
25
?J
0
i 5 -
-25
Trot 50
r
8OOr
L
Time(s)-
0.2
10
Fig. 3. Representative records of longitudinal stress acting in the dorsal (D) and palmar (P) cortices of the metacarpus at a walk, trot and canter based on the in uivo bone strain recordings (solid curves) and compared with equivalent records (dashed curves) calculated to occur for each frame of film from the film/force analysis. The net bending (Fb) and compressive (F,) components of force calculated to be acting on the bone are also shown based on each technique.
during
locomotion
569
calculated to occur in each frame from the film/force analysis. The net bending (Fb) and compressive (F,) forces determined to be acting on the bone are also shown. Because the forces exerted by the extensor muscles acting about the metacarpo-phalangeal joint are parallel to the longitudinal axis of the metacarpus, such forces only contribute to the compressive component of force acting on the bone. Consequently, the bending force is due entirely to the transverse component of the ground reaction force (R,) acting on the bone (Fb = R,). The net compressive component of force is equal to the axial component of the ground reaction force (R,) acting on the bone plus the force exerted by extensor musculature (P2) acting about the metacarpo-phalangeal joint (F, = R, + Pz). In the palmar cortex the pattern of stress determined by each method is similar for each gait despite a large difference in the peak magnitude developed. For the dorsal cortex, however, the development and pattern of stress calculated to occur by each analysis is quite different. The bone strain recordings show that stress in the dorsal cortex is compressive at each gait with its peak occuring early in the support phase at a walk, but midway at a trot, and late at a canter. In contrast, the film/force analysis shows stress in the dorsal cortex to be tensile for most of the time the foot is on the ground. Only during the latter half of the support phase of a walk does this analysis show that the ratio between compression and bending is sufficient to place both cortices in net compression. The peak tensile stress in the dorsal cortex using this analysis is shown to occur early in the support phase at all gaits, coinciding with the peak compressive stress in the palmar cortex. The reason for this discrepancy in the stress regime indicated for the metacarpus by these two analyses becomes apparent if the net bending and compressive forces are considered. F, is very similar both in its amplitude and in the pattern of its rise and fall for each method of analysis, but F, varies considerably. Based on the bone strain recordings, F, is found to be initially negative at a walk. indicating a bending force acting to place the dorsal cortex in compression, and then swings positive, acting to place the dorsal cortex in tension. At higher speeds the negative phase of F, is small or absent and bending is in the palmar direction for most of the support phase. The film/force analysis, however, shows F, to be 2-4 times larger and always remains positive with a peak soon after the foot lands on the ground. The fact that differences in bending forces have a greater influence on stress values than similar differences in compressive forces explains both the discrepancy observed in the magnitude of stress as well as the discrepancy in stress regimes shown by the two analyses. In fact, the difference in peak F, correlates exactly with the differences in peak stress determined by each method. Figure 4 similarly compares representative records of the longitudinal stress acting in the cranial and caudal cortices of the radius, as well as the bending and compressive forces acting on the bone determined
571
Bone stress in the horse forelimb during locomotion
Table 1. Peak longitudinal bone stress (MN m-‘). A comparison of the peak stress developed in the dorsal and palmar cortices of the metacarpus and in the cranial and caudal cortices of the radius at four different speeds determined by two independent methods of analysis. The ratio of peak stress in the pahnar (or caudal) cortex for each analysis is indicated (values are for the three horses grouped together)
Metacarpus
Bone strain analysis peak stress ( f S.D.) N = 10 Dorsal/palmar
Film/force analysis peak stress ( f S.D.) N = 5 Dorsalipalmar (k9.5)
Ratio (palmar stress)
-6.3 (kl.l)/-16.8
(k3.9)
32.3 (*8.3)/-49.0
Slow trot (1.7-2.3 ms-‘)
-8.9 (_f9.1)/-17.7
(k3.8)
27.0 (+ 10.8)/ -48.7 (+ 13.3)
1:?.8
Fast trot (3.84.6ms-‘)
-11.4 (*6.1)/-21.3
(+6.5)
39.6 (+ 14.0)/-60.6 (i 15.6)
1z2.9
Canter (nl) (4.5-5.8 m s- ‘)
-14.0 (*3.1)/-25.1
(k2.8)
38.1 (+ 18.5)/-68.5 (+ 16.7)
1:2.7
Canter (lead) (4.5-5.9 m s _ ’ )
-13.8 (*2.9)/-21.4
(f4.3)
60.4 I* 14.3)/-84.0 (k23.5)
1:4.0
Cranial/caudal
Ratio (caudal stress)
Radius
Cranial/caudal
Walk (1.4-1.8 ms-‘)
14.2 (*3.2)/-21.2
(k5.7)
32.9 (&2.7)/-45.2
(k3.0)
1:2.1
Slow trot (1.7-2.2 ms-‘)
17.1 (+4.9)/--34.6
(f3.8)
31.3 (a5.0)/-45.2
(f3.0)
1:1.4
Fast trot (3.9-4.5 ms- r)
20.6 (*6.3)/-40.8
(k4.1)
38.5 (10.8)/ -60.0
(+ 10.6)
1:1.5
Canter (nl) (4.45.6 m s _ ’ )
16.1 (+5.6)/-36.1
(k4.4)
53.2 (+15.6)/-76.4
(k18.2)
1:2.1
Canter (lead) (4.65.8 ms-‘)
12.7 (_+4.6)/-28.9 (k2.6)
48.1 (+6.8)/-65.8
(k7.2)
1:2.3
da1 direction) acting on the radius during the second half of the support phase, whereas the film/force method suggests that this peak occurs early in the support phase at a time similar to the rise in Fb determined for the metacarpus. In fact, based on the bone strain recordings, it is the bending moment exerted by the compressive force about the midshaft of the radius due to its caudally concave curvature which causes the peak in stress due to bending that is observed at each gait. The level of F, in fact passes through zero at this time. During the first half of the support phase F, is initially negative, acting to counteract the bending engendered by the curvature of the radius caudally, but during the second half of the support phase it swings positive and then augments the bending due to F, acting about the curvature of the bone at its midshaft. The peak longitudinal stress data (mean &-S.D.) determined by each technique for the metacarpus and radius in the three horses combined are compared in Table 1. The four speeds shown represent three different gaits. At each gait and speed the values of peak stress in the metacarpus (palmar cortex) calculated from the film/force analysis are about three times greater than those from the bone strain recordings (again, this is largely due to the film/force analysis showing that the metacarpus is loaded in cranio-caudal
_
1:2.9
Walk (I.41.7ms-‘)
bending when in fact the strain gauge results show that it is not). In the radius (caudal cortex) the discrepancy in peak stress is less; the film/force results showing an overestimate varying from 1.5-2 times that shown by the strain gauges. The values of peak stress shown in Table 1 have relatively large standard deviations, indicating that there is a significant amount of variation between individuals. This is especially true for the metacarpus. In two individuals the component of stress due to bending was low, whereas in the third it was much higher. Within an individual, however, the stress acting in each cortex of a bone at a given speed is consistent. Table 1 also shows that the values determined from the bone strain recordings are less variable than those calculated from the film/force analysis, although fewer determinations were made from the latter method. Both sets of data show a general increase in peak stress in each bone as speed increases. However, the proportional increase in peak stress from that at a walk to the maximum value at either a fast trot or a canter is relatively small. Over a four-fold increase in speed the bone strain analysis shows a proportional increase in peak stress of 1.5 times walking values for the metacarpus (non-lead leg, canter) and 1.9 times for the radius (fast trot). The film/force analysis predicts a proportional increase of 1.7 times for each bone (lead
572
A. A. BIEWENER,J. THOMSON, A. G~~DSHIP and L. E. LANYON
leg, canter for the metacarpus and non-lead leg, canter for the radius). In contrast to the trot in which the movement of contralateral limbs is symmetrical, the canter is an asymmetrical gait. In a trot the contralateral pair of fore (and hind) limbs land approximately 180” out of phase with one another, with an aerial phase in between. The movements and ground contact time of each forelimb are similar during their respective support phases. In a canter, or a gallop, the two forelimbs land slightly out of phase with one another, with an aerial phase after, but not in between, the support phases of the forelimbs. The non-lead leg is the first to land, and it is the lead forelimb, which lands second, that throws the animal into the next aerial phase. The phase difference between the two limbs is speed dependent and decreases as speed increases from a slow canter to a fast gallop (Gambaryan, 1974; Hildebrand, 1965). Since the two forelimbs land at different times and act asymmetrically in the stride, during a canter and gallop values of peak stress are given for the experimental limb when it was used as the non-lead, as well as the lead foreleg. For a canter at a given speed the bone strain analysis shows that peak stress is greater in both the metacarpus and radius of the non-lead leg (on which the animal lands from the aerial phase) than on the lead leg (which throws it into the air). This difference between the lead and non-lead legs is slightly greater in the radius (207,$ p c 0.01) than in the metacarpus (15 y,, p < 0.1). The fact that the difference in the non-lead and lead leg metacarpus is not significant for the data from all three horses combined is misleading. Within an individual horse the difference in the magnitude of stress between its non-lead and lead legs is highly significant (p c 0.01) for both the metacarpus and radius. Values of peak stress acting in the palmar cortex of the metacarpus and the caudal cortex of the radius for each of the two horses for which data was obtained from the non-lead and lead legs of a canter are shown in Table 2. The results presented here are similar to those reported for the difference in principal
strain acting in the caudal cortex of the lead and nonlead radius of a horse while changing gaits from a trot to a canter during treadmill locomotion (Lanyon and Rubin, 1980). The film/force analysis agrees with the strain gauge data by showing a similar difference in stress for the lead and non-lead radius. In the metacarpus however the two methods disagree. In this bone the film/force analysis shows that the magnitude of stress is greater in the lead leg. Does the loading regime change with gait?
Representative records of longitudinal stress determined from the in oiuo bone strain recordings for the cranial and caudal cortices of the radius and the dorsal and palmar cortices of the metacarpus at each gait are shown in Fig. 5. The non-lead and lead forelimbs of a canter are again presented separately. The vertical component of the ground force (F,) is included as well. These records are regular both in pattern and magnitude. The standard deviation of the mean of the peak stress for a series of such records (N = 10) for an individual horse at a given speed is typically less than 87; of the mean. The manner of loading of both the radius and metacarpus remains the same at different gaits, with a general increase in the magnitude of peak stress as speed increases. The radius is subjected predominantly to sagittal bending, acting to place its caudal cortex in compression and its cranial cortex in tension. The metacarpus however is loaded primarily in axial compression. The small amount of sagittal bending that occurs contributes to the compressive stress in the palmar cortex of the bone but is insufficient to place the dorsal cortex into tension. Despite the differences which occur in the magnitude of stress with speed, gait change and limb sequence, the loading patterns are remarkably consistent. Moreover, changes in the magnitude of peak stress observed in both cortices of the metacarpus and radius at each gait, and between the non-lead and lead legs of a canter, are closely correlated with similar changes in the vertical component of the ground force.
Table 2. A comparison of the peak palmar (or caudal) stress acting in the non-lead versus lead metacarpus and radius of two individual horses at a canter. The level at which the difference between the means of peak stress in the non-lead versus lead leg metacarpus and radius are significant, based on a two-tailed r-test (9 degrees of freedom), is given to the right in each case. The data shown are based on the in uioo bone strain recordings of each bone
Metacarpus Horse A Horse B Radius Horse A Horse B
Lead leg Non-lead leg Peak palmar stress + SD. (MN m- ‘) -23.1 f 1.6 -28.4k2.1
-17.5k2.4 -25.4kO.5
Peak caudal stress + S.D. (MN m - ‘) - 38.7 f 2.9 -37.1 k4.0
-28.5 + 3.5 -29.3k1.2
N = 10 p < 0.01 p < 0.01 N= 10 p = 0.001 p < 0.01
Bone stress in the horse forelimb during locomotion Walk ? E % m-
zlil 82 “;,
Trot
Canter
3020 IO-
c,
0 -to c,
-20 -3o-
-4o-
Non- lead
Time
lead
u 0.5 s
573
A more quantitative analysis of how the orientation and loading vary in both of these bones at different gaits is obtained by comparing the stress due to bending (a,) with that due to compression (a,). The magnitude of shear stress is indicative of the orientation of the load since a torsional load and/or an eccentric load applied will cause the axis of the principal stress to deviate from a longitudinal orientation. These data are presented in Table 3. The only significant shift in the loading regime of either bone occurs at the walk-trot transition. As the horse changes gait the magnitude of the compressive component of stress increases and bending is reduced. After this transition the ratio of bending to compression is maintained over the range of speeds measured, including the change of gait from a trot to a canter. The angles of peak principal stress show that the orientation of the principal stresses do not change more than 4” within the cortices of either bone over the range of speeds observed. At each gait the longitudinal shear stress that does occur in both bones is due primarily to torsion. Accordingly, the values of peak longitudinal shear stress in the cortices of each bone do not change significantly during normal locomotion and remain quite low. DISCUSSION
Fig. 5. Representative records of longitudinal stress determined from the in uivo bone strain recordings for the cranial (C,) and caudal (C,) cortices of the radius and the dorsal (0) and palmar (P) cortices of the metacarpus at three different gaits. Records of stress for the non-lead and lead legs of a canter are presented as well for comparison. The vertical component of the ground reaction force (F,) is also shown in each case.
The two independent methods used in this study to determine the pattern and magnitude of locomotory stresses acting in the radius and metacarpus of the horse forelimb yielded significantly conflicting results. These differences were more pronounced for the metacarpus but in each bone depended largely on
Table 3. A comparison of the ratio of bending stress (uJ to compressive stress (a,) at four different speeds and three gaits for each bone. The shear stress and the angle of the principal stress to the bone’s longitudinal axis in the dorsal (0,) and palmar (0,) cortices of the metacarpus and the cranial (0,) and caudal(f7,) cortices of the radius are also presented. The consistency of each set of data illustrates that the loading regime is uniformly maintained in each bone despite a change of gait Ratio Qb:b, Metacarpus Walk Slow trot Fast trot Canter (nl) Canter (lead)
l:I.4 1:4.3 1:3.7 1:4.1 114.2
Radius Walk Slow trot Fast trot Canter (nl) Canter (lead)
Shear stress (MNm‘*) Dorsal/palmar -l.6/-0.7 -0.8/-1.8 -1.31-2.0 -2.1/-1.7 -2.O/- 1.8
edieP (degrees)* - 10.3/-2.5 -9.5/-3.7 -9.o/-3.1 -7.91-2.1 -ll.l/-2.9
Cranial/caudal 5.2: 1 3.2: 1 3.1:1 2.8:1 3.1:1
+ l.l/+ 1.6 +1.8/+1.8 +1.8/+1.7 f 1.2,‘+2.4 i t.2/+2.4
i2.4/+5.3 +4.5/+3.1 +5.0/+3.1 + 1.3/+4.8 +0.8/+4.3
* ( - ) indicates distal lateral rotation and ( + ) indicates distal medial rotation relative to the proximal end of the bone.
514
A. A. BIEWENER,J. THOMASON,A. GOODSHIPand L. E. LANYON
differences in the antero-posterior forces calculated to act on the bone. Greater sagittal bending was indicated by the film/force technique than was determined from the in uiuo bone strain recordings. Both analyses agreed that the radius was loaded predominantly in bending, but the film/force technique predicted peak stresses that were 1.5-2 times greater in magnitude than those derived from the bone strain data. For the metacarpus, not only did the film/force analysis yield peak stresses more than two-fold greater in magnitude, but it indicated that the metacarpus was subjected primarily to bending; whereas the in vivo bone strain data showed that, during steady state locomotion, it was maintained in axial compression. Determination of the bending component of force (Fb) acting on a bone by the film/force method is susceptible to error because measurement of the net compressive and bending components of force depends not only on joint position and the point of application of the ground reaction force, both of which are subject to error, but the orientation and assumed activity of muscles acting across the joint. Though the error in bending force may be comparatively small, the difference it makes to the magnitude and distribution of stress calculated toact at the bone’s midshaft may be quite large. Based on the bone strain recordings, F, ranged from 100 N at a walk to 200 Nat a canter in the metacarpus, and from 200 to 400N in the radius. These values were more than an order of magnitude less than the corresponding values of F,, which ranged from - 5000 to - 9000 N in the metacarpus and from - 4000 to - 8500 N in the radius. It can be seen, then, that in terms of absolute magnitude, a given error in the value of F, compared to a similar error in the value of F, will not have nearly as significant an effect on the stress regime calculated to occur in a bone. Thus, a difference as small as 400 N calculated for F, by each method (which can occur from an error of only a few degrees in the orientation of the ground force and muscle force vectors, see Figs 3 and 4) may result in the prediction of an entirely different stress distribution. This was presumably the case for the discrepancy in stress calculated for the metacarpus. These potential sources of error in a relatively ‘simple’ mechanical system close to the measured ground reaction force brings into question the use of this technique for more complex locations. The large discrepancy in metacarpal loading obtained by the two methods, however, can be diminished to some extent if the model includes the force exerted by the extensor tendons, via the sesamoid bones, on the distal end of the metacarpus. The magnitude of this force depends on the angle of the distal portion of the extensor tendons, which insert on the distal phalangeal bones, to the longitudinal axis of the metacarpus. This transverse loading of the metacarpus (in the dorsal direction), however, will be resisted, at least in part, by a reaction force transmitted by the first phalangeal bone to the metacarpus across the metacarpo-phalangeal joint. Because the orien-
tation of the distal extensor tendons to the metacarpus is difficult to measure accurately, and there is uncertainty in the determination of their net contribution to bending, we chose not to estimate the force exerted by the sesamoids on the distal end of the metacarpus. Clearly, with a knowledge of what the stress distribution at the bone’s midshaft is meant to be, it is possible to progressively modify the model to obtain a good fit. However, the bone-bonded strain gauge technique, although providing a more direct and detailed measurement of the strain (and hence, the stress) acting in a bone’s cortex, also has its limitations. Most importantly, the strain data provided by each gauge are only relevant to the area of bone to which the gauge is adherent. To extrapolate to a loading regime for the bone’s diaphysis requires the assumptions that the site where strain is measured is representative of the manner in which the bone is loaded, and that cortical bone is homogeneous in its properties along the bone’s length. When considering the midshaft of the radius and metacarpus, we beheve these assumptions to be reasonable and that the strains recorded at the cranial and caudal gauge sites represent the peak strains transmitted through the length of the bone during normal locomotion. However, more detailed studies of the strain distribution along a bone’s length and around its circumference must be conducted to more fully substantiate these assumptions. Another disadvantage of the strain gauge technique, borne out by the difference between the pre- and post-operative force plate recordings, is that normal function can be impaired by the surgery involved. In this study, however, whereas the magnitude of peak stress was reduced by approximately 20’;,,, it seems unlikely that this significantly affected the pattern and orientation of stress in either bone. Given that errors in determining the bending forces acting on a limb bone are due to estimates of the orientation of limb segments relative to the direction of ground force, this suggests that the differences in loading obtained by each method should be greater for more proximal bones, further away from the point of application of the ground force. However, the closer agreement obtained in this study for the horse radius at each gait is due to the inherent concave curvature of the radius along its caudal aspect, which results in the compressive component of force (F,) exerting a bending moment about the bone’s midshaft. At each gait the magnitude of this bending moment was typically two to three times greater than the moment exerted by F,. Consequently, because the magnitude of F, measured by each technique was similar, each analysis showed the radius to be subjected predominantly to bending, despite differences in the magnitude and pattern of F,. The importance of a bone’s curvature engendering a significantly large bending moment about its midshaft, expecially one greater than that due to the transverse component of the reaction force acting at a joint, has not been generally recognized. Alexander (1974, 1975) does not take bone curvature into account in his
Bone stress in the horse forehmb during locomotion
analysis of bone stress in the tibia and femur of a dog jumping or the kangaroo hopping. In the present work, if the curvature of the radius had not been considered, the film/force analysis would have predicted a much smaller magnitude of bending. as well as a more variable pattern in the bending moment about the bone’s midshaft. This would have suggested a totally different distribution within the cortices of the bone. A study of functional stresses within the skeleton is of interest not only because of their mathematical relationship to the loads involved but also because bone is a tissue which is capable of remodelling in response to changes in the mechanical load applied to it. However, it is not clear whether the precise stimulus for remodelling is the magnitude of the stresses developed (Goodship et al., 1979), their distribution (Lanyon et nl., 1982) or the rate at which they change (Lanyon and O’Connor, 1980). In the case of an animal, such as a horse, which uses different gaits during locomotion, the orientation and manner of loading might have been expected to vary significantly. If this had been the case it would have important implications for the design of a bone; requiring greater bone mass more generally arranged about its longitudinal axis in order to resist equally the diverse range of loads imposed. However if the applied loading regime remains uniform, the bone’s design requirement is simplified, thus permitting a structure with the minimum tissue needed to support the load effectively. This may be particularly important for the horse, the evolution of whose limbs has been directed to an increase in the length but a reduction in the number of its distal elements. The records of stress determined from the bone strain recordings show that the metacarpus is loaded primarily in axial compression and the radius in sagittal bending and that the pattern and orientation of stresses within each bone remains uniform despite changes of gait and speed. Our results agree with those reported for single element strain gauge recordings during walking and trotting in each of these bones (Turner et al.. 1975). In this study gauges were attached to not only the cranial and caudal cortices but the medial and lateral cortices as well. Since the stimulus for bone remodelling is assumed to arise from the strains developed within the bone when it is loaded (Bassett, 1968). a bone’s architecture in relation to the manner of loading would be the result of these repetitively imposed strain patterns. The difference in peak stress values developed in the radius (-41 MNm-‘) and in the metacarpus ( - 25 MN m-*) when considering steady state locomotion would appear to contradict the view that a bone’s remodelling objective is to achieveand maintain a uniform safety margin to failure. However, this apparent discrepancy in peak functional stress is eliminated if acceleration and deceleration are considered to be common enough occurences to influence bone architecture (Rubin and Lanyon, 1981). The peak
515
stress in the palmar cortex of the metacarpus during acceleration ( - 40.6 F 3.5 MN m- ‘) is substantially higher than during steady state locomotion. and is nearly equal to that which occurs in the caudal cortex of the radius during steady state locomotion. The radius on the other hand does not show an increase in peak stress during acceleration or deceleration ( - 37.8 IS.8 MN m- * in the caudal cortex). The question remains, however, as to why the radius develops a caudally concave curvature while the metacarpus does not. If the goal of designing a structure is to minimize the stresses acting within it, to maximize its safety margin, then this could most easily be achieved if the radius were a much straighter bone. Instead. by developing a curvature, the peak stresses at the radial midshaft during steady state locomotion are elevated to a similar level as those developed in the metacarpus during acceleration and deceleration. Since bone loss would be expected in either bone as a result of a decrease in its individual functional strain level. this suggests that perhaps there is an optimum range of peak functional strain. The upper value of this range would be determined by the appropriate safety factor to either monotonic or fatigue failure. and the lower value by some benefit conferred on the bone tissue by being intermittently strained at or above a ‘threshold’ level. Such a benefit could be related to the more effective movement of fluid, transporting nutrients and waste products throughout the tissue (Piekarski. 1981), to the generation of strain related electrical potentials, or to some other advantage conferred to the cells or their matrix. While this proposal must be speculative at this time, it is in agreement with the evidence which we have presented here. although it is contrary to conventional ideas which have assumed that most features of bone architecture are developed to maximize safety margins or economy of tissue. Acknow[rdyements-This work was supported by NIH Training Grant T32GM07117-04 awarded to Harvard University, NIH Grant AM1814Oawarded to Dr. C. Richard Taylor, and by a grant awarded to the University of Bristol Veterinary School by the Horse Race Betting Levy Board. The assistance of Mrs. Maureen Redcoe and Mr. Peter Walker was greatly appreciated.
REFERENCES
Alexander, R. McN. (1974) The mechanics of jumping by a dog (Canisfamilioris) J. zool. Res. 173, 549-573. Alexander. R. McN. and Vernon. A. (1975) The mechanics of hopping by kangaroos (Macropodidae). J. :ooi. Res. 177, 265-303. Bassett, C. A, L. (1968) Biological significance of piezoelectricity. Co/c. Tiss. Res. 1, 252-272. Beers, F. P. and Johnston, E. R. (1977) Vector Mrchunicsjor Engineers. 3rd Ed. McGraw Hill, New York. Biewener, A. A., Alexander, R. McN. and Heglund, N. C. (1981) Elastic energy storage in the hopping%f kangaroo rats (Dipodomys spectabilis). J. zoo/. Res. 195. 369-383. Carter, D. R. (1978) Anisotropic analysis of strain rosette information from cortical bone. J. Biomechanics 11, 199-202.
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BIEWENER,J. THOMASON, A. G~~DSHIP and
Carter, D. R., Vasu, R., Spengler, D. M. and Dueland, R. T. (1981) Stress fields in the unplated and plated canine femur calculated from in uioo strain measurements. J. Biomechanics 14, 63-70. Dally, J. W. and Riley, W. F. (1965) Experimental Stress Analysis. McGraw-Hill. New York. Gambaryan, P. PI (1974) How Animals Run. John Wiley, New York. Goodship, A. E., Lanyon, L. E. and McFie, H. (1979) Functional adaptation of bone to increased stress. J. Bone Jt Surg. 6lA, 539-546. Hildebrand, M. (1965) Symmetrical gaits of horses. Science 150, 701-70s. Hylander, W. L. (1979) Mandibular function in Galago crassicaudatus and Macacafasicularis: an in uiuo approach to stress analysis of the mandible. J. Morph. 159.253-296. Lanyon, L. E. (1976) The measurement of bone strain in oiuo. Acia orthop. be/g. 42 suppl. 1, 98-108. Lanvon. L. E.. Manee. P. T. and Bannott. D. G. (19791The reiationship’of ft.&ional stress and-&in to the processes of bone remodelling. An experimental study on the sheep radius. J. Biomechanics 12, 593-600. Lanyon, L. E. and Bourn, S. (1979) The influence of mechanical function on the development and remodelling of the tibia. An experimental study in sheep. J. Bone .Jt Surg. 6lA, 263-273. Lanyon, L. E. and O’Connor, J. A. (1980) Adaptation of bone artificially loaded at high and low physiological strain rates. J. Physiol., Lond. 303, 36P.
Lanyon, L. E. and Rubin, C. T. (1980) Loading of mammalian long bones during locomotion. J. PhysioL, Lo&. 303,72P. Lanyon, L. E., Goodship, A. E., Pie, C. P. and McFie, H. (1982) Mechanically adaptive bone remodelling. J. Biomechanics 15, 141-154. Lanyon, L. E., Hampson, W. G., Goodship, A. E. and Shah, J. S. (1975) Bone deformation in uiuo from strain gauges attached to the human tibia1 shaft. Acta orthop. stand. 46, 256268. Morrison, J. B. (1968) Bio-engineering analysis of force actions transmitted by the knee joint. Biomed. Engng 3,164. Morrison. J. B. (1970) The mechanics of muscle function in locomotion. i Biomechanics 3, 431451. Paul, J. P. (1966) The biomechanics of the hip and its clinical relevance. Proc. R. Sot. Med. 39, 10. Paul, J. P. (1967) Forces transmitted by joints in the human body. Proc. Instn. mech. Engrs 181, 3. Paul, J. P. (1973) Design aspects of endo-prostheses for the lower limb. Perspectiues in Biomaterial Engineering, (Edited by Kenedi, R. M.), pp. 91-94. Macmillan, London. Piekarski, K. (1981) Mechanically enhanced perfusion of bone. Mechanical Properties of Bone, Vol. 45, pp. 185-191. ASME symposium.
L. E. LANYON
Reilly, D. T. and Burstein, A. H. (1975) The elastic and ultimate properties of compact bone tissue. J. Biomechanics 8, 39345.
Rubin, C. T. and Lanyon, L. E. (1981) Bone remodelling in response to applied dynamic loads. Trans. orrhop. Res. Sot., 55,237-238.
Schryver, H. F. (1978) Bending properties of cortical bone in the horse. Am. J. vet. Res. 39. 2>28. Turner, A. S., Mills, E. J. and Gabel, A. A. (1975) In oioo measurement ofbone strain in the horse. Am. J. uet. Res. 36, 1573-l 579.
APPENDIX Equations used to calculate stress due to (Al) compressive forces and (A2) bending forces. cc =
Rcos0+Pcosa A
(Al)
where R is the resultant vector of the ground force acting on the distal end of the bone, 6 is the angle between the vector R and the bone’s longitudinal axis, P is the resultant vector of force exerted by extensor muscles, and a is the angle between the vector P and the bone’s longitudinal axis. In the case of the radius and metacarpus, a = 0. Substituting R, = R cos 0, (Al) reduces to R,+P
6, = -
A
e~=[(RsinB+Psina)l+(RcosB+Pcosa)r]c
,
WI
1
where R, 8, P, and a are defined as above, I is the orthogonal distance from the distal end of the bone to its midshaft, and r is the orthogonal distance bisecting the chord drawn from the proximal to the distal end of the bone (along which the axial forces were defined to be acting), to the center of the bone at its midshaft. In the case of the radius, a = 0. Substituting R, = R sin fl and R, = R cos 0, (A2) reduces to
u = b
[RJ+R.+P)rlc I
In the case of the metacarpus, a = 0 and r = 0. Again, substituting R, = R sin 9 and R, = R cos& (AZ) reduces to: R, Ic ub=---. I
16. No
8. pp 565-576.
OOZI-9290 83 13.00 +
1983 g
Prmted m Great Bntam
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1983 Pergamon Press Ltd.
BONE STRESS IN THE HORSE FORELIMB DURING LOCOMOTION AT DIFFERENT GAITS: A COMPARISON TWO EXPERIMENTAL METHODS
OF
A. A. BIEWENER Museum of Comparative Zoology, Harvard University, Cambridge, MA, U.S.A. J. THOMASON Royal Ontario Museum, University of Toronto, Ontario, Canada A. GOODSHIP Department of Veterinary Science, University of Bristol, U.K. and L. E. LANYON Department of Anatomy, Tufts University, Boston. MA, U.S.A. Abstract-Longitudinal stresses acting in the cranial and caudal cortices of the radius and the dorsal and palmar cortices of the metacarpus in the horse were determined using two independent methods simultaneously. One approach involved the use of rosette strain gauges to record in uiuobone strain; the other involved filming the position of the horse’s forelimb as it passed over a force plate. Agreement between the two analyses was better for the radius than for the metacarpus. Both methods showed the radius to be loaded primarily in sagittal bending acting to place the caudal cortex in compression and the cranial cortex in tension. At each gait the magnitude of peak stress in each cortex based on the film/force analysis was 1.5-2 times higher than that determined from the bone strain recordings. In the metacarpus, the magnitude of stress in each cortex calculated from the film/force method was 2-3 times greater at each gait than that shown by the bone strain recordings. However, whereas the film/force analysis indicated that the metacarpus was loaded in sagittal bending (acting to place the pahnar cortex in compression and the dorsal cortex in tension), the bone strain recordings showed the metacarpus to be loaded primarily in axial compression at each gait. Because the film/force method depends on an accurate measure of limb segment orientation relative to the direction of ground reaction force, comparatively small errors in calculations of bending moments may lead to a significant difference in the level and distribution of stress determined to act in the bone’s cortices. The discrepancy in metacarpal loading obtained by the two methods may be explained in part by the simplicity of the biomechanical model which, for instance, neglected the force exerted by the sesamoids on the distal end of the metacarpus. The records of stress determined from the in oiuobone strain recordings showed that each bone was subjected to a consistent loading regime despite changes of gait. Such a consistent strain distribution should allow these bones to maximize economy in the use of tissue required to support the dynamic loads applied. Peak stresses measured from the bone strain recordings in the radius during locomotion at constant speed ( - 40.8 k 4.1 MN m-‘) were significantly larger than those in the metacarpus ( - 25.1 f 2.8 MN m- *), regardless of speed and gait. During acceleration and deceleration, however, peak stress rose dramatically in the metacarpus ( -40.6+ 3.4 MNm-*) but remained constant in the radius ( - 37.8 f 5.8 MN me2). This suggests that when thecommonlyencountered loadingconditions likely to cause the highest strains are taken into account. both bones have similar safety factors to failure.
NOMENCLATURE F, (N) RI, (N)
R (NJ R, (N) R, (N) M, (N m) M,, (N m)
PI (N) P, (N)
vertical component of ground reaction force horizontal component of ground reaction force resultant ground reaction force acting on distal end of bone axial component of ground force acting on bone transverse component of ground force acting on bone moment exerted by ground force about carpal joint moment exerted by ground force about metacarpo-phalangeal joint
Receiced 28 Augur
1981: in recisrdform
18 January
x1 (m) x2 (m) 4 (m’) 1 (m4) c (m) u, (MN m-‘) a, (MN m-‘) 0 (degrees)
1983. 565
force exerted by extensors about carpal joint force exerted by extensors about metacarpophalangeal joint moment arm of extensors at carpal joint moment arm of extensors at metacarpophalangeal joint cross-sectional area of bone at its midshaft area moment of inertia for bending in the antero-posterior plane at the midshaft of the bone maximum distance from neutral plane of bending to the periosteal surface of the cortex at midshaft stress due to compression stress due to bending angle of maximum principal stress to the longitudinal axis of the bone
566
A. A. BIEWENER, J. THOMASON, A. G~~DSHIPand L. E. LANYON INTRODUCTlON
At present two methods are available for assessing the forces transmitted through bones and across joints. One technique involves determining the animal’s limb position from high speed tine film and using a force plate to measure the components of force exerted by each animal’s foot while it is in contact with the ground. The external forces and moments acting at each joint and the resulting bone stresses can then be calculated using the geometry of the limb and the material properties of the bone. The second approach is to measure directly in uiuo strain developed within the bone using rosette strain gauges attached to the bone’s surfaces. From these strain measurements, the stresses and hence the forces within the bone can be calculated using the same procedure as in the previous analysis, only in reverse. Both approaches have their limitations. Strain gauges can only respond to the deformation of the bone’s surface to which they are directly attached. The calculation of local stresses from local strains is based on experimental values for the physical properties of bone tissue. By using published values of longitudinal and transverse elastic moduli, shear modulus and Poisson’s ratio, the anisotropy of the bone can be taken into account and reasonable confidence placed in the stress values obtained. Extrapolation to the loading of the whole bone involves assumptions not only of its material and structural properties but also simplification of its manner of loading. If gauges are attached to the midshaft of the bone it is reasonable to assume that there are no stress concentrations such as could arise from local muscle insertions. To calculate forces from stresses and vice versa, in both the strain gauge and the film/force method it is assumed that the bone is homogeneous throughout its cross-section and along its length. The film/force method also assumes that the point of application of the ground force acting on the foot is known. While it is evident that details of the loading regime and the resulting stresses that act in a bone may be more accurately determined from rosette strain gauges, the film/force analysis has the advantage of being non-invasive and applicable to animals which are small or unsuitable for surgery. This technique has been used by a variety of investigators interested in the forces and stresses developed in limb muscles, tendons and bones in different animals such as: kangaroo rats, dogs, kangaroos and humans (Biewener et al., 1981; Alexander, 1974; Alexander and Vernon, 1975; Morrison, 1970) and in the forces acting across the knee and hip joints of humans (Paul, 1966, 1967; Morrison, 1968), as well as experimental hip prostheses in humans (Paul, 1973). The rosette strain gauge approach has been used to measure in vivo bone strain in the radius and tibia of sheep (Lanyon et 01., 1979; Lanyon and Bourn, 1979), the radius of pigs (Goodship et al., 1979), the tibia and radius of horses and dogs (Lanyon and Rubin, 1980), the tibia of man
(Lanyon et al., 1975). the femur of dogs (Carter et al., 1981), as well as the mandible of primates (Hylander, 1979). A knowledge of the distribution and magnitude of forces and stresses within the skeleton is a necessary prerequisite to an understanding of how bones, or joints, are designed to resist the loads applied. Moreover, a knowledge of the manner in which forces are transmitted across joints, and the stresses which are developed in the articulating bones, is relevant to an understanding of the mechanical factors that may influence the deterioration and ‘ultimate failure of articular cartilage or bone, as well as to the design of replacement prostheses. In this study we addressed two questions relevant to the structural design of the radius and metacarpus in the horse: (1) Are the manner of loading and amplitude of stress similar for these two bones at a given speed and gait and (2) does the stress distribution within each bone change with gait and speed? In attempting to answer these questions, we employed both the strain gauge method and the film/force method simultaneously. This study provides the only opportunity to date for comparing these two methods. Since the anatomy of the horse forelimb is less complex than in other species, the comparison between the film/force analysis and direct strain measurement we employed for the radius and metacarpus should provide the best chance for agreement between the two techniques. In more proximal locations and for more complex limbs, such as the hip and knee in humans, the assumptions required by the film/force analysis are more extensive and the possibility of error greater.
MATERIALS AND METHODS
Data collection
Three small horses weighing 268, 281 and 291 kg were trained to be led at a constant speed over a Kistler force plate mounted in a rubber covered, concrete track 45 m in length. The force plate was designed to withstand a peak force in the vertical direction of 49 kN. Passes over the force plate were made at three gaits and at four different speeds: walk, slow trot, fast trot and canter. Velocity was determined from an electric clock triggered by two photocells mounted 10 m apart, before and after the force plate. The horses were filmed in lateral view using light cinematography at 64 f s- i as they moved over the plate. The animals’ hooves were whitened for contrast against the black rubber matting and a white background was used for contrast against the animals’ limbs. The vertical, horizontal, and lateral components of force, as well as the raw strain data, were recorded using an FM tape recorder and were synchronized with the film using a clock placed in the camera’s field of view which marked pulses on the tape at one second intervals.
567
Bone stress in the horse forelimb during locomotion Once a series of pre-operative
force recordings
had
been obtained, rosette strain gauges were attached to the cranial and caudal cortices of the right radius and to the dorsal and palmar cortices of the right metacarpus of each animal. The gauge preparation and operative procedure was similar to that previously described by Lanyon (1976). Anaesthesia was induced with thiopentone sodium and maintained, after intubation. with a halothane/oxygen/nitrous oxide mixture. The areas to which gauges were attached were approached from the medial side of the limb. Surgical interference of the soft tissues caused some postoperative lameness. especially in the case of the metacarpus where the suspensory ligament had to be retracted to attach the palmar gauge. Phenylbutazone was included in the animals’ diet as a post-operative analgesic. A measure of the relative degree of lameness suffered by the animals due to surgery was that the post-operative records of vertical force were 75-80 “/, of the pre-operative forces recorded at each gait. Following the surgery, two or three days were allowed for recovery, after which simultaneous bone strain, film and force recordings were made on each of four days. The lengths of the bones, the position and orientation of the strain gauges relative to the bone’s longitudinal axis, the moment arms of tendons about joints, the moment arms due to the curvature of the bones, and at the bone’s midshaft: the cross-sectional area, area moment of inertia, and maximum distance from the neutral plane of bending to the periosteal surface were determined from radiographs or from post-mortem dissection.
The film records were analyzed using a Vanguard motion analyzer which digitized the coordinates of the joints for each frame. The corresponding vertical and horizontal components of the ground force for each frame were also determined (the lateral component of the ground force was neglected because its magnitude was small compared to that of the vertical and horizontal components of the ground force). Figure 1 shows a drawing of the forelimb position and corresponding ground force vector acting at the hoof for three frames of film of a representative stride over the force plate at a trot (4.3 m s ‘). The vertical (F,) and horizontal (Fh) components of the ground force are shown below. Both sets of data were entered into a PDP-11 microprocessor to calculate the moments and forces acting on each bone. The resulting stresses developed at the midshaft were then calculated based on the bones’ physical dimensions and an assumed
c Rodlus
1 1
b
Metocorpus
L
Fig. 1. A drawing of the lateral aspect of the radius articulating with the metacarpus. The transverse (R,) and axial (R,) components of the resultant ground force (R) acting on the distal end of each bone are shown. The forces exerted by the extensor muscles acting about the carpal (PI ) and metacarpophalangeal (P2) joints that are necessary to counteract the moments exerted by the ground force about each of these joints (M, and kf,,,, respectively) are also shown along with their correspondmg moment arms (x1 and x,). The orientations of P, and P2, determined from radiographs and dissection, were found to act parallel to the longitudinal axis of each bone. The position of the limb corresponds to the second frame (no. 5) drawn in Fig. 1.
longitudinal elastic modulus of 18.2 GN m-2. This value represents the mean ( _t S.D. = 0.20) for equine cortical bone determined by Schryver (1978).
frame number 5 shown in Fig. 1. The forces exerted by the extensor muscles acting about the carpal (PI) and metacarpo-phalangeal (P2) joints to counteract the moments exerted by the ground reaction force about each of these joints (M, and Mm,,, respectively) are also shown along with their moment arms (x1 and xz). No activity was assumed in antagonistic muscles during the support phase. Moreover, inertial forces due to angular accelerations of these limb segments were not calculated and were considered to be relatively insignificant. The stress due to axial compression in the case of radius is then given by
A drawing of the radius and metacarpus showing the transverse (R,) and axial (R,) components of the ground force (R) acting at the distal end of each bone is presented in Fig. 2. The orientation of the bones relative to the ground force vector corresponds to
where A is the cross-sectional area of the radius at its midshaft; and the stress due to sagittal bending is
A. A. BIEWENER,J. THOMSON, A. G~~DSHIP and L. E. LANYON
568 Fast
(4 3 ms-I)
trot
Frame
I 0
1
I
01
02
Time
1
03
(s)
Fig. 2. A drawing of the forelimb position of a horse for three frames of film during a stride over the force plate at a trot of 4.3 m s- ‘. Recordings of the vertical (F,) and horizontal (Fk) components of the ground reaction force acting at the hoof arc shown below. The dashed lines indicate the values of F, and F, corresponding to each frame shown.
sectional geometry at the midshaft to determine A and I rather than assuming an ideal geometrical shape. The measurement of r was made directly from radiographs and represents the perpendicular distance taken from the chord, drawn from the proximal to the distal end of the bone, to the center of the bone at its midshaft. The measurement of A and I was made from photographs of bone cross-sections at the midshaft which were magnified and traced along the endosteal and periosteal surfaces, using a digitizing table to enter the geometric data into the microprocessor. The bone strain data were entered into the microprocessor from the magnetic tape using an A/D converter. The magnitude and orientation of each principal strain at the gauge site was then calculated using standard formulae (Dally and Riley, 1965). The corresponding stresses in the longitudinal and transverse material directions of each bone were determined based on the anisotropic analysis of principal bone strain data described by Carter (1978). This procedure considers bone to be an orthotropic material that is transversely isotropic and assigns values for the transverse elastic modulus (12.2 GN m-‘) and shear modulus (3.4 GN m-‘) in proportion to the longitudinal elastic modulus of bone according to the results of Reilly and Burstein (1975). The values of the transverse elastic and shear moduli were calculated from the values of Poisson’s ratio determined for compact bone by these workers (0.46 f 0.14 when maximum principal stress is in the longitudinal direction and 0.31 f0.09 when it is in the transverse direction), using a value of 18.2 GN m-* for the longitudinal elastic modulus of compact bone (Schryver, 1978). The components of bending and compressive force acting on the radius and metacarpus were then calculated and compared with the values obtained from the film/force plate method.
given by d = b
[RJ+(Ra+P,)rlc I
(2)
where 1is the moment arm of the ground force about the midshaft of the radius, r is the moment arm of the force exerted by the extensors about the midshaft due to bone curvature, c is the distance from the neutral plane of bending to the surface of the cortex and I is the area moment of inertia (for bending in the anteroposterior direction) at the midshaft. The line of action of the extensor muscles and their tendons was found to be closely parallel to the longitudinal axis of both the radius and metacarpus. Consequently the force exerted by them was not considered to contribute to the net transverse component of force acting on each bone (see Appendix). However, in the case of the radius the axial component of the ground force (R,) and the force exerted by the extensor muscles (PI) do, in fact, induce a bending moment about the bone’s midshaft due to its curvature. The analysis described here is similar to that of Alexander (1974), except that it accounts for the curvature of each bone and uses the actual cross-
RESULTS
Figure 1 shows the anatomical relations of the radius and metacarpus in the forelimb. The ulna, which is fused to the caudal aspect of the radius, is greatly reduced so that in the middle forearm the radius provides the only resistance to axial force between the elbow and carpal joints. The third metacarpal (or ‘cannon’ bone) is similarly the single weight bearing bone between the carpal and metacarpo-phalangeal joints. The first and fifth metacarpals are not present and the second and fourth are small and attached along the sides of the third metacarpal to which the gauges were attached. From a lateral view the radius shows a slight caudally concave curvature whereas the metacarpus is relatively straight. Representative records of longitudinal stress determined to occur in the dorsal (D) and palmar (P) cortices of the metacarpus at each gait based on the bone strain recordings are presented in Fig. 3 and compared with equivalent records (dashed curves)
Bone stress in the horse forelimb
Metacarpus Walk
25
?J
0
i 5 -
-25
Trot 50
r
8OOr
L
Time(s)-
0.2
10
Fig. 3. Representative records of longitudinal stress acting in the dorsal (D) and palmar (P) cortices of the metacarpus at a walk, trot and canter based on the in uivo bone strain recordings (solid curves) and compared with equivalent records (dashed curves) calculated to occur for each frame of film from the film/force analysis. The net bending (Fb) and compressive (F,) components of force calculated to be acting on the bone are also shown based on each technique.
during
locomotion
569
calculated to occur in each frame from the film/force analysis. The net bending (Fb) and compressive (F,) forces determined to be acting on the bone are also shown. Because the forces exerted by the extensor muscles acting about the metacarpo-phalangeal joint are parallel to the longitudinal axis of the metacarpus, such forces only contribute to the compressive component of force acting on the bone. Consequently, the bending force is due entirely to the transverse component of the ground reaction force (R,) acting on the bone (Fb = R,). The net compressive component of force is equal to the axial component of the ground reaction force (R,) acting on the bone plus the force exerted by extensor musculature (P2) acting about the metacarpo-phalangeal joint (F, = R, + Pz). In the palmar cortex the pattern of stress determined by each method is similar for each gait despite a large difference in the peak magnitude developed. For the dorsal cortex, however, the development and pattern of stress calculated to occur by each analysis is quite different. The bone strain recordings show that stress in the dorsal cortex is compressive at each gait with its peak occuring early in the support phase at a walk, but midway at a trot, and late at a canter. In contrast, the film/force analysis shows stress in the dorsal cortex to be tensile for most of the time the foot is on the ground. Only during the latter half of the support phase of a walk does this analysis show that the ratio between compression and bending is sufficient to place both cortices in net compression. The peak tensile stress in the dorsal cortex using this analysis is shown to occur early in the support phase at all gaits, coinciding with the peak compressive stress in the palmar cortex. The reason for this discrepancy in the stress regime indicated for the metacarpus by these two analyses becomes apparent if the net bending and compressive forces are considered. F, is very similar both in its amplitude and in the pattern of its rise and fall for each method of analysis, but F, varies considerably. Based on the bone strain recordings, F, is found to be initially negative at a walk. indicating a bending force acting to place the dorsal cortex in compression, and then swings positive, acting to place the dorsal cortex in tension. At higher speeds the negative phase of F, is small or absent and bending is in the palmar direction for most of the support phase. The film/force analysis, however, shows F, to be 2-4 times larger and always remains positive with a peak soon after the foot lands on the ground. The fact that differences in bending forces have a greater influence on stress values than similar differences in compressive forces explains both the discrepancy observed in the magnitude of stress as well as the discrepancy in stress regimes shown by the two analyses. In fact, the difference in peak F, correlates exactly with the differences in peak stress determined by each method. Figure 4 similarly compares representative records of the longitudinal stress acting in the cranial and caudal cortices of the radius, as well as the bending and compressive forces acting on the bone determined
571
Bone stress in the horse forelimb during locomotion
Table 1. Peak longitudinal bone stress (MN m-‘). A comparison of the peak stress developed in the dorsal and palmar cortices of the metacarpus and in the cranial and caudal cortices of the radius at four different speeds determined by two independent methods of analysis. The ratio of peak stress in the pahnar (or caudal) cortex for each analysis is indicated (values are for the three horses grouped together)
Metacarpus
Bone strain analysis peak stress ( f S.D.) N = 10 Dorsal/palmar
Film/force analysis peak stress ( f S.D.) N = 5 Dorsalipalmar (k9.5)
Ratio (palmar stress)
-6.3 (kl.l)/-16.8
(k3.9)
32.3 (*8.3)/-49.0
Slow trot (1.7-2.3 ms-‘)
-8.9 (_f9.1)/-17.7
(k3.8)
27.0 (+ 10.8)/ -48.7 (+ 13.3)
1:?.8
Fast trot (3.84.6ms-‘)
-11.4 (*6.1)/-21.3
(+6.5)
39.6 (+ 14.0)/-60.6 (i 15.6)
1z2.9
Canter (nl) (4.5-5.8 m s- ‘)
-14.0 (*3.1)/-25.1
(k2.8)
38.1 (+ 18.5)/-68.5 (+ 16.7)
1:2.7
Canter (lead) (4.5-5.9 m s _ ’ )
-13.8 (*2.9)/-21.4
(f4.3)
60.4 I* 14.3)/-84.0 (k23.5)
1:4.0
Cranial/caudal
Ratio (caudal stress)
Radius
Cranial/caudal
Walk (1.4-1.8 ms-‘)
14.2 (*3.2)/-21.2
(k5.7)
32.9 (&2.7)/-45.2
(k3.0)
1:2.1
Slow trot (1.7-2.2 ms-‘)
17.1 (+4.9)/--34.6
(f3.8)
31.3 (a5.0)/-45.2
(f3.0)
1:1.4
Fast trot (3.9-4.5 ms- r)
20.6 (*6.3)/-40.8
(k4.1)
38.5 (10.8)/ -60.0
(+ 10.6)
1:1.5
Canter (nl) (4.45.6 m s _ ’ )
16.1 (+5.6)/-36.1
(k4.4)
53.2 (+15.6)/-76.4
(k18.2)
1:2.1
Canter (lead) (4.65.8 ms-‘)
12.7 (_+4.6)/-28.9 (k2.6)
48.1 (+6.8)/-65.8
(k7.2)
1:2.3
da1 direction) acting on the radius during the second half of the support phase, whereas the film/force method suggests that this peak occurs early in the support phase at a time similar to the rise in Fb determined for the metacarpus. In fact, based on the bone strain recordings, it is the bending moment exerted by the compressive force about the midshaft of the radius due to its caudally concave curvature which causes the peak in stress due to bending that is observed at each gait. The level of F, in fact passes through zero at this time. During the first half of the support phase F, is initially negative, acting to counteract the bending engendered by the curvature of the radius caudally, but during the second half of the support phase it swings positive and then augments the bending due to F, acting about the curvature of the bone at its midshaft. The peak longitudinal stress data (mean &-S.D.) determined by each technique for the metacarpus and radius in the three horses combined are compared in Table 1. The four speeds shown represent three different gaits. At each gait and speed the values of peak stress in the metacarpus (palmar cortex) calculated from the film/force analysis are about three times greater than those from the bone strain recordings (again, this is largely due to the film/force analysis showing that the metacarpus is loaded in cranio-caudal
_
1:2.9
Walk (I.41.7ms-‘)
bending when in fact the strain gauge results show that it is not). In the radius (caudal cortex) the discrepancy in peak stress is less; the film/force results showing an overestimate varying from 1.5-2 times that shown by the strain gauges. The values of peak stress shown in Table 1 have relatively large standard deviations, indicating that there is a significant amount of variation between individuals. This is especially true for the metacarpus. In two individuals the component of stress due to bending was low, whereas in the third it was much higher. Within an individual, however, the stress acting in each cortex of a bone at a given speed is consistent. Table 1 also shows that the values determined from the bone strain recordings are less variable than those calculated from the film/force analysis, although fewer determinations were made from the latter method. Both sets of data show a general increase in peak stress in each bone as speed increases. However, the proportional increase in peak stress from that at a walk to the maximum value at either a fast trot or a canter is relatively small. Over a four-fold increase in speed the bone strain analysis shows a proportional increase in peak stress of 1.5 times walking values for the metacarpus (non-lead leg, canter) and 1.9 times for the radius (fast trot). The film/force analysis predicts a proportional increase of 1.7 times for each bone (lead
572
A. A. BIEWENER,J. THOMSON, A. G~~DSHIP and L. E. LANYON
leg, canter for the metacarpus and non-lead leg, canter for the radius). In contrast to the trot in which the movement of contralateral limbs is symmetrical, the canter is an asymmetrical gait. In a trot the contralateral pair of fore (and hind) limbs land approximately 180” out of phase with one another, with an aerial phase in between. The movements and ground contact time of each forelimb are similar during their respective support phases. In a canter, or a gallop, the two forelimbs land slightly out of phase with one another, with an aerial phase after, but not in between, the support phases of the forelimbs. The non-lead leg is the first to land, and it is the lead forelimb, which lands second, that throws the animal into the next aerial phase. The phase difference between the two limbs is speed dependent and decreases as speed increases from a slow canter to a fast gallop (Gambaryan, 1974; Hildebrand, 1965). Since the two forelimbs land at different times and act asymmetrically in the stride, during a canter and gallop values of peak stress are given for the experimental limb when it was used as the non-lead, as well as the lead foreleg. For a canter at a given speed the bone strain analysis shows that peak stress is greater in both the metacarpus and radius of the non-lead leg (on which the animal lands from the aerial phase) than on the lead leg (which throws it into the air). This difference between the lead and non-lead legs is slightly greater in the radius (207,$ p c 0.01) than in the metacarpus (15 y,, p < 0.1). The fact that the difference in the non-lead and lead leg metacarpus is not significant for the data from all three horses combined is misleading. Within an individual horse the difference in the magnitude of stress between its non-lead and lead legs is highly significant (p c 0.01) for both the metacarpus and radius. Values of peak stress acting in the palmar cortex of the metacarpus and the caudal cortex of the radius for each of the two horses for which data was obtained from the non-lead and lead legs of a canter are shown in Table 2. The results presented here are similar to those reported for the difference in principal
strain acting in the caudal cortex of the lead and nonlead radius of a horse while changing gaits from a trot to a canter during treadmill locomotion (Lanyon and Rubin, 1980). The film/force analysis agrees with the strain gauge data by showing a similar difference in stress for the lead and non-lead radius. In the metacarpus however the two methods disagree. In this bone the film/force analysis shows that the magnitude of stress is greater in the lead leg. Does the loading regime change with gait?
Representative records of longitudinal stress determined from the in oiuo bone strain recordings for the cranial and caudal cortices of the radius and the dorsal and palmar cortices of the metacarpus at each gait are shown in Fig. 5. The non-lead and lead forelimbs of a canter are again presented separately. The vertical component of the ground force (F,) is included as well. These records are regular both in pattern and magnitude. The standard deviation of the mean of the peak stress for a series of such records (N = 10) for an individual horse at a given speed is typically less than 87; of the mean. The manner of loading of both the radius and metacarpus remains the same at different gaits, with a general increase in the magnitude of peak stress as speed increases. The radius is subjected predominantly to sagittal bending, acting to place its caudal cortex in compression and its cranial cortex in tension. The metacarpus however is loaded primarily in axial compression. The small amount of sagittal bending that occurs contributes to the compressive stress in the palmar cortex of the bone but is insufficient to place the dorsal cortex into tension. Despite the differences which occur in the magnitude of stress with speed, gait change and limb sequence, the loading patterns are remarkably consistent. Moreover, changes in the magnitude of peak stress observed in both cortices of the metacarpus and radius at each gait, and between the non-lead and lead legs of a canter, are closely correlated with similar changes in the vertical component of the ground force.
Table 2. A comparison of the peak palmar (or caudal) stress acting in the non-lead versus lead metacarpus and radius of two individual horses at a canter. The level at which the difference between the means of peak stress in the non-lead versus lead leg metacarpus and radius are significant, based on a two-tailed r-test (9 degrees of freedom), is given to the right in each case. The data shown are based on the in uioo bone strain recordings of each bone
Metacarpus Horse A Horse B Radius Horse A Horse B
Lead leg Non-lead leg Peak palmar stress + SD. (MN m- ‘) -23.1 f 1.6 -28.4k2.1
-17.5k2.4 -25.4kO.5
Peak caudal stress + S.D. (MN m - ‘) - 38.7 f 2.9 -37.1 k4.0
-28.5 + 3.5 -29.3k1.2
N = 10 p < 0.01 p < 0.01 N= 10 p = 0.001 p < 0.01
Bone stress in the horse forelimb during locomotion Walk ? E % m-
zlil 82 “;,
Trot
Canter
3020 IO-
c,
0 -to c,
-20 -3o-
-4o-
Non- lead
Time
lead
u 0.5 s
573
A more quantitative analysis of how the orientation and loading vary in both of these bones at different gaits is obtained by comparing the stress due to bending (a,) with that due to compression (a,). The magnitude of shear stress is indicative of the orientation of the load since a torsional load and/or an eccentric load applied will cause the axis of the principal stress to deviate from a longitudinal orientation. These data are presented in Table 3. The only significant shift in the loading regime of either bone occurs at the walk-trot transition. As the horse changes gait the magnitude of the compressive component of stress increases and bending is reduced. After this transition the ratio of bending to compression is maintained over the range of speeds measured, including the change of gait from a trot to a canter. The angles of peak principal stress show that the orientation of the principal stresses do not change more than 4” within the cortices of either bone over the range of speeds observed. At each gait the longitudinal shear stress that does occur in both bones is due primarily to torsion. Accordingly, the values of peak longitudinal shear stress in the cortices of each bone do not change significantly during normal locomotion and remain quite low. DISCUSSION
Fig. 5. Representative records of longitudinal stress determined from the in uivo bone strain recordings for the cranial (C,) and caudal (C,) cortices of the radius and the dorsal (0) and palmar (P) cortices of the metacarpus at three different gaits. Records of stress for the non-lead and lead legs of a canter are presented as well for comparison. The vertical component of the ground reaction force (F,) is also shown in each case.
The two independent methods used in this study to determine the pattern and magnitude of locomotory stresses acting in the radius and metacarpus of the horse forelimb yielded significantly conflicting results. These differences were more pronounced for the metacarpus but in each bone depended largely on
Table 3. A comparison of the ratio of bending stress (uJ to compressive stress (a,) at four different speeds and three gaits for each bone. The shear stress and the angle of the principal stress to the bone’s longitudinal axis in the dorsal (0,) and palmar (0,) cortices of the metacarpus and the cranial (0,) and caudal(f7,) cortices of the radius are also presented. The consistency of each set of data illustrates that the loading regime is uniformly maintained in each bone despite a change of gait Ratio Qb:b, Metacarpus Walk Slow trot Fast trot Canter (nl) Canter (lead)
l:I.4 1:4.3 1:3.7 1:4.1 114.2
Radius Walk Slow trot Fast trot Canter (nl) Canter (lead)
Shear stress (MNm‘*) Dorsal/palmar -l.6/-0.7 -0.8/-1.8 -1.31-2.0 -2.1/-1.7 -2.O/- 1.8
edieP (degrees)* - 10.3/-2.5 -9.5/-3.7 -9.o/-3.1 -7.91-2.1 -ll.l/-2.9
Cranial/caudal 5.2: 1 3.2: 1 3.1:1 2.8:1 3.1:1
+ l.l/+ 1.6 +1.8/+1.8 +1.8/+1.7 f 1.2,‘+2.4 i t.2/+2.4
i2.4/+5.3 +4.5/+3.1 +5.0/+3.1 + 1.3/+4.8 +0.8/+4.3
* ( - ) indicates distal lateral rotation and ( + ) indicates distal medial rotation relative to the proximal end of the bone.
514
A. A. BIEWENER,J. THOMASON,A. GOODSHIPand L. E. LANYON
differences in the antero-posterior forces calculated to act on the bone. Greater sagittal bending was indicated by the film/force technique than was determined from the in uiuo bone strain recordings. Both analyses agreed that the radius was loaded predominantly in bending, but the film/force technique predicted peak stresses that were 1.5-2 times greater in magnitude than those derived from the bone strain data. For the metacarpus, not only did the film/force analysis yield peak stresses more than two-fold greater in magnitude, but it indicated that the metacarpus was subjected primarily to bending; whereas the in vivo bone strain data showed that, during steady state locomotion, it was maintained in axial compression. Determination of the bending component of force (Fb) acting on a bone by the film/force method is susceptible to error because measurement of the net compressive and bending components of force depends not only on joint position and the point of application of the ground reaction force, both of which are subject to error, but the orientation and assumed activity of muscles acting across the joint. Though the error in bending force may be comparatively small, the difference it makes to the magnitude and distribution of stress calculated toact at the bone’s midshaft may be quite large. Based on the bone strain recordings, F, ranged from 100 N at a walk to 200 Nat a canter in the metacarpus, and from 200 to 400N in the radius. These values were more than an order of magnitude less than the corresponding values of F,, which ranged from - 5000 to - 9000 N in the metacarpus and from - 4000 to - 8500 N in the radius. It can be seen, then, that in terms of absolute magnitude, a given error in the value of F, compared to a similar error in the value of F, will not have nearly as significant an effect on the stress regime calculated to occur in a bone. Thus, a difference as small as 400 N calculated for F, by each method (which can occur from an error of only a few degrees in the orientation of the ground force and muscle force vectors, see Figs 3 and 4) may result in the prediction of an entirely different stress distribution. This was presumably the case for the discrepancy in stress calculated for the metacarpus. These potential sources of error in a relatively ‘simple’ mechanical system close to the measured ground reaction force brings into question the use of this technique for more complex locations. The large discrepancy in metacarpal loading obtained by the two methods, however, can be diminished to some extent if the model includes the force exerted by the extensor tendons, via the sesamoid bones, on the distal end of the metacarpus. The magnitude of this force depends on the angle of the distal portion of the extensor tendons, which insert on the distal phalangeal bones, to the longitudinal axis of the metacarpus. This transverse loading of the metacarpus (in the dorsal direction), however, will be resisted, at least in part, by a reaction force transmitted by the first phalangeal bone to the metacarpus across the metacarpo-phalangeal joint. Because the orien-
tation of the distal extensor tendons to the metacarpus is difficult to measure accurately, and there is uncertainty in the determination of their net contribution to bending, we chose not to estimate the force exerted by the sesamoids on the distal end of the metacarpus. Clearly, with a knowledge of what the stress distribution at the bone’s midshaft is meant to be, it is possible to progressively modify the model to obtain a good fit. However, the bone-bonded strain gauge technique, although providing a more direct and detailed measurement of the strain (and hence, the stress) acting in a bone’s cortex, also has its limitations. Most importantly, the strain data provided by each gauge are only relevant to the area of bone to which the gauge is adherent. To extrapolate to a loading regime for the bone’s diaphysis requires the assumptions that the site where strain is measured is representative of the manner in which the bone is loaded, and that cortical bone is homogeneous in its properties along the bone’s length. When considering the midshaft of the radius and metacarpus, we beheve these assumptions to be reasonable and that the strains recorded at the cranial and caudal gauge sites represent the peak strains transmitted through the length of the bone during normal locomotion. However, more detailed studies of the strain distribution along a bone’s length and around its circumference must be conducted to more fully substantiate these assumptions. Another disadvantage of the strain gauge technique, borne out by the difference between the pre- and post-operative force plate recordings, is that normal function can be impaired by the surgery involved. In this study, however, whereas the magnitude of peak stress was reduced by approximately 20’;,,, it seems unlikely that this significantly affected the pattern and orientation of stress in either bone. Given that errors in determining the bending forces acting on a limb bone are due to estimates of the orientation of limb segments relative to the direction of ground force, this suggests that the differences in loading obtained by each method should be greater for more proximal bones, further away from the point of application of the ground force. However, the closer agreement obtained in this study for the horse radius at each gait is due to the inherent concave curvature of the radius along its caudal aspect, which results in the compressive component of force (F,) exerting a bending moment about the bone’s midshaft. At each gait the magnitude of this bending moment was typically two to three times greater than the moment exerted by F,. Consequently, because the magnitude of F, measured by each technique was similar, each analysis showed the radius to be subjected predominantly to bending, despite differences in the magnitude and pattern of F,. The importance of a bone’s curvature engendering a significantly large bending moment about its midshaft, expecially one greater than that due to the transverse component of the reaction force acting at a joint, has not been generally recognized. Alexander (1974, 1975) does not take bone curvature into account in his
Bone stress in the horse forehmb during locomotion
analysis of bone stress in the tibia and femur of a dog jumping or the kangaroo hopping. In the present work, if the curvature of the radius had not been considered, the film/force analysis would have predicted a much smaller magnitude of bending. as well as a more variable pattern in the bending moment about the bone’s midshaft. This would have suggested a totally different distribution within the cortices of the bone. A study of functional stresses within the skeleton is of interest not only because of their mathematical relationship to the loads involved but also because bone is a tissue which is capable of remodelling in response to changes in the mechanical load applied to it. However, it is not clear whether the precise stimulus for remodelling is the magnitude of the stresses developed (Goodship et al., 1979), their distribution (Lanyon et nl., 1982) or the rate at which they change (Lanyon and O’Connor, 1980). In the case of an animal, such as a horse, which uses different gaits during locomotion, the orientation and manner of loading might have been expected to vary significantly. If this had been the case it would have important implications for the design of a bone; requiring greater bone mass more generally arranged about its longitudinal axis in order to resist equally the diverse range of loads imposed. However if the applied loading regime remains uniform, the bone’s design requirement is simplified, thus permitting a structure with the minimum tissue needed to support the load effectively. This may be particularly important for the horse, the evolution of whose limbs has been directed to an increase in the length but a reduction in the number of its distal elements. The records of stress determined from the bone strain recordings show that the metacarpus is loaded primarily in axial compression and the radius in sagittal bending and that the pattern and orientation of stresses within each bone remains uniform despite changes of gait and speed. Our results agree with those reported for single element strain gauge recordings during walking and trotting in each of these bones (Turner et al.. 1975). In this study gauges were attached to not only the cranial and caudal cortices but the medial and lateral cortices as well. Since the stimulus for bone remodelling is assumed to arise from the strains developed within the bone when it is loaded (Bassett, 1968). a bone’s architecture in relation to the manner of loading would be the result of these repetitively imposed strain patterns. The difference in peak stress values developed in the radius (-41 MNm-‘) and in the metacarpus ( - 25 MN m-*) when considering steady state locomotion would appear to contradict the view that a bone’s remodelling objective is to achieveand maintain a uniform safety margin to failure. However, this apparent discrepancy in peak functional stress is eliminated if acceleration and deceleration are considered to be common enough occurences to influence bone architecture (Rubin and Lanyon, 1981). The peak
515
stress in the palmar cortex of the metacarpus during acceleration ( - 40.6 F 3.5 MN m- ‘) is substantially higher than during steady state locomotion. and is nearly equal to that which occurs in the caudal cortex of the radius during steady state locomotion. The radius on the other hand does not show an increase in peak stress during acceleration or deceleration ( - 37.8 IS.8 MN m- * in the caudal cortex). The question remains, however, as to why the radius develops a caudally concave curvature while the metacarpus does not. If the goal of designing a structure is to minimize the stresses acting within it, to maximize its safety margin, then this could most easily be achieved if the radius were a much straighter bone. Instead. by developing a curvature, the peak stresses at the radial midshaft during steady state locomotion are elevated to a similar level as those developed in the metacarpus during acceleration and deceleration. Since bone loss would be expected in either bone as a result of a decrease in its individual functional strain level. this suggests that perhaps there is an optimum range of peak functional strain. The upper value of this range would be determined by the appropriate safety factor to either monotonic or fatigue failure. and the lower value by some benefit conferred on the bone tissue by being intermittently strained at or above a ‘threshold’ level. Such a benefit could be related to the more effective movement of fluid, transporting nutrients and waste products throughout the tissue (Piekarski. 1981), to the generation of strain related electrical potentials, or to some other advantage conferred to the cells or their matrix. While this proposal must be speculative at this time, it is in agreement with the evidence which we have presented here. although it is contrary to conventional ideas which have assumed that most features of bone architecture are developed to maximize safety margins or economy of tissue. Acknow[rdyements-This work was supported by NIH Training Grant T32GM07117-04 awarded to Harvard University, NIH Grant AM1814Oawarded to Dr. C. Richard Taylor, and by a grant awarded to the University of Bristol Veterinary School by the Horse Race Betting Levy Board. The assistance of Mrs. Maureen Redcoe and Mr. Peter Walker was greatly appreciated.
REFERENCES
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APPENDIX Equations used to calculate stress due to (Al) compressive forces and (A2) bending forces. cc =
Rcos0+Pcosa A
(Al)
where R is the resultant vector of the ground force acting on the distal end of the bone, 6 is the angle between the vector R and the bone’s longitudinal axis, P is the resultant vector of force exerted by extensor muscles, and a is the angle between the vector P and the bone’s longitudinal axis. In the case of the radius and metacarpus, a = 0. Substituting R, = R cos 0, (Al) reduces to R,+P
6, = -
A
e~=[(RsinB+Psina)l+(RcosB+Pcosa)r]c
,
WI
1
where R, 8, P, and a are defined as above, I is the orthogonal distance from the distal end of the bone to its midshaft, and r is the orthogonal distance bisecting the chord drawn from the proximal to the distal end of the bone (along which the axial forces were defined to be acting), to the center of the bone at its midshaft. In the case of the radius, a = 0. Substituting R, = R sin fl and R, = R cos 0, (A2) reduces to
u = b
[RJ+R.+P)rlc I
In the case of the metacarpus, a = 0 and r = 0. Again, substituting R, = R sin 9 and R, = R cos& (AZ) reduces to: R, Ic ub=---. I