female. thereby confirming the widely held perception that "the bigger they are, the harder they fall ... Introduction. Hip fractures are a health problem of ...
Dynamic Models for Sideways Falls From Standing Height A. J. van den Kroonenberg
W. C. Hayes Orthopaedic Biomechanics Laboratory. Charles A. Dana Research Institute, Department of Orthopaedic Surgery, 8eth Israel Hospital and Harvard Medical School,
Boston, MA 02215
T. A. McMahon Division of Applied Sciences,
Harvard University, Cambridge, MA 02138
Despite our growing understanding oJthe importance offall mechanics in the etiology of hip fracture, previous studies have largely ignored the kinematics and dynamics of falls from standing height. Beginning from basic principles, we estimated peak impact force on the greater trochanter in a sideways fall from standing height. Using a one degree~ofl'reedom impact model, thi.tforce is detennined by the impact velocity of the hip, the effective mass of that part of the body that is moving prior to impact, and the overall stiffness of the soft tissue overlying the hip. To determine impact velocity and effective mass, three different paradigms of increasing complexity were used: 1) a falling point mass or a rigid bar pivoting at its base; 2) tvvo-link models consisting of a leg segment and a torso; and 3) three-link models including a knee. The total mechanical energy of each model before falling was equated to the total mechanical energy just prior to impact in order to estimate the hip impact velocity. In addition. the configuration of the model just before impact was used to estimate the effective mass. Our model predictions were compared with the results of an earlier experimental study with young subjects falling on a 10-inch thick mattress. Values from literature were used to estimate the soft tissue stiffness. For the models, predicted values for hip impact velocity and effective mass ranged from 2.47 to 4.34 mls and from 15.9 to 70.0 kg, respectively. Predicted valuesfor the peakforce applied to the greater trochanter ranged from 2.90k to 9.99k N. Based on comparisons to the experimental falls. impact velocity and impact force were best predicted by a simple tvvo-link model with the trunk at 45 degrees to the vertical at impact. A threelink model with a quadratic spring incorporated in the knee of the model was the best predictor of effective mass. Using our most accurate model. the peak impact force was 2.90k N for a 5th percentile female and 4.26k N for a 95th percentile female. thereby confirming the widely held perception that "the bigger they are, the harder they fall . ..
Introduction Hip fractures are a health problem of enonnous proportions. In the United States, about 250,000 hip fractures occur annually, resulting in estimated annual costs of medical and nursing services of over 7 billion dollars [1, 9, 21]. Over 90 percent of these hip fractures occur among people over age 70 [7J. Continued growth in the elderly population can be expected to increase the total number and associated costs of hip fractures dramatically. To date, hip fracture prevention efforts have been directed primarily toward the amelioration of bone loss [26]. However, since 90 percent of all hip fractures are caused by falls [1], an alternative and complementary approach to the prevention of hip fractures is to focus on fall prevention or on reducing the injury potential of those falls that do occur [13, 17,24,27]. Falls have proven remarkably resistant to prevention [14 J, however, and the use of trochanteric padding to reduce the injury potential of a fall is in the early phases of development [17, 24 J . With respect to the mechanics of falling, extensive research has been conducted on fall initiation [10]. Besides the instability phase that results in loss of balance, a fall consists of three additional phases [11]: 1) a descent phase; 2) an impact phase; and 3) a post-impact phase during which the subject comes to rest. Little is known about the kinematics and dynamics of these last three phases. Since falling to the side and impacting on the hip raises the risk of hip fracture significantly [11, 20] and yet only about 2
Contributed by the Bioengineering Division for publication in the JOURNAL ENGINEERING. Manuscript received by the Bioengineering Division September 21, 1993; revised manuscript received July 26, 1994. Associate Technical Editor: S. A. Goldstein.
OF BIOMECHANlCAL
Journal of Biomechanical Engineering
percent of all falls result in hip fracture [25], a fundamental variable which detennines fracture risk is the force applied to the greater trochanter in a sideways fall. Given the lack of information on the mechanics of the descent and impact phases, we do not know whether the peak impact force associated with a sideways fall exceeds femoral breaking strength by a small margin or a large one. If the force is barely sufficient to break the hip, trochanteric padding to reduce the force to a safe level becomes a reasonable approach to hip fracture prevention [24]. To address these issues, we have developed simple rigidbody models that simulate sideways falls from standing height. Our objectives were: 1) to predict the ranges of hip impact velocity and effective mass as a result of a sideways fall from standing height; 2) to define the lower and upper bounds for the peak force applied to the greater trochanter; and 3) to assess the validity of the predictions and the consequences of increasing the realism of the calculations by comparison with experi~ mental data from young athletes who fell voluntarily on a thick foam mattress f 16] .
Methods In the following, two categories of models are described: an impact model and several rigid-link models. The impact model is used to predict the hip impact force, based on: 1) hip impact velocity; 2) the effective mass of that part of the body that is moving in the vertical direction prior to impact; and 3) soft tissue stiffness. The rigid-link models are used to predict hip impact velocity and effective mass. Impact Model. In order to estimate the peak force applied to the greater trochanter in a fall, the body is modeled as an undamped singJe-degree-of-freedom system (Fig. 1). The mass AUGUST 1995, Vol. 117 / 309
m
x
Fig. 1 Single-degree-of-freedom impact model. M represents the mass of that part of the body that is moving in the vertical direction prior to impact. V is the velocity of M just before the body hits the ground. K is the stiffness of a linear spring representing the soft tissue overlying the hip and the flexural stiffness of the body, including the action of muscles.
M represents the mass of that part of the body that contributes to the impact force on the hip (Eq. 1 et seq.); we will call this the "effective mass", According to this single-degree-offreedom model, the effective mass M is moving in the vertical direction with velocity V prior to impact. The spring, with linear spring constant K, accounts for the soft tissue overlying the hip and the flexural stiffness of the body, including the action of muscles. Damping is not considered in this model, since Robinovitch et a1. [23 J found that the damping ratio for the impact motion is small (damping ratio S = 0.2). From this simple impact model, the peak impact force applied to the hip is equal to the maximum force in the spring, which is defined by F_ = 16:"",. The value for F p,'" is given by [18J: (1)
with:
Fig. 2 Point-mass model. The mass of the model is equal to the body mass, its initial height is taken equal to the height of the center of gravity of the body.
is calculated by equating the available potential energy before the fall to the potential and kinetic energy just before hip impact occurs. The available potential energy is equal to mghcc' where m and heg are the total body mass and the height of the center of gravity of the body, respectively. and g is the gravitational constant, g = 9.81 mfs2. Note thatm and M are separate parameters. The effective moving mass, M. is equal to the total body mass, m, for the point-mass model. In most other cases to be considered, M and m are not the same. The simplest model to simulate a fall from standing height is one in which the point mass m falls from a height equal to the height of the center of gravity of the body (Fig. 2). Equating the available potential energy before the fall to the translational kinetic energy before impact results in:
(2)
(4) V
and:
(3) The physical meaning of the different terms in the above equations can be clarified by considering a few special cases. For example, if the hip impact velocity V is set to zero, the' 'impact number," given by the dimensionless term Vwn/g, becomes zero also, and the mass overshoots the equilibrium position by a factor of two. The resulting peak value in force, F peak/ Mg, is equal to two, and this occurs when the phase in radians, given by (Wnt)maXl is 7r radians. When V > 0, the peak in force occurs at a smaller phase, but the overshoot is larger and therefore Fpeak/Mg becomes larger. The effective mass M for a given configuration can be calculated by expressing the vertical impact force applied to the hip of the model in terms of the vertical acceleration of the hip during impact (Z·hip, which is unknown) and then dividing this by (i"p + g). From Eqs. ( 1 ) - (3) it follows that the parameters K, V, and M uniquely define the peak impact force. In the following, values for V and M are detennined using different rigid-link models of the human body. Values from the literature were used to estimate the soft tissue stiffness, K ["23]. In addition, for each set of values for V and M, we calculated the following parameters: 1) the dimensionless "impact number", given by Vw n / g; 2) the phase in radians at which the peak in force occurred, given by (wnt)max; 3) the normalized peak force at the hip, given by Fhlp/ Mg; and 4) the absolute peak force F hip in kN. One-Link Models. These models are based on energy conservation, and no energy loss during the fall is assumed. In addition, no energy storage in muscles is accounted for The value for the impact velocity of the hip (V in Eqs. ( 1 ) an'... , :::! ») 310 I Vol. 117, AUGUST 1995
= hgh" = 4.43.Jh:,.
(5 )
A slightly more complex one-link model includes the effect of rotational energy. A rigid, slender rod representing the body falls by pivoting at its base (Fig. 3A). T., mass and height of the bar are equal to the total body rna.'>, and body height, respectively. The location of the "hip" of the model is chosen at half the height of the bar. Equating the potential energy before the fall to the rotational kinetic energy before impact gives:
(6) V
= .J~gh" = 3.84.Jh:"
(7)
h
A
Fig.3 Rigid uniform slender bar pivoted at base (front view), The hip of the model is located at the center of the bar and coincides with the center of gravity of the bar. J is the mass moment of inertia of the bar with respect to the pivot point (A). During impact it is assumed that the hinge is not connected to the floor (6).
Transactions of the ASME
::r -:T.
m, I,
ffi
1
I"1
Considering first the vertical lack-knife fall (Fig. SA), both rotational and translational kinetic energy, as well as potential energy, are present after the fall:
h h1=h'=i I ml=
3m
m2=
~
m
3 I 2 Il='3 m1h1 o
with:
I 2 I, =IZm,h, h eg ,
= _h .
( 10)
4'
Flg.4 Two-link model in front view. The legs and trunk are modeled as two uniform slender bars connected by a frictionless hinge located at the hip. f, is the mass moment of inertia of link 1 with respect to the pivot point. 12 is the mass moment of inertia of link 2 with respect to tts center of gravity.
and:
2V
=;;
( II )
= 3.24.Jh,
(12)
W!
so that: where w is the angular velocity of the rod about the pivot and 1= mh 2 /3 is the moment of inertia of the rod about the pivot. Assuming that the rod is not connected to the hinge during impact, and recognizing the fact that the hip of the model coincides with the center of gravity of the rod, the force applied to the hip of the model during impact is given by (Fig. 3B):
(8) and therefore the effective mass M is equal to the total mass m. Two-Link Models_ The simplest two-link model (Fig. 4) consists of two slender bars: a leg segment connected to the floor by a hinge, and a trunk segment. The links are interconnected by a hinge located at the hip. They are chosen to be equal in length, but not in mass. Based on an experimentally-observed range of trunk orientations prior to impact from _8° to 54°, obtained from a study with young subjects falling on a thick foam mattress [16]. the following two cases are considered (Fig. 5). In a mode we call the "vertical Jack-knife" fall, the trunk is vertical just prior to impact (hence, the trunk resembles the blade of a lack-knife fixed at right angles to the sheath). A variation (the "45 deg Jack-knife") assumes that the trunk angle (defined as the angle between the trunk and the vertical) is 45 deg just before impact occurs. A limitation of these twolink models (to be discussed later) is that the impact configurations in Fig. 5 are anatomically realistic only when knee flexion is allowed, and yet knee flexion must be excluded to keep such models physically determinate. As with the one-link models, the hip impact velocity V is calculated by equating the available potential energy before the fall to the kinetic and potential energy just before impact. Again, we assume there is no energy storage in muscles. Ail implicit assumption is that any change in available energy due to initial conditions on the positions and velocities of the links makes a negligible contribution to the total available energy before the fall. The validity of this assumption is tested in Appendix C for sets of small initial angles and angular velocities of the links that result in each of the two prescribed impact configurations. For the vertical lack-knife fall, an additional assumption is that the rotational energy of the trunk just before impact is small, and therefore can be neglected. This assumption is also tested in Appendix C for the same sets of initial conditions. Although these sets of initial conditions are certainly not unique, the calculations in this appendix indicate that neglecting the effects of initial conditions in the calculation of hip impact velocity is justified. Journal of Biomechanical Engineering
v~
1 15
14
gh
where Wj is the angular velocity of link I; Vz is the linear velocity of link 2; h~g and h~g are the initial heights of the centers of gravity of links 1 and 2; and h = hl + h',! = total body height. For the 45 deg lack-knife model, the energy just before impact consists of the potential and kinetic energy of the trunk and the kinetic energy of the legs. Energy conservation results in:
I
= -
2
I
I?w7 + -
2
12w~
I
+-
2
m2v~
h, 2
+ m2g --= cos 45 deg,
(13)
where WI and W2 are the angular velocities of the leg and trunk segments, respectively, and V2 and h jll and Mil are as defined before. In order to solve for the hip impact velocity V, we assumed that the angular velocity of the legs is 1.38 times the angular velocity of the trunk of the model just before impact. This choice was obtained from a numerical integration showing that the final conditions on the angles of the leg and trunk segments can indeed be reached from the assumed initial conditions (Appendix C). Substituting w, = L38w, in Eq. (13), the hip impact velocity is given by:
v~
2.72.Jh,
(14 )
with h as defined before.
A
B Fig.5 Front view of impact configurations for two-link models. (A) vertical Jack-knife model: the trunk is vertical at impact. (B) 45-degree Jackknife model: the trunk is at 45 degrees to the vertical.
AUGUST 1995, Vol. 117/311
The equations of motion for this model can be presented in matrix fonn as functions of the generalized coordinates q:
H(q)q + C(q, q) + G(q) = T,
(IS)
where
q
=
hip
U:J [~ J
(19)
H is the inertia tensor, C is the tensor representing coriolis and coupling tenns, G is the gravity vector, and T is the external torque vector: y
x
T=
SIDE VIEW Fig. 6
REAR VIEW
Three-link model. The model consists of a shank (link 1), a thigh
(link 2) and a trunk (link 3). This model has three degrees of freedom: a, 0 and y. To ensure that the model falls sideways, the hip does not move forward or back, and the angle K is fixed (arbitrarily) at 30 degrees.
Each of the two impact configurations in Fig. 5 has a different effective mass. In Appendix A, we derive the effective mass of a link at an angle (J to the vertical. The results are:
(3 = 0": (3
= 45": (15)
where mlink is the mass of one link. Assuming that the legs are not connected to the hinge during impact, the total effective mass for each case can be obtained by summation of the effective masses of the two links separately using the above equations. The effective mass for the vertical lack-knife model is:
U:J .
(20)
The expressions for H, C, and G were obtained using Lagrangian Dynamics [2, 6]. The complete derivation is given in Appendix B. As opposed to the one- and two-link models, in this case the impact configuration is not specified in the statement of the problem. Given a set of initial conditions, qo and 40, the differential equations are solved using a fourth-order RungeKutta numerical integrator. Once the angles and their derivatives are calculated, the vertical velocity of the hip joint of the model is evaluated when a comes within a prescribed tolerance (0.5 percent) of 90 deg. As with the one~ and two-link models, the effective mass at impact is estimated by calculating the vertical impact force at the hip of the model and dividing by the vertical acceleration of this point, including the acceleration due to gravity. In order to solve for the hip impact force, the shank and thigh segment (links 1 and 2) are considered to be rigidly connected during impact. Therefore, for the purpose of calculating the effective mass only. the shank and thigh segment are treated as a single link during impact. In Appendix A, the effective mass for the combination of links 1 and 2 and for link 3 are derived. Referring to the symbols in Figs. 6 and A2, the results are: M 1cg,
=
__ -,let,,,,+,,'1..'_ _
with:
d' + ---.:I£!. "'" +',,-'_
i
The result for the 4SO lack-knife model is:
2
(3 +
312 I Vol. 117, AUGUST 1995
m2
m,) m,
cos
e
d=-----
(21)
I, _ _ ----,,_..2.
(22)
m, +m,
(17)
Three-Link Models. In these models, the body is represented as an open chain of three rigid links connected by frictionless hinges (Fig. 6). The three links are: 1) a shank segment connected to the floor by a hinge; 2) a thigh segment, and 3) a trunk segment. The hip joint has two rotational degrees of freedom representing hip flexion/extension and hip abduction! adduction. The knees are modeled as a one-degree-of-freedom joint, representing knee flexion I extension. Rotation between shank and floor and internal! external rotation in the hip joint are not considered. The degrees of freedom (Fig. 6) are: 1) cr, between the vertical and the projection of the leg segment on the y-z plane; 2) 8, between the shank segment and the line connecting the hinge at the floor with the hip joint; and 3) 'Y, between the vertical and the projection of the trunk segment on the y-z plane. Angle K between the trunk and the line through the hinge at the fioor and the hip joint is not a degree of freedom, but is assumed to have a fixed value, set at 30°. The trajectory of the hip is in the two-dimensional y-z plane because horizontal motion of the hip in the anterior/posterior direction is not included.
+
ml
(16)
M
trunk-
L2
1
- sin 2 4
f3 + -1.. m,
The trunk angle (3 between the vertical and the trunk segment is determined by: cos
/3
=
cos
K
cos 'Y,
(23)
with K and 'Y defined as earlier. For this three-link model, three different cases are considered. First, a simulation is performed using values for initial conditions (listed in Appendix C, case 2) obtained from an experimental fall study with human subjects [16], and taking T = 0 in Eq. (18). This case (model 3A) represents a situation in which no active or passive forces act at the joints. Next, the same initial conditions are used but instead a quadratic rotational spring is incorporated in the knee joint of the model to account for muscle forces acting on this joint (model 3B). The characteristics of the spring are obtained by calculating the torques Ta, Te and T.., from Eq. (18) for given functions of a, Transactions of the ASME
Results for one-link models
Table 1 Case
M
V
Model
(% 'tile)
(kg)
(mls)
Vwn/g
(deg)
Fhll'lMg
(leN)
lk point mass
5 95 5 95
45.3 70.0 45.3 70.0
3.91 4.16 3.39 3.60
15.8 13.5 13.7 11.7
93.6 94.2 94.2 94.9
16.8 14.5 14.7 12.7
7.47 9.99 6.54 8.74
IB: rigid bar
B and 'Y and their derivatives (listed in Appendix D), which were obtained from the same experimental fall study. These values represent a typical fall with close to average values for hip impact velocity and trunk angle at impact. We fit a secondorder polynomial to the resulting TfJ VS, curve using a leastsquare method and then performed a simulation with To estimated by the obtained function and To = T'f = O. Finally, in order to provide a simulation as close as possible to a real fall, the same experimentally-obtained kinematic data are converted into linear displacements and used to calculate the impact velocity of the hip of the model (model 3C). Also, the resulting impact configuration in terms of the angle y and the knee flexion angle () is used to calculate the total effective mass (Eqs. 21, 22,23). Because this model (modeI3C) is specified by directlymeasured experimental values for the hip impact velocity and trunk angle, we believe it predicts values for the effective mass M and hip impact force Filip that are the most representative of a real fall. Therefore, the results of this model are used for comparison with the other models.
e
Segment Parameters. We used two different sets of segment properties corresponding to a 5 th percentile (short and light) and a 95th percentile (tall and heavy) female. Using an interactive computer program [3], the segment properties of an existing 15-link model [22J were obtained for given values of body height and weight (expressed in percentiles). The data stored in the 15-link program were based on 32 body measurements of 2420 male flying personnel of the u.s. Air Force and 1905 women in the U.S. Air Force [4]. Values for segment lengths and segment masses for our models were obtained by lumping the segment parameters of this i5-link model. Segment mass moments of inertia for the three-link model were obtained from the values for the individual mass moments of inertia and the segment lengths of the i5-link model, and by using the parallel axis theorem. The segment properties of all the models are given in Appendix E. Results Our first objective was to predict the ranges of hip impact velocity V and the effective mass M for falls to the side using one-, two-, and three-link models. Values for M ranged from 15.9 to 45.3 kg for the 5th percentile female and from 24.5 to 70.0 kg for the 95th percentile female (Tables 1-3). Our results indicate percentage increases in the effective mass ranging between 51 and 59 percent for a 95 th percentile compared to a 5 th percentile female. While the values for the effective mass are fairly constant for the one- and three-link models, we found a large difference in M between the vertical and the 45 deg Jack-knife fall models (Table 2, models 2A, 2B). According to our. calculations, the effective mass is approximately doubled Table 2
2A:
Jack~knjfe
vertical trunk
2B: Jack-knife 45-deg. trunk
for a trunk angle of zero degrees compared to 45 degrees. The most realistic value for M was calculated using an experimentally determined value for y of 35.1 degrees (Table 4, model 3C). This resulted in values for M ranging between 17.1 and 26.2 kg (Table 3, model 3C), which corresponds to about 38 percent of the total body mass. Values for V ranged from 2.47 to 4.00 mls and from 2.93 to 4.34 mis for the 5 th and 95 th percentile cases, respectively (Tables 1-3). Slightly larger values for V were obtained for a 95th percentile compared to a 5th percentile female. Values for V were not different by more than six percent for the comparisons based on our one~ and two-link models, but rose to 19% for the several comparisons based on model 3C in which experimentally-obtained data were used. Values for V calculated using the detenninistic models were generally larger than the range of V (2.47 to 2.93 mis) obtained from experimental kinematic data (Table 3, model 3C). Using a value for K of 71k N/m [23], predicted values for peak impact force ranged from 2.90k to 7.47k N and from 4.26k to 9.99k N, for the 5th and 95th percentile females, respectively (Tables 1-3). Model 3C, which we presume to be the most realistic because it reflected experimental values for the impact configuration and velocity, gave values for the peak impact force ranging between 2.90k and 4.26k N (Table 3, model 3C). Predicted values for the normalized hip impact force FhiplMg were always smaller for a 95 th percentile female than for a 5 th percentile case (Tables 1-3). The reason for this is the lower impact number Vwnl g for the 95 th percentile case. The smaller value for W n , due to the larger effective mass M, dominates the higher value for the impact velocity V in the impact number. Even though we found that the nonnalized impact force is higher for the 5 th percentile case than it is for the 95 th percentile case, our results predict an increase in total hip impact force ranging between 33 and 47 percent for a 95 th percentile female compared to a 5 th percentile female. This corresponds to an increase of 55 percent in mass and an increase of 75 percent in potential energy available for the fall. The phase in radians after which the peak force was reached, (Wnt)mlU' was approximately constant at a value somewhat greater than 90 degrees (Tables 1-3). This implies that the peak force is reached after about a quarter of a full vibration cycle of the spring-mass system. Using Eg. (I), neglecting 1.0 compared to (~I g), and with (w,t)_ ~ 90 deg, F_ is approximately VVKM. Finally, we investigated the effect of increasing the realism of our calculations, including increasing the number of links and incorporating the action of muscles about the joints. Comparing the other models to the model in which experimentallyobtained kinematic data were used (model 3C) indicates that adding links usually improved the perfonnance of the models in terms of the effective mass M (Tables 1-3). The reason
Results for two-link models
M
V
(% 'tile)
(kg)
(mls)
Vwn/g
(deg)
FhtplMg
(leN)
5 95 5 95
34.0 52.5 15.9 24.5
4.00 4.25 3.35 3.57
18.6 15.9 22.8 19.6
93.1 93.6 92.5 92.9
19.7 17.0 23.8 20.6
6.56 8.74
Case Model
F hip
(wnt)mn
Journal of Biomechanical Engineering
F htp
(wnO max
3.72
4.96
AUGUST 1995, Vol. 117 I 313
!
"
Table 3 Model
Results for three-link models
Case (% 'tile)
M (kg)
(m/s)
Vwn/g
(wMt)ma, (deg)
5 95 5 95 5 95
18.8 28.4 16.5 25.6 17.1 26.2
3.98 4.34 3.60 3.94 2.47 2.93
24.9 22.1 24.1 21.2 16.2 15.6
92.3 92.6 92.4 92.7 93.5 93.7
3k no springs 3B: knee-spring 3C: kin. from expo
V
for this is that adding links results in more realistic impact configurations and therefore in better estimates for effective mass. In particular, the three-link model with the knee-spring resulted in impact configurations and effective mass very close to the values found through experiments (Tables 3, 4). However, adding links does not necessarily improve the prediction of the hip impact velocity Var the peak impact force F hip (Tables 1-3). For the three-link models, the incorporation of a quadratic knee spring to account for muscle forces acting at this joint resulted in a small improvement in the predictions for M, V, and F hip '
Discussion Using one-, two~. and three-link rigid-body models. we obtained ranges for the predicted hip impact velocity and effective mass of that part of the body that is moving prior to impact as a result of a sideways fall from standing height. The predicted values were compared with the results of an experimental study with human subjects falling to the side on a thick foam mattress. We found that increasing the complexity of our models did not necessarily result in an improvement in the predictions. One of our simple two-link models (the 45° lack-knife fall) predicted values for hip impact velocity that were closest to but still higher than values obtained using kinematic data from experiments. However, the effective mass was predicted most accurately by our most complicated model, a three-link model with a spring in the knee joint. This study represents the first in-depth theoretical analysis of the kinematics and dynamics of a fall from standing height. Even though our models are very simple, they provide substantial improvements over a point mass falling from the height of the center of gravity of the body. We were also able to compare the predictive fidelity of models of increasing complexity and found that a simple two-link model consisting of a leg-segment and a trunk at 45 degrees to the vertical (the 45° lack-knife fall) resulted in better predictions of the hip impact velocity and peak impact force than our more complicated three-link models. Although the relative simplicity of our models can be regarded as a strength of this study, it required a large number of assumptions and simplifications and therefore can also be considered a limitation. Regarding the level of realism of our rigid body models, we mention three limitations. First, we used energy conservation principles for all the calculations of the hip impact velocities. During a real fall, some work is done on active muscle as they are forced to lengthen, and this is likely
Table 4
Impact configurations for
Model 3A: no springs 3B: knee-spring 3C: kin. from expo
three~link
models
Case
a
e
(% 'tile)
(deg)
(deg)
y (deg)
5 95 5 95 5 95
90.0 90.3 90.1 90.2 89.8 89.8
58.7 52.9 35.4 33.2 33.8 33.8
25.6 26.5 39.0 37.5 35.1 35.1
314 / Vol. 117, AUGUST 1995
Fh,p
FhiplMg
(lu'l)
26.0
4.79 6.45 4.06 5.57 2.90 4.26
23.1 25.1 22.2 17.3 16.6
to be the primary reason why the predicted values for impact
velocities are higher than for the experimental case (model 3C). Second, except for the three-link model with the knee-spring, the action of muscles was not included in our models, and therefore any twisting or bending moments acting about joints' are not included. The third limitation concerns the body configuration of the vertical lack-knife model at impact. We realize that it would be anatomically impossible to fall with a vertical trunk and the legs extended horizontally. However, the experimental fall study showed that with knee flexion added, the trunk angle in front view (Fig. 5) could become as small as -8 degrees (16]. Regarding the level of realism of the impact model, the most important limitation is that the effects damping were not included in the calculations of hip impact force. Including damping (damping ratio S ~ 0.1, [23]) in our model decreases the peak impact force by about 19 percent. Because the impact model (Fig. 1) is separate from the fall models predicting M and V (Figs. 2-6), introducing light damping into the impact model makes no change in the predictions for M and V and no change in the relative ranking of the results for F hip (Tables 1- 3). Therefore, and for the sake of simplicity in our calculations, damping was not considered. We made some additional approximations in the calculation of the effective mass. First, we neglected the horizontal component of the hip impact force compared to the vertical component. However, if the friction between the impact surface and the impact site of the body is small, this is a reasonable assumption. Second, for the three-link modeL the thigh and lower leg segments were considered to be rigidly coupled during impact. This assumption was necessary to solve for the hip impact force. Finally, for calculation of the effective mass of the trunk segment at an angle f3 with the vertical (Appendix A, case 1, 3), we assumed that (32 ~ lijl at the time the peak force occurred. In the experimental fall study with human subjects [16], we found that p ~ 0.26 radls and therefore p' ~ 0.068 rad 2 / s 2, just before impact. During impact, the value for {3 rapidly decreases to zero. Values for jj during impact can be estimated by dividing the experimentally-obtained values for (3 just before impact by the approximate time it tak~~ to reach the peak force (tpeak)' This results in an estimate.for (3 of 8.67 radl s 2, assuming [peak ::::: 30 msec [23]. Thus, fJ 2 is less than one percent of jj, thereby confirming the validity of the approximation. As a final limitation, we mention the use of anthropometric data from young women in our rigid body models to simulate falls of elderly women. For our three-link models, we used a computerized data base which produced segment dimensions, masses, and mass moments of inertia about the three prinCipal axes for a given body height and weight (in percentiles). To our knowledge, there exists no such in~depth data based on elderly people. However, Jensen et a1. [15} recently performed measurements on 12 females and 7 males (age ranging between 63 and 75 years) and calculated masses of 16 body segments. Comparing the results of Jensen et a1. to the values used in our three-link models suggests that the mass of the trunk should be increased by 6.6%, the shank by 2.6% and the mass of the thigh should be decreased by 12%. Using this new anthropometric data set, and scaling segment mass moments of inertia by the Transactions of the ASME
same factors, resulted in very small changes in hip impact veloc~ ity and impact configuration. For example, for model 3A, we found a decrease in hip impact velocity of less than 1 percent (5th percentile: 3.97 versus 3.98 rnIs; 95th percentile case: 4.34 versus 4.31 rnI s). Regarding the impact configuration, values for angles B and y (which are determinants of the effective mass) increased by less than 0.5 degree (5 th percentile case: 58.7 and 25.6 degrees versus 59.2 and 25.7 degrees, respectively; 95th percentile case: 52.9 and 26.5 versus 53.5 and 26.7 degrees, respectively). Therefore, based on these comparisons, we believe it is justified to use anthropometric data from young adults in our three-link models and appJy the results to the elderly. While the lack of previous studies on the kinematics and dynamics of falls preclude comparison to the results of other workers, two previous studies suggest that falling with the muscles relaxed reduces the injury potential of a fall. Robinovitch et a1. r23] found that the muscle-active state at impact significantly increased the stiffness K, and therefore the impact force at the hip, compared to the muscle~relaxed state. In addition, through an experimental fall study with young athletes, van den Kroone~ nberg et al. [16] found that falling relaxed resulted in a 7 percent lower hip impact velocity than was measured in muscle-active falls. However, in the same study, we found that muscle-relaxed falls resulted in a 38% smaller trunk angle (defined as the angle of the trunk with the vertical) than in muscle-active falls. According to predictions made in the present paper, the more erect position of the trunk in muscle-reJaxed falls would increase the effective mass at impact and therefore increase the hip impact force. Given these competing influences, we cannot, from these simple models, answer the question whether falling relaxed would be safer than falling in a muscle-active state. However, based on our calculations of the effective mass, we now have a better insight into the effect of the configuration of the body on the hip impact force. For example, we found that a fan with a vertical trunk results in more than a two-fold increase in effective mass compared to an impact configuration in which the trunk is at 45 degrees with the vertical. Assuming no difference in V, this would result in an increase in the peak impact force of about 40 percent, since we found that Fpcar. is approximately v.fKM. Our models also have implications for understanding the eti~ ology of hip fracture and for the design of effective prevention efforts. Although we found that a tall and heavy person can be expected to experience a larger force applied to the greater trochanter after a fall to the side than a short and light person, this does not necessarily imply that tall and heavy people are at higher risk for fracture, since they may have stronger femurs. However, to shed further light on fracture etiology, we can compare our predicted values for the peak force applied to the greater trochanter with data on strength of young and old femurs in a loading configuration simulating a fall to the side with impact on the greater trochanter. Courtney et al. [5] performed in-vitro fracture tests of9 younger (mean age 32.7 :::+.: 12.8 [SD] years) and 8 older femurs (mean age 73.5:': 7.4 [SD] years) in this special loading configuration. The mean strength for the younger femurs was 7.21k :': 1.09 [SD] N and for the older femurs 3.44k :': 1.33 [SD] N. The majority of our predicted values for the peak impact force are below the strength of a young femur but above the strength of an old femur (Tables 1-3). More accurately, we can compare the average value for peak impact force predicted by the three-link model with 'its segment trajectories specified using kinematic data from experiments (model 3C), and decreased by 19 percent to correct for exclusion of damping, to these strength data (2.90 versus 3.44k N (old); 2.90 versus 7.21k N (young». This comparison confirms that a fall to the side can result in a hip impact force close to or even larger than the force required to fracture an old femur, but clearly not enough to fracture a young femur. Journal of Biomechanical Engineering
Acknowledgments This study was supported by grants from the National Institutes of Health, (AR40321, CA41295), and by the Maurice E. Mueller Professorship of Biomechanics at the Hanrard Medical School (WCH).
References 1 Anonymous. "National Center for Health Statistics. Advance Data from Vital and Health Statistics: 1985 Summary: National Hospital Discharge Survey," 1986, PHS 86-1250, Hyattsville. MD. 2 Asada, H., and Slotine. J.-J. E., Robot AlUll.vsis and Control, 1986, WHeylnterscience, New York. 3 Baughman. D. L., "Development of an Interactive Computer Program to Produce Body Description Data," 1983, Air Force Aerospace Medical Research Laboratory, Aerospace Medical Division, Air Force Systems Command, WrightPatterson Air Force Base, AFAMRL-TR-83-058. OH 45433. 4 Clauser, C. E., Tucker, P. E .. McConville, J. T .. Churchill. Eo. Laubach, L. L.. and Reardon. J. A .. "Anthropometry of Air Force Women." 1972. Aerospace Medical Research Laboratory, Wright-Patterson Air Force Base, AMRL-TR-705. OH 45433. 5 Counney. A. C, Wachtel. E. F., Myers, E. R .. and Hayes, W. c.. ·'Agerelated Reductions in the Strength of the Femur tested in a Fall Loading Configuration," J Bone Joint Surg. [Am). 1995, Vol. 77-A, no. 3:387-395. 6 Crandall. H. S. H., Kamopp. D. c., Kurtz Jr., E. F.. and Pridmore-Brown. D. c.. DynamiCS of Meclumical and Electromechanical Systems. 1968, Krieger. Malabar. FL. 7 Cummings. S. R.. Kelsey, J. L., Nevitt, M. c.. and O'Dowd, K. J .. "Epidemiology of Osteoporosis and Osteoporotic Fractures." Epidemiol Rev, 1985. 7: 178-208. 8 Grunhofer, H. 1.. and Kroh, G., "A Review of Anthropometric Dala of German Air Force and United States Air Force Personnel 1967-1968,'· 1975, Advisory Group for Aerospace Research and Development. AGARD-AG-205, 7 Rue Ancelle 92200, Neilly SUf Seine, France. 9 Holbrook, T. L.. Grazier, K., Kelsey. 1. L., and Stauffer, R. N., "The Frequency of Occurrence, Impact, and Cost of Selected Musculoskeletal Conditions in the United States," J984. American Academy of Orthopaedic Surgeons, Chicago, IL. 10 Horak. F. B., Shupert. C. L., and Mirka. A., "Components of Postural Dyscontrol in the Elderly: A Review," NeurobiolAging. 1989, Vol. 10. pp. 727738. 11 Hayes, W. C.. Myers, E. R., Morris, J. N., Gerhart, T. N., Yett, H. S. and Lipsitz, L A., "Impact Near the Hip Dominates Fracture Risk in Elderly Nursing Home Residents Who Fall." CalcifTissue Int, 1993,52:192-198. 12 Hayes, W. c.. Piazza. S. J .. and Zysset. P. K .. "Biomechanics of Fracture Risk Prediction u&ing Quantitative Computed Tomography". In: Radiological Clinics of North America, Rosenthal, D. L, ed, 1991, W.B. Saunders Phil., 29:1-18. 13 Hayes, W. C, Robinovitch, S. N., McMahon. T. A., "Energy-shunting Hip Padding System Reduces Femoral Impact Forces from a Simulated Fall to Below Fracture Threshold," 1993. Proc. of Third Injury Prevention Through Biomechanics CDC Symp .. Wayne State Univ .. Detroit, MI. 14 Hindmarsh, 1. J., Estes. E. H. Jr., "Falls in older Persons: Causes and Interventions," Arch Intern Med. 1989, 149:2217-2222. 15 Jensen. R. K., and Fletcher. P .. "Distribution of mass to the segments of elderly males and females," J Biomech, 1994. Vol. 27, pp. 89-96. 16 Kroonenberg. A. J. van den. Munih, P.. Weigent-Hayes, M., and McMahon, T. A., "Hip Impact Velocities and Body Configurations fOf Experimental Falls from Standing Height," J Biomech, 1995. in revision. 17 Lauritzen, J. B., Petersen. M. M., and Lund. B., "Effect of External Hip Protectors on Hip Fractures," Lancet, 1993. Vol. 341, pp. 11-13. 18 McMahon. T. A., Valiant, G., and Frederick, E. c., "Groucho Running," J Appl Physio/. 1987.62(6):2326-37. 19 Myers. E. R., Robinovitch, S. N., Greenspan. S. L., and Hayes. W. C. "Factor of Risk is Associated with Frequency of Hip Fracture in Case-control Study." Trans 40 th Annual Meeting. Orthopaedic Research Society, 1994. Vol. 19, p. 526. 20 Nevitt, M. c., and Cummings, S. R., "Falls and fractures in older women." In: Falls. Balance and Gait Disorders in the Elderly, Vellas, B., Toupet. M., Rubenstein, L, Albarede, J. L., Christen. Y .. eds., 1992, Elsevier, Paris, pp. 6980. 21 Nickens, H., "Intrinsic Factors in Falling Among the Elderly." Arch Intern Med. 1985. Vol. 145, pp. 1089-1093. 22 Obergefell. L. A .. Gardner, T. M., Kaieps, 1., and Fleck, J. T., "Aniculated Total Body Model Enhancements." 1988. AAMRL-TR-88-043 1-2, Wright-Patterson Air Force Base, OR 23 Robinovitch, S. N .. Hayes, W. C .• and McMahon, T. A., "Prediction of Femoral Impact Forces in Falls on the Hip," AS ME JOURNAL OF B10MECHAN1CAL ENGINEERING, 1991. Vol. 113, pp. 366-374. 24 Robinovitch. S. N.. McMahon, T. A.. and Hayes, W. c., "Energy-shunting hip padding system improves femoral impact force attenuation in a simulated fall," ASME JOURNAL OF BIOMECHANICAL ENGINEERlNG, in press. 25 Tinetti. M. E., SpeechJey, M., and Ginter. S. F, "Risk Factors for Falls Among Elderly Persons Living in the Community," N. Eng. 1. Med .. 1988, Vol. 319, pp. 1701-1707.
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APPENDIX A Calculatious of Effective Mass for Two- and ThreeLiuk Models In the following, we derive the effective mass of the individuallinks of the two- and three-link models. The effective mass for each link is calculated by expressing the vertical impact force applied at the hip in terms of the vertical acceleration of this point during impact (Z'hip) and then dividing this by Zmp + g, where g is the acceleration of gravity. The effective mass of the total models can be obtained by summation of the effective masses of the individual links.
1) Leg or Trunk Segment of Two-Link Model. Torque and force equilibrium result in the following (Fig.
knee
\ hip
\ x Fig. A2
Top view of shank and thigh of threeMlink model during impact
parallel-axis theorem, and neglecting the mass moment of inertia about the longitudinal axis of each of the two links, the following expression for the mass moment of inertia of the total leg about an axis passing through its own center of mass and parallel to the x-axis is obtained (Fig. A2) 1(1+2)
= cos 2
()
A] ): I
...
- F hip sin fj 2
= I{3
and
-munkg
+ F hip = mlinktcK'
(A.I, A.2)
where mhnk is the mass of one link. Also, from Fig. Al it follows that: Zhip
'2I cos (3,
(A3)
,; .. '2I (p2 cos (3 + f3 sin (3).
(AA)
= Zcg
-
and therefore, Z"hip
= Zeg
+
where 11 and 12 are the mass moments of inertia of links 1 and 2 about axes perpendicular to the individual links passing through their own centers of mass. Using Eqs. (A.l). (A.2), and (AA), but with 1I2 replaced by d (Fig. A2) and (3 = 90 deg, the following expression for the effective mass can be derived: M
10 + 2) = ---'-'-'''"''-d'
rJ2
Combining Eqs. A.I, A.2 and A4 and neglecting compared to iJ in the above equation, the following expression for the effective mass can be derived:
M=~=
to;, +
g
mlmk
3(sin (3)' + I .
(A5)
2) Leg and Thigh Segment of Three-Link Model. Since the reaction force at the knee joint of the model is unknown, we can not solve for the hip impact force by considering torque equilibrium of the thigh segment. Therefore, we assume that the leg and the thigh segments are rigidly interconnected during impact; i.e. for the purpose of calculating the effective mass only, links I and 2 are treated as a single link and the equations derived above can be used. In this case, the mass moment of inertia is not given by ~mhnkz2 but instead is defined by values from the literature [3, 4, 22]. Using the
with d given by (Fig. BI):
(A.7)
+ 10 + 2 )
m,) cosO
-I ( 3 + 2 m,
d=----1 + m2
(A.S)
m, 3) Trunk Segment of Three-Link Model. As before, angle f3 is defined between the vertical and the trunk. The calculations for the two-link model (case 1 of this Appendix) can be used again. Using the same thre~ equa~jons (Eqs. (A.]), (A2) and (AA» and assuming that (3'
