Body Segment Moments of inertia of the Elderly

Transcription

1 JOURNAL OF APPLIED BIOMECHANICS, 1993,9, O 1993 by Human Kinetics Publishers, Inc. Body Segment Moments of inertia of the Elderly Robert K. )ensen and Paula Fletcher The segment principal moments of inertia of a sample of 7 elderly males and 12 elderly females were estimated using a model based on stacked elliptical cylinders at 2cm intervals. Apart from the thigh, all male parameters were larger than female parameters. The largest differences were for the lower trunk and hand and for the forearm. The inertia parameters of the thigh for the males were about 12% smaller than the females. Nonlinear estimations of segment principal moments were then determined. The effect of the differences was tested by cross validating cadaver results against the elliptical model results. The regressions were then cross validated using an independent sample of 6 subjects. The standard errors of fit given as a percentage of the mean, Sf, were smaller than the cross validation results for the cadaver regressions and the differences were attributed to differences between cadavers and living subjects. With aging, humans become more susceptible to accidental death and injury. In the United States, the increasing population of elderly adults are having to deal with problems that stem from mobility impairment; declining joint flexibility, muscle strength, power, and postural balance; and daily activities such as chair transfers and gait (Schultz, 1992). According to 1989 Canadian figures (Statistics Canada, 1991), falls in particular are a leading cause of death after 70 years (Table 1). Also, although motor vehicle deaths show a steady decline from the 20to29 year cohort, the figures include motor vehicle traffic and nontraffic deaths, and the latter are probably higher for the elderly. In falls and motor accidents, the segments of the body are subjected to large angular accelerations. These are usually a result of external force, such as an impact, or of changes in an external force, such as friction. In some cases, net joint moments that attempt to bring the accelerations under control are produced. The net joint moments are in part volitional, due to the contraction of muscles about the joint. The problem is controlling the angular momentum of the segments. The volitional response is the magnitude and timing of the moments about the joints, and it is recognized in general terms that the joint moments decline with age (Gabbard, 1992). The segment principal moments of inertia are primary variables required for the Robert K. Jensen and Paula Fletcher are with the Centre for Research in Human Development and School of Human Movement, Laurentian University, Sudbury, ON, Canada P3E 2C6.

2 288 Jensen and Fletcher Table 1 Accidental Deaths From Motor Vehicle Accidents and Falls in Canada, 1989 Age range (years) All accidents Motor vehicle Falls Note. Based on Statistics Canada (1991), pp analysis and simulation of angular momentum problems. The purpose of this study was to examine the segment principal moments of inertia of elderly males and females and to predict the principal moments using standard anthropometry and a rationale based on nonlinear regression ( & Morlock, 1989). An accurate knowledge of the segment principal moments of inertia is a prerequisite for the modeling of human movement. The equations for net moments relative to a principal axes coordinate system embedded in the segment (Ounpuu, Davis, Tyburski, & Gage, 1988) are obtained with Euler equations of motion, where I,,, I,,, and I, are the principal moments of inertia approximately about the anteroposterior (AP), transverse (TVS), and longitudinal (LON) axes. The components of the sum of the external moments applied to the segment M, the segment absolute angular acceleration a, and the segment absolute angular velocity o are also included in the equations. If cadaver moments of inertia are being measured, the body is dissected into segments and the segment mounted in a pendulum. Lightweight specimen holders are used with six swing axes to determine the moments and products of inertia (Lephart, 1984) from which the principal moments and the directional angles of the principal axes are decided. Chandler, Clauser, McConville, Reynolds, and Young (1975) have produced the most comprehensive cadaver study of segment moments of inertia. Six male embalmed cadavers stored for at least 1 year were dismembered using segmentation planes similar to those reported by Clauser, McConville, and Young (1969). The mean age was 54.3 years with a standard deviation of 7.4 years. Principal moments of inertia were calculated for the segments of the upper and lower extremities, the head, and the torso. These cadaver data from Chandler et al. (1975) could be used for the simulation of movements for each of the six individuals. However, the extrapolation of these results to the elderly (60 years and older) population would depend

3 Moments of Inertia 289 upon the external validity of the segment statistics. The number of cadavers is small and there are inconsistencies between the physical sizes of the cadavers and the elderly population. The mean stature of the sample was 1.72 m and this can be expressed as a 55th percentile based on the age of the sample and results from a crosssectional study of 6,142 males and 7,549 females conducted by Nutrition Canada (Desmirjian, 1980). The mean body mass of 65.2 kg gave a 22nd percentile, however, which was probably due to the ailment prior to death and the extended period of storage. Reduced mass could have an effect on segment moment of inertia by affecting the total mass being distributed, f dm and the distribution, 12 about a reference axis, such as through the segment center of mass. Concern has also been expressed about the magnitude of the principal moments, I,,, I,,, I,,, reported by Chandler et al. (1975). and Morlock (1989) found that of the six torso principal moments all but one failed to meet the inequality condition In their study of the estimation of segmental inertia parameters, they converted all segment principal moments to a pseudo data set of moment of inertia values, I,, I,, I,, about the AP, TVS, and LON anatomical axes. Chandler et al. (1975) cautioned that their data should not be interpreted as reflecting population parameters due to the restricted number of specimens. However, because of the limited data available, others have used the study to predict segment principal moments. Hinrichs (1985) used the anthropometric measures from Chandler et al. as predictor variables for the transverse and longitudinal principal moments but with the multiple linear regressions restricted to two predictor variables because of the small number of specimens. However, theoretical relationships can be used to form nonlinear regressions that are more appropriate. and Morlock (1989) presented a rationale for a nonlinear approach to the estimation of segment inertia parameters. They argued that theoretical relationships between quantities are known and can be used to form nonlinear regression equations that are more appropriate than linear multivariable regression equations. For the head and upper and lower extremity segments, two constants kl and k2 are determined from simple linear regression. These are then used in nonlinear equations to predict longitudinal moment of inertia (I,) and ''transverse' ' moment of inertia (I,), an average of I, and I,, using a weighted perimeter p and the length of segment h. The trunk requires three constants, cl, c2, and c3, which are determined by

4 290 Jensen and Fletcher solving a set of simultaneous equations. The three moments of inertia for axes that pass through the segment center of mass are predicted from using segment width w, depth d and length h. This rationale was used with the pseudo data set of segment moments of inertia and the length, perimeter, and width data from the Chandler et al. (1975) study. The nonlinear equations were cross validated by comparing left limb data to equations based on the right limb. In addition, the equations were evaluated outside the sample range from anthropometry on a 10yearold male together with criteria values obtained by the stadium mathematical model of (1990). Based on these favorable preliminary results it was suggested that the nonlinear equations should be tested with values calculated from inertia models as criteria. Nonlinear regressions that use data from wholebody gamma mass scanning have been presented by Zatsiorsky, Seluyanov, and Chugunova (1990). Their sample consisted of 100 healthy young adult males. All segments were considered as cylinders in the geometrical model proposed for the nonlinear regressions and four coefficients, K, K,, K, and KI determined for each segment. Principal moment of inertia parameters can be estimated using where I,, If, Il are the sagittal, frontal, and longitudinal axes moments of inertia; length L and circumference C are anthropometric measurements; and the mass m = KLC. (14) The coefficient K takes into consideration the differences in shape and density between the scanned segment and its cylindrical model. For subjects within the range of the original study (age 23.9 years, SD = 6.2 years; stature m, SD = m; mass 73.0 kg, SD = 9.1 kg), the nonlinear regressions were found to be more accurate than height and mass multiple regressions but less accurate than linear predictions using segment anthropometric independent variables. Also, results for those with significant fat deposits were not particularly accurate. The size of the sample makes these regressions appealing but the results may not extrapolate to females, other races and cultures, and younger and older males. Nonlinear regression should increase the probability of accurate predictions, however. The Chandler et al. (1975) sample was all male, and the differences in the shape of segments between males and females could affect the estimation of principal moments in accordance with Equation 4. Changes in body shape with aging for males and females are different, and the survey by Nutrition Canada shows that although there is little change in stature, body mass continues to increase throughout the adult years (Desmirjian, 1980). By 5059 years mean

5 Moments of Inertia 291 body mass increases by 22% for females and 7.5% for males. Gender differences in shape for the upper trunk and thigh, in particular, could affect the principal moments (Jensen, Fletcher, & Abraham, 1991). Some indication of differences between male and female segment moments of inertia prior to the slow accumulation of mass throughout the adult years is given in the study of young Japanese adults by Ae, Tang, and Yokoi (1991). Samples of 215 male subjects (mean age 19.8 years, SD = 1.1) and 80 female subjects (mean age 19.9 years, SD = 1.2) were evaluated using the elliptical cylinder method (Jensen, 1978; Yokoi, Shibukawa, & Ae, 1986). Radius of gyration about the three anatomical axes through the segment center of mass were given as a proportion of segment length. The most noticeable differences were for the head, where the male radii for all three axes were larger, and the thigh, where the female radii were larger. Other interesting results included a larger longitudinal axis radius (k,) for female upper trunk and forearm, a larger female arm segment (k, and k,), and a larger male radius for the transverse axis (k,) of the lower trunk. Multiple regression equations with anatomical axes moments of inertia (I,, I,, I,) for males and females as dependent variables and segment length and body mass as predictor variables gave multiple correlations ranging from.31 to.96. The application of the elliptical cylinder model has made it clear that there are differences between individuals in segment inertias and these differences include sex differences. However, these differences have not been shown adequately for the elderly population principal moments of inertia. The first objective of this study was to (a) describe the segment principal moments of inertia of elderly males and females and (b) compare the segment principal moments of inertia of elderly males and females. and Morlock (1989) have provided a method for predicting segment principal moments. However, their predictions are limited by the size, composition, and circumstances of the Chandler et al. (1975) sample. The elliptical cylinder model provides a noninvasive shapesensitive estimation of principal moments. The second objective of the study was to (a) determine nonlinear estimations of segment inertia parameters using the and Morlock (1989) rationale and the combined sample of elderly males and females, (b) compare these predictions with the regressions based on the Chandler et al. (1975) sample of male cadavers, and (c) make recommendations for the estimation of the segment principal moments of inertia of elderly males and females. Method The subjects in this study of the description and prediction of the principal moments of inertia of the elderly were 12 females age 63 to 75 years (mean 67.4 years) and 7 males age 63 to 75 years (mean 69.5 years). The 19 subjects were volunteers recruited from the elderly population of the Region of Sudbury, Ontario. The mean stature of the sample was 1.60 m (SD = 0.20 m) for the females and 1.70 m (SD = 0.30 m) for the males. The mean mass was 64.7 kg (SD = 8.0 kg) for the females and 79.4 kg (SD = 8.9 kg) for the males. The mean stature and mass were also converted to percentiles corresponding to age and gender (Desmirjian, 1980) and these established that the females were of average stature (56.6 %ile) and mass (51.2 %ile). The males were above average

6 292 lensen and Fletcher for both stature (66.3 %ile) and mass (80.2 %ile). All subjects were of a healthy disposition. A 16segment stacked elliptical cylinder mathematical model (Jensen, 1978; Jensen & Nassas, 1988) with a cylinder height of 2 cm was used to estimate the segment principal moments of inertia of the sample of elderly males and females. The centroidal principal moments of inertia of an elliptical cylinder of uniform density are 1 I,, = m(r; + r:) 4 where the major and minor axes r, and ry and the cylinder height d, are averaged from adjacent pairs of cross sections of the longitudinal (z) axis. The cylinder mass m is calculated from the density p and the dimensions of the cylinder, The segment density was assumed to be uniform and as reported by Clauser et al. (1969). However, the trunk was subdivided at the level of the xiphoid process and upper and lower trunk densities were based on Dempster (1955). Segment moments of inertia with respect to the horizontal (x and y) and vertical (z) axes are determined by applying the parallel axis theorem and summing across the elliptical cylinders representing the segment, n 4 = [In + m(l?; + If)] (19) I n I; = C [Iyy + dl;l + If)] (20) where l,, l,, and 1, are the components of the distance between the cylinder centroidal axes and the three parallel reference axes through the segment centroid, and there are n elliptical cylinders in the segment. The shape of each segment with respect to the horizontal and vertical axes produces products of inertia P,,, P,,, and P,,. The segment centroidal axes can be rotated, however, to a position where the products of inertia vanish. These axes are the principal axes and the corresponding moments of inertia are the principal moments. Eigenvalues give the principal moments and eigenvectors give the direction cosines of the principal axes of the segment. If required, the moments of inertia for the three principal axes through the body center of mass or any other point can be obtained in similar fashion. The data for the model were taken from orthogonal front and side 35mm single lens reflex camera photographic images recorded simultaneously. The

7 Moments of Inertia 293 cameras were 7 m from the subject midfrontal and midsagittal planes, and 200mm lenses were used. The subject was placed at the intersection of the two view lines and in an upright position, similar to the anatomical orientation. The forearms were pronated, the hands were extended with fingers and thumb together, the feet were plantar flexed by having the subject stand on an inclined base, and the neck was extended slightly to expose the anterior surface. Small paper disks on the skin aided in the location of the joint centers and boundaries of the segments. The subject was clothed in his or her bathing costume and most were of a tight fit. However, some were a loose fit and judgment of body outline was necessary during analysis. A rubber swim cap was essential to flatten the hair of some subjects and long hair was tied back. A calibration scale for each of the two images consisted of a pair of meter sticks forming a set square within the midsagittal or midfrontal plane. The scale was adjustable so that it was vertical and could be placed close to the subject. A cluster of lamps to illuminate the subject and a contrasting screen behind provided a sharp image. The body image was projected and front and side body segment outlines digitized. During the outlining with a graphics digitizing tablet (Scriptel), a computer program automatically determined coordinates at 2cm intervals. A further program calculated the major and minor axes and used these to compute the inertias of the elliptical cylinders and the segments. The and Morlock (1989) procedure for determining nonlinear regressions was used to address the second objective of the study. These regressions were preferred to those of Zatsiorsky et al. (1990) because the Chandler et al. (1975) sample (mean 54.3 years) was much closer in age to the elderly population and the trunk was defined by depth, width, and height, rather than by a cylinder model. The 29 anthropometric measurements needed for the regressions (Chandler et al., 1975; & Morlock, 1989) were recorded using standardized techniques and a stadiometer, anthropometer, and tape. Two measurements needed for the two segment trunk were added. Upper trunk length (h) was measured from the acromion to the level of the xiphoid. Lower trunk length (h) was measured from the level of the xiphoid to the trochanter. Perimeter and width for the upper trunk were at the level of the nipple. Perimeter and width for the lower trunk were averaged from the measures at the umbilicus and hip levels. The constants for Equations 6 to 10 were determined and compared to the constants from the Chandler et al. sample. The 19 subjects (Sample I), 12 females and 7 males, comprised the validation sample for the predictions based on nonlinear regressions. Unbiased standard error of fit was estimated for each segment and moment of inertia in accordance with the procedures presented by and Morlock (1989), n Ii)2/(n k)]0.5 (22) sf = [C(J; I where Ji is the inertia estimates, Ii is the inertia data for the n subjects, and k is the number of constants used in the equation. Next, a set of standard errors was calculated using the constants determined by and Morlock (1989) and the elderly males and females () as a cross validation sample. The work by and Morlock (1989) with a subject outside the range of their primary sample suggested that various subjects could be combined, and it was decided to use male and female subjects together for the nonlinear regressions.

8 294 lensen and Fletcher An additional 6 subjects () were selected from the same population as an independent cross validation sample of 4 females, mean age 63.8 years (SD = 1.6 years) and 2 males, age 68 and 73 years. The mean stature was 1.73 m (SD = 0.16 m) for females and 1.69 m and 1.68 m for the males. Body mass for the females had a mean of 70.6 kg (SD = 8.0 kg), and the males were 82.1 and 65.2 kg. The female subjects were younger by 3.5 years and had mass and stature that were less than the validation sample, but the means were within 1 standard deviation. The males were of similar age and stature to the validation sample and their masses were within f 1 standard deviation of. It was expected that the nonlinear regression predictor and criterion variables for the cross validation sample would fall within the range of the validation sample. Although the sample was small it was the same size as the Chandler et al. (1975) sample of 6 males. Results Accuracy of the total body mass, estimated by summing the segment masses and comparing the measured mass, was determined for the 7 males and 12 females in the sample. The mean error was 0.05% with a standard deviation of 2.96%. Originally, 5 additional subjects were analyzed, but the estimated body masses were in error by more than 6%, approximately 2 SD of the error reported by Jensen (1989), and were considered to be outliers. The errors for the rejected subjects were not systematically positive or negative. When the outlines were redigitized, the changes for each subject were minimal. Accuracy and precision across segments for volume has been evaluated using standard objects (Jensen & Fletcher, in press); intraobserver common variance (R2) ranged from.996 to.998 and interobserver common variance ranged from.992 to.998. Analysis of variance was used to establish that nonsignificant differences for operator repeated measures were consistent with operator reliability, and nonsignificant differences between three observers were indicative of operator objectivity. Mean and standard deviation summary statistics for the three principal moments of inertia of the nine segments are presented in Table 2. The means of the moments of inertia for the males and females were then used to examine gender differences (Table 3). However, the data from all 19 subjects were used in the nonlinear regressions. The three constants c,, c2, and c3 required for the trunk (Equations 8, 9, and 10) are given in Table 4. The limited anthropometry and moments of inertia from Chandler et al. (1975) restricted the first column of constants, taken from and Morlock (1989), to the whole trunk. The same rationale was used to determine the constants for the upper and lower trunk using the elliptical cylinder method data as the criterion. In each of the three cases the constants are characteristic of the morphology and density distribution of the segment. The two constants k, and k2 required for the nonlinear regressions (Equations 6 and 7) are given for the head and upper and lower extremity segments in Table 5. The corresponding constants provided by and Morlock (1989) and based on Chandler et al. (1975) are also tabulated. To assist the interpretation of the magnitudes of the standard errors, comparisons can be made with standard errors for segment mass and transverse moment of inertia estimates (Table 6) based on proportions from Dempster (1955) as presented by Winter (1979).

9 Moments of Inertia Table 2 Male and Female Principal Moments of lnertia Statistics for the Nine Segments (in kg m2) Segment Male I, Female I, Male 1, Female 1, Male, 1 Female I, Head Upper trunk Lower trunk Arm Forearm Hand Thigh Leg Foot Table 3 Percentage Differences Between Males and Females in the Mean Principal Moments of lnertia for the Segments Segment Mean I, Mean I, Mean I, Mean Head Upper trunk Lower trunk Arm Forearm Hand Thigh Leg Foot Mean

10 lensen and Fletcher Table 4 Trunk Segment Constants Based on Chandler et al. (1975) and the Upper and Lower Trunk Constants Determined From Constant Chandler Upper trunk Lower trunk Table 5 Determined Coefficients k, and k, for and the Corresponding Coefficients From and Morlock (1989) Segment k2 k, (Chandler) k, (Chandler) Head Arm Forearm Hand Thigh Leg Foot Mean Table 6 Standard Errors for Segment Mass and Transverse Moment of Inertia (in %) Based on Demgster (1955) Proportions Segment Sf (Mass) Sf (Its) Head Arm Forearm Hand Thigh Leg Foot Mean

11 Moments of Inertia 297 The elliptical cylinder model I, and I, results were used as the criterion values for the validations and cross validations of the head and appendage segments predictions (Tables 7 and 8). and Morlock (1989) regressions were cross validated with, and the regressions were cross validated with. A third set of constants was determined for, and the standard errors are presented for comparison in the tables under Validation. Table 7 is of the longitudinal (I,) standard errors and Table 8 is of the "transverse" (I3 standard errors. Four figures are presented to show the validation and the cross validation of the predictions. One segment, the forearm, was selected. Figure 1 illustrates the data and regression for the longitudinal moment. The regression based on the and Morlock (1989) constant is added and illustrates the cross validation of their prediction. Figure 2 is the corresponding illustration for the transverse moment. It should be noted that the forearm standard error of fit for the and Morlock (1989) regression was the largest of the five segments (Table 7). Figure 3 shows the longitudinal moment data and regression and the regression as a cross validation. Figure 4 is the corresponding illustration for the forearm transverse moment of inertia. With the exception of the and Morlock (1989) regression (Table 7), the forearm standard errors were approximately the median errors for the regressions. Table 7 Standard Errors of Fit for Validation and Cross Validation Longitudinal (I,) Principal Moments Estimates Sample size Validation Cross validation Segment Sample. (4?/') (yo) Head Upper arm Forearm Thigh Lower leg Note. Data obtained from and Morlock (1989), and regressions.

12 298 Jensen and Fletcher Table 8 Standard Errors of Fit for Validation and Cross Validation Transverse (I,) Principal Moments Estimates Sample size Validation Cross validation Segment Sample (fl)?/o) W) Head Upper arm Forearm Thigh Lower leg Note. Data obtained from and Morlock (1989), and regressions. Discussion The accuracy of the total body mass estimation, with a mean error of 0.05% and standard deviation of 2.96%, for the 7 males and 12 females was acceptable and consistent with values reported in comparable studies (Ae et al., 1991; Finch, 1985; Jensen, 1989; Sanders, Wilson, & Jensen, 1991; Yokoi et al., 1986). Actual measures of segment moments of inertia of living subjects are probably not feasible and accuracy has to be judged from indirect sources, such as body mass and the volumes of inanimate objects. The inaccurate results for the 5 subjects treated as outliers were attributed to loosefitting bathing suits. Many of the subjects were sensitive to close observation and wore their own bathing costumes. It is possible that the number of outliers could have been reduced, and it is recommended that subjects be provided with more appropriate clothing prior to the testing session. The summary statistics (Table 2) give a description of the magnitude of the three principal moments of inertia for each of the body segments. The means can be compared across the three axes for males and females. For the arm, forearm, thigh, leg, and foot, all of which approximate a cylinder, the I,, and I, values, principal moments about the A/P and TVS axes, were similar but the y axis values were consistently slightly larger. The I, means, principal moments about the LON axis, were much less and ranged from 10% to 23% of the I,, and

13 Moments of Inertia 299 Figure 1 Prediction of the elliptical cylinder model forearm longitudinal principal moments data using the determined and and Morlock (1989) k, constants. In keeping with Equation 6, the variables are k,p4h for the xaxis and I, (kg. m2) for the yaxis. O Moments; + Regression; 0 Regression I,, values. The head, which is approximately spherical in shape, had I,, values greater than I,, values by approximately 3% for males and 14% for females. The I, means were less than the principal moments for the other axes and ranged from approximately 70% to 88%. It appears that the moments produced by the shape of the head are not consistent with a sphere, and the determination of principal moments and axes requires detailed shape analysis. The lower trunk, upper trunk, and hand had I,, values, which are between 9% and 30% greater than the I,, values. Compared to the other axes moments the I,, means were approximately 30% for the hand and 68% for the lower trunk. In the case of the

14 300 lensen and Fletcher Figure 2 Prediction of elliptical cylinder model forearm transverse principal moments data using the determined and and Morlock (1989) k2 constants. In keeping with Equation 7, the variables are 0.51, + k2p2h3 for the xaxis and I, (kg. m2) for the yaxis. 0 Moments; + Regression; 0 Regression upper trunk, the relationships between principal moments were much less regular. The male I,, mean was larger than the I,, and I,, values by 7% and 55%, respectively. For the females, the longitudinal moment was 14% smaller than the mean for I,, but 14% greater than I,,. These differences within segments indicate the irregularities of mass distribution and the need to accommodate these differences by capturing shape effectively and developing an appropriate rationale for prediction. The comparison of the percentage differences between males and females for the three principal moments of inertia (Table 3) revealed that all of the body

15 Moments of Inertia 301 Figure 3 Cross validation of the predicted forearm longitudinal principal moments using data. In keeping with Equation 6, the variables are klp4h for the xaxis and I, (kg. mz) for the yaxis. LI Moments; + Nonlinear Regression; 0 Regression segments, excluding the thigh, were larger in the male; the lower trunk, forearm, hand, and foot in particular were larger in the male. These differences are similar to the mass proportion differences that have been reported (Jensen & Fletcher, in press). Desmirjian (1980) found that on average beyond 60 years of age males have greater stature and mass (mean height of crn and mean mass of kg) than females (mean height of cm and mean mass of kg). The greater stature and mass explain in part the larger segment moments of inertia. Results showed that on average the male inertia values are 53% larger than female values for the I,, and I,, principal moments of inertia (Table3) and 66%

16 302 jensen and Fletcher OXn3 0.m2 om1 aot 0.m 0.m 0.m 0.m om I I I I I I I I I > 0,0000 rn 11#140 am Qrn Figure 4 Cross validation of the predicted forearm transverse principal moments using data. In keeping with Equation 7, the variables are 0.51, + k,pzh3 for the xaxis and I, (kg. mz) for the yaxis. O Moments; + Nonlinear Regression; 0 Regression larger for the longitudinal (I,) principal moment, where the greatest difference was for the forearm. In particular, the more readily available male inertias should not be used for females. The magnitude of the differences in segment inertias between males and females would be expected to introduce substantial errors into the analysis and simulation of motion. The thigh was the only segment where the female had larger inertia values than the male for all three principal moments of inertia. The anthropometry results revealed that the mean length for the female thigh (M = 0.39 m, SD = 0.2 m) was greater than the male thigh (M = 0.37 m, SD = 0.04 m) and a longer

17 Moments of Inertia 303 thigh should result in larger inertia values for the segment. The weighted mean perimeters for the females and males were the same (M = 0.48 m, SD = 0.3 m and 0.02 m, respectively) suggesting that the length of the segment was the prime determinant of the gender difference. Desmirjian (1980) reported comparable length anthropometry. He found a mean female lower extremity length of m and a mean male lower extremity length of m, and that lower extremity length relative to stature for females over the age of 60 was greater than the values obtained for males. 's and Morlock's (1989) constants for the trunk segment pertain to the trunk as a whole. For the elliptical cylinder method constants were determined for the upper trunk and the lower trunk separately. When the Chandler constants are compared to the lower trunk constants (Table 4) they are more similar than the upper trunk constants. This is consistent with the lower trunk being much longer than the upper trunk (M = 0.39 m, SD = 0.03 m and M = 0.18 m, SD = 0.02 m, respectively). These predictions have not been cross validated. Constants for the remaining segments of the body were also calculated (Table 5). Overall, the differences appear to be small between the Chandler et al. (1975) constants and the elliptical model constants. Individual comparisons, however, show that the constants are different and thus reflect a distinct density distribution. The differences were explored by cross validation of the predictions. For the segments listed in Table 7, the standard error of prediction of I, was less in each case. For example, the standard error for the forearm decreased from 57.38% to 34.14% when the elliptical model constants were substituted for the Chandler constants. The results suggest that the k, constants based on the elliptical cylinder calculations when used in the nonlinear regressions (Equation 6) are somewhat better predictors than the Chandlerbased constants. However, it should be noted that there are several reasons for the differences in the standard errors. First, the populations are different, with the Chandler et al. sample representing elderly male cadavers and representing living elderly males and females. Second, there are errors associated with both methods of estimating the moments of inertia, with inequality problems (Equation 5) in the Chandler procedure and some uncertainties about the effects of the assumptions on the elliptical cylinder procedure. It is submitted that the differences in standard errors are primarily due to the differences between the two populations. The difference between the two predictions for the forearm is illustrated in Figure 1, where the effect of the different constants can be seen. The cross validation of the I, predictions (n = 19) using the second sample (n = 6) resulted in lower standard errors (Sf) for three of the five predictions (Table 7). For example, when the elliptical model constants were used to predict forearm I, with the smaller sample, the Sf decreased from 34.14% to 26.41%. The cross validation Sfs should be considered in relation to the validation Sfs for, in this case 13.3%. However, as with the Chandler sample, because of the small sample the Sf can be unduly influenced by a single subject with high values for both the predictor and criterion variables. On the other hand, the cross validation Sfs have to be compared to the validation Sfs which were smaller (Table 7). These validation Sfs can also be compared to the Sfs for the Dempster (1955) mass proportions (Table 6), and it can be seen that the errors and the mean error are very similar. These results support the cross validation and the contention that the constants and regressions

18 304 lensen and Fletcher are suitable for application to samples of elderly males and females similar to the validation sample. For the transverse moment of inertia (I,), the validation standard errors were smaller than the and Morlock (1989) Sfs (Table 8, Figure 2). The Sfs for the Dempster (1955) I,,, proportions, with mass used as a proportion of total mass and transverse axis radius of gyration as a proportion of segment length, were much larger than the I, Sfs and had a mean error of 76.94% when the foot and hand were included (Table 6). The cross validation of the I, regressions using produced very low Sfs for the thigh and leg, which were similar to the Sfs for the validations (Table 8) and reflect the similar regression coefficients (Table 5). The other three segment cross validations were also reasonable as were the comparisons with the smaller validations (Table 8, Figure 4). and Morlock (1989) suggested in their paper that the nonlinear equations be tested for a range of subjects using values calculated from an inertia model as criteria. The results of this study of a range of elderly males and females using the elliptical cylinder model calculations of principal moments as criteria show some differences from regressions based on elderly cadavers (Chandler et al., 1975). This was anticipated because of cadaver sample limitations and concerns about the external validity of the mass distribution estimations. The magnitude of the Sfs reported by and Morlock (1989) for the cadaver sample was very similar to the Sfs for the cross validation sample (Tables 7 and 8), which was of the same size (n = 6). This suggests that the equations are equally appropriate for the living sample. The differences between the two studies become evident when the regression constants (Table 5) and cross validations (Tables 7 and 8, Figures 1 to 4) are examined. Although the difference could be due in part to the need to use the moments of inertia rather than the principal moments from the Chandler et al. (1975) study, it is more likely that there are fundamental differences in mass distribution between the cadavers and the living. It is recommended that because of the wide diversity of body shapes in the elderly, the elliptical cylinder model should be applied directly. However, if predictions are preferred, it is recommended that the nonlinear regressions developed by and Morlock (1989) and based on 3 1 anthropometric dimensions and the constants developed in the this study should be used. References Ae, M., Tang, H., & Yokoi, T. (1991). Body segment parameters of Japanese adults. Proceedings of the 12th Annual Meeting of the SOBIM Japan (pp ). Japan: SOBIM. Chandler, R.F., Clauser, C.E., McConville, J.T., Reynolds, H.M., & Young, J.W. (1975). Investigation of inertial properties of the human body (Technical Report, AMRL TR74137). WrightPatterson Air Force Base, OH. Clauser, C.E., McConville, J.T., & Young, J.W. (1969). Weight, volume and center of mass of segments of the human body (AMRL Technical Report, TR6970). Wright Patterson Air Force Base, OH. Dempster, W.T. (1955). Space requirements of the seated operator (WADC Technical Report, 55159). WrightPatterson Air Force Base, OH. Desmirjian, A. (1980). Anthropometry report. Ottawa: Health Promotions Directorate.

19 Moments of Inertia 305 Finch, C.A. (1985). Estimation of body segment parameters of college age females using a mathematical model. Unpublished master's thesis, University of Windsor, ON. Gabbard, C. (1992). Lifelong motor development. Dubuque, IA: Brown. Hinrichs, R.N. (1985). Regression equations to predict segmental moments of inertia from anthropometric measurements: An extension of the data of Chandler et al. (1975). Journal of Biomechanics, 18, Jensen, R.K. (1978). Estimation of the biomechanical properties of three body types using a photogrammetric method. Journal of Biomechanics, 11, Jensen, R.K. (1989). Changes in segment inertia proportions between four and twenty years. Journal of Biomechanics, 22, Jensen, R.K., & Fletcher, P. (in press). Distribution of mass to the segments of elderly males and females. Journal of Biomechanics. Jensen, R.K., Fletcher, P., & Abraham, C. (1991). Body shape changes in the elderly and the influence of density assumptions on segment inertia parameters. In R.E. Herron (Ed.), Biosteriometric technology and applications (pp , SPIE Vol. 1380). Bellingham, Washington: International Society for Optical Engineering. Jensen, R.K., & Nassas, G. (1988). Growth of segment principal moments of inertia between four and twenty years. Medicine and Science in Sports and Exercise, 20, Lephart, S.A. (1984). Measuring the inertial properties of cadaver segments. Journal of Biomechanics, 17, Ounpuu, S., Davis, R.B., Tyburski, D.J., & Gage, J.R. (1988). Threedimensional lower extremity joint moments. In C.E. Cotton (Ed.), Proceedings of the Fifth Biennial Conference of the Canadian Society for Biomechanics (pp ). Ottawa: Canadian Society for Biomechanics. Sanders, R.H., Wilson, B.D., & Jensen, R.K. (1991). Accuracy of derived ground reaction force curves for a rigid link human body model. International Journal of Sport Biomechanics, 7, Schultz, A.B. (1992). Mobility impairment of the elderly: Challenges for biomechanics research. Journal of Biomechanics, 25, Statistics Canada. (1991). Morta1it)rSummary list of causes 1989 (Suppl. 1, Vol. 3, No. 1). Ottawa, ON: Supply and Services Canada. Winter, D.A. (1979). Biomechanics of human movement. New York: Wiley., M.R. (1990). The simulation of aerial movement. Part 11: A mathematical inertia model of the human body. Journal of Biomechanics, 23, 6774., M.R., & Morlock, M. (1989). The appropriate use of regression equations for the estimation of segment inertia parameters. Journal of Biomechanics, 22, Yokoi, T., Shibukawa, K., & Ae, M. (1986). Body segment parameters of Japanese children. Japanese Journal of Physical Education, 31, Zatsiorsky, V., Seluyanov, V., & Chugunova, L. (1990). In vivo body segment inertial parameters determination using a gammascanner method. In N. Berme & A. Capozzo (Eds.), Biomechanics of human movement: Applications in rehabilitation, sports and ergonomics (pp ). Worthington, OH: Bertec Corporation. Acknowledgments This research was supported in part by a grant from the Natural Sciences and Engineering Research Council of Canada (NSERC).