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The Effect of Walking Speed on the Joint Displacement Patterns and Forces and Moments Acting on the Above-Knee Amputee Prosthetic Leg

S.A. Hale, M.Sc., C.O.

Abstract

Above-knee (AK) amputees generally walk slower than normal subjects. To understand why this occurs, the kinetics (forces and moments) of walking, at different walking speeds, must be examined. The purpose of this study was to determine the effect of walking speed on stride length, stride and swing times, the knee and hip displacement and torque patterns, and the roles of the gravitational forces and interactive moments associated with thigh motion. Three subjects were filmed walking at three self-selected walking speeds (slow, natural, fast). The swing phase was analyzed in detail using the inverse dynamics approach. As walking speed increased, the stride length increased while stride and swing times decreased. The maximum knee flexion and initial hip flexion peaks increased as walking speed increased. As walking speed increased, the knee flexor torque during terminal swing increased, and this was necessary to slow the leg down in preparation for the subsequent weightbearing stage. The principle moment limiting knee flexion was the gravitational moment, which was assisted by the interactive moments associated with thigh acceleration and velocity (L-AT and L-VT). The hip muscular torque pattern was characterized by four phases - initial flexor and extensor, final flexor and extensor. The initial extensor torque decelerated the thigh, which in turn contributed to the term L-AT. L-AT assisted in accelerating the leg and resisted knee flexion and initiated knee extension. The final hip extensor torque slowed down the thigh and maintained the knee in extension, through the term L-AT in preparation for footstrike.

Introduction

It is well documented that AK amputees walk slower than normal subjects, although the AK amputee can achieve walking speeds similar to that of normal subjects (10,13,14). In order to understand why the AK amputee walks slower, one must thoroughly examine the swing phase, because it is the period when the majority of the stride length is attained. Walking speed is dependent upon stride length and time (Equation 1).

Walking Speed Stride Length/Stride Time

The stride time consists of a stance and swing time (Figure 1) .

There is little difference between the sound limb and normal swing times, whereas the prosthetic leg swing time is considerably longer. This difference amounts to more than half of the increase in stride time (13,14). When AK amputees walking at a fast pace are compared with normal subjects walking at the same pace, that speed being free walking speed, the AK amputee takes a longer stride length and a longer stride time, which is brought about primarily by a substantial increase in the swing time. It has, therefore, been suggested that the time to recover the prosthetic limb may be a critical factor in dictating and perhaps limiting the walking speed of the AK amputee (8,17).

Factors potentially influencing the prosthetic swing time, and which have been identified as possible causes of the reduced walking speed include: (1) alterations in the knee and ankle function, (2) hip musculature and (3) inertial parameters - mass, mass center location and moment of inertia of the prosthesis (2,6,8,9,13,14,17).

AK amputees altered their hip motion, and these alterations were in the form of: (1) an abrupt reversal from hip extension to flexion just prior or during early swing, (2) a rapid reversal from hip flexion to extension during late swing, (3) a decrease in the total hip excursion (6,13,14). While these alterations are readily observed, their importance in terms of assisting the amputee to recover the prosthetic limb are not well understood.

In order to understand how the swing limb is moved through space, an examination of the forces and moments acting on the segments is necessary. Generally, there are externally applied forces, such as gravitational and ground reaction forces. During the swing period, the ground reaction forces do not exist. Other forces applied to individual segments are represented collectively as resultant joint force (RJF) and a resultant joint moment (RJM) associated with one or both ends of the segment (Figure 2). The RJF represents the contact forces applied across or through the joint (e.g. bone-on-bone contact forces). The RJM largely comprises the moments of forces exerted by muscles crossing the joint.

In normal gait, considerable attention has been placed on the RJM-time histories, while less emphasis has been paid to the RJF role (2,20,21). It has been shown in normal and pathological gait studies that the RJF plays an important role in segment motion during the swing phase (9,11,16,22). It has been documented that gravity has an important function during the early swing stages (16,22).

The prosthetic limb can be modeled as a compound pendulum, consisting of a fixed proximal (hip) joint and a joint between the thigh and leg segments. Frigo and Tesio (6) and Murray et al. (13) have referred to pendular action to describe the kinematics of the AK prosthesis during the swing phase. When the proximal segment is accelerated forward, the distal segment moves backward, resulting in joint flexion (Figure 3a) . The opposite occurs when the proximal segment is accelerated backwards, resulting in the distal segment swinging forward and the joint going through extension (Figure 3b) .

Putnam (15), using inverse dynamic approach, describes this segmental interaction in terms of the RJF. The RJF is broken down to describe the swing motion of a segment in terms of the adjacent segment motion, gravity and hip joint linear acceleration. The advantage of this type of analysis is that it allows quantification of the effect of the hip RJM on the knee and leg motion through the adjacent segment velocity and acceleration terms and segment mass on the swing motion of the prosthetic limb.

The purposes of this study were to determine the effect of walking speed on: (1) stride length, stride and swing times, (2) selected knee and hip displacement peaks, (3) peak knee and hip RJM and (4) the roles of the gravitational and interactive moments on the prosthetic leg motion of above-knee amputees.

Methodology

Subjects: Three subjects, two males and one female, volunteered to participate in the study. Each was required to read an outline of the study and give written consent before participating. Subject descriptions are presented in Table 1 .

Prostheses: The prostheses used by the subjects are described in Table 2 . The subjects were dressed in shorts and T-shirts. Markers were positioned on the subjects' limb to delineate the position of the hip joint (the greater trochanter), the knee joint (axis of rotation of the knee unit) and the heel and toe of the shoe. As the subjects walked, a Locam camera recorded their gait at a rate of 75 frames/second. Two trials for three self-selected walking speeds for each subject were collected. Self-selected means the subjects were asked to walk at a slow, normal and fast pace, and the individuals selected their own walking speed for that pace.

For each filmed trial, the swing period was defined by a toe-off and footstrike frame. Five points (origin, hip, knee, heel and toe markers) in each frame within the swing period were digitized. The relative coordinates for each point were stored in a computer. From this data, the joint and segment displacement data were derived, then smoothed using a zero lag, low pass, second order Butterworth digital filter to reduce the effect of noise (19). The smoothed data was then twice differentiated using a finite difference routine to generate the joint and segment linear and angular velocities and accelerations.

Segment Inertial Parameters (SIP): To determine the SIP of the prosthesis, the prosthesis was taken apart at the center of rotation of the knee. The prosthetic thigh section consisted of the socket and upper half of the prosthetic knee unit. The prosthetic leg section consisted of the lower half of the prosthetic knee unit, the shank and the prosthetic foot. The mass, mass center location and moment of inertia of the prosthetic thigh and leg sections were determined by weighing each section, balancing each on a knife-edge beam and oscillating them in a specially designed pendulum.

To determine the SIP of the residual thigh, the mass of the anatomic thigh was required. This was calculated using data from Dempster's cadaver study (4) and correcting for the subjects' height and weight. The mass of the residual thigh was measured as a percentage of the remaining thigh length. To determine the amount of inertia and mass center location, the residual thigh was modeled as a right frustrum of a cone. Known equations were used to mathematically calculate the mass center location and moment of inertia of the residual thigh (1). The SIP of the prosthetic thigh section were combined with the residual thigh SIP to provide the SIP for the thigh segment.

Prosthetic Limb Model: The forces and moments acting on each segment were calculated using the inverse dynamics procedure. The outcome of the forces, the segment and joint displacement can be measured from the film, and the segment inertial parameters are mathematically determined; from this information the forces and moments can be calculated.

To calculate the leg kinetics, the prosthetic lower limb was modeled as a rigid two link system (Figure 4) . Standard Newtonian equations of motion were written for each segment. With appropriate equation substitution, the following equations were derived for the analysis.

LEG:
L-NET = RJMK + L-AT + L-VT + L-AH + L-G = L-G

THIGH:
T-NET = RJM_h + T-AT + T-VT + T-AL + T-VL + T-AH + T-G

(The terms are defined in the appendix.)

In order to describe the roles of the leg interactive moments for the three walking speeds, four periods were defined:

• Period 1-early swing (the first .04 seconds after toe-off) where the initial conditions of the swing phase may influence the maximum knee flexion attained;
• Period 2-post maximum knee flexion (.04 seconds after maximum knee flexion) is the period when the amputee initiates control of knee extension;
• Period 3-the period, .02 seconds before and after (total of .04 seconds) the maximum positive thigh angular acceleration, when the thigh changes its direction of rotation (determine the effect of the thigh acceleration on the leg motion);
• Period 4-the period, .02 seconds before and post the maximum negative thigh acceleration, when the thigh is decelerated. During these defined periods, the average was determined for the moments acting on the leg and the kinematic variables in question.

Statistics: A simple linear regression was performed (18). This regression was performed to quantify the strength of a relationship between selected variables. The correlation between two variables indicated whether an increase in one variable was associated with an increase, or decrease, of another variable. All subject data was grouped to see if general patterns existed despite differences in subjects' prostheses.

Results

Time related data and stride kinematics: Temporal and stride kinematic data, for each subject, for the prosthetic side, are summarized in Table 3 . Two subjects did not attain three distinct speeds, as defined by Murray et al. (14). As walking speed increased, the prosthetic stride length increased (r = .6905), stride and swing times decreased (r = - .7770 and - .7225) (Figure 5) .

Joint angular displacement: The knee and thigh (hip) angle curves for each individual trial at each speed, expressed as a percent swing time, are presented in Figures 6a and 6b . The knee and thigh angles were measured as illustrated by the stick figures. The means and standard deviations for selected knee and hip angular displacement variables are presented in Table 4 .

The knee curves (Figure 6a) demonstrated a normal pattern of flexion followed by extension and the knee held in hyperextension for a period before footstrike. Maximum knee flexion ([theta]_kM) attained increased as walking speed increased (r = .8195), but its time of occurrence (T[theta]_kM)) did not correlate as well (r = .4221).

The thigh (hip) was in an extended position at the time of toe-off (Figure 6b) . The thigh was flexed until a maximum hip flexion angle ([theta]_hM1) was attained and this angle increased as walking speed increased (r = .7875). A weak correlation occurred between the time at which maximum hip flexion (T([theta]_hMl)) occurred and walking speed (r = .5931). The thigh then extended attaining a minimum flexion angle ([theta]_hM2). No correlation existed between minimum hip flexion angle and walking speed (r = .3570), while its time of occurrence (T([theta]_hM2)) was delayed as walking speed increased (r = .7168). The thigh was briefly flexed again reaching a final peak flexion (P_hM3). This peak and its time of occurrence (T([theta]_hM3)) did not correlate with walking speed (r = .4125 and .0898). Finally the hip was extended at the end of the swing phase for the slow and free walking speeds. Kinetic data - moment contributions to leg motion: Figure 7 (subject A, slow speed trial 33) illustrates the various moments contributing to the net moment acting on the leg, which represents the leg acceleration. The moment contributions for each section are presented in Figure 8 .

Knee and hip moments acting on the prosthetic limb: The averaged knee RJM curves for each walking speed are seen in Figure 9 . Generally there was a very low extensor (positive) moment (torque) for the initial half of the swing phase. There was no correlation between the maximum knee extensor torque and walking speed (r = .0719, Figure l0a ). The latter half of the swing phase was characterized by an increasing knee flexor moment as walking speed increased (r = .7847, Figure l0b ).

The averaged hip RJM curves for each speed are presented in Figure 9b . Generally, there were four distinct phases: (1) an initial flexor (H1), (2) an extensor (H2), (3) a final flexor (H3) and (4) a final extensor moment (H4). The initial peak hip flexor (positive) torque (Figure 11a) and final hip extensor (negative) torque (Figure l1d) did not correlate well with walking speed (r = .1377 and - .1221, respectively). There was a weak correlation between the initial hip extensor (H2) torque (r = .6822, Figure 11b ) and walking speed, and the final hip flexor (H3) torque (r = .5823, Figure 11c ) and walking speed, meaning that as walking speed increased, the hip extensor and hip flexor torques increased as walking speed increased, although this was not consistent in all trials.

Discussion

The walking speeds of the three subjects were within one standard deviation of the previously recorded AK walking speed classification (13,14). The AK subjects walked slower than normal subjects, based on the relative speed, and this supported previous findings (3,10,13,14,23). Based on the data from Murray et al. (14), two AK amputees (subjects B and C) were able to attain normal free walking speeds while walking at their fast pace.

The change in walking speed was a result of increased stride length and a decrease in the stride time. Both variables correlated well with relative walking speed. This trend was similar to that seen in normal subjects (7). It has been suggested that the reduced walking speeds of AK amputees may be related to the prosthetic swing time (5,14,17). As walking speed increased, the absolute swing times of the AK amputees decreased. However, the swing times for the AK amputee were longer than for the normal subject when similar walking speeds were compared. This difference in swing time may be the result of an altered knee angular displacement pattern, changes in the function of the residual hip musculature and prosthesis SIP.

It is important to examine the maximum knee flexion angle since excessive flexion during swing is one of the most frequently seen gait deviations of AK amputees (6,8,9,13,14,17,23). There was a strong negative correlation between maximum knee flexion and walking speed. This increase was consistent with the pattern seen in normal subjects (12,21). The increase in swing time has been attributed to the longer time needed to cover a larger knee range of motion because of the increased knee flexion and the hyperextension (8). There are several factors that may influence the maximum knee flexion attained during the swing phase: (1) the initial conditions of the swing phase which includes (a) the knee angle at toe-off, (b) the velocity of the knee at toe-off (this is dependent on the leg and thigh velocities; (2) the time allowed to attain maximum knee flexion; and (3) the forces and moments acting on the segments and joint.

The thigh angular kinematics were not similar to normal hip swing motion (12,21). The average thigh (hip) patterns exhibited three patterns: (1) slow-double periods of flexion/extension; (2) free-flexion/extension/flexion; and (3) fast-flexion/extension. It is believed that because of an insufficient and/or ill-responsive knee unit, the thigh angular kinematics were altered. Murray et al. (14) found that the rapid reversal from flexion to extension at the end of the swing phase (slow speed example) was a necessity to ensure the prosthetic knee was fully extended in time for the subsequent footstrike. Frigo and Tesio (6) stated that the final hip extension led the knee unit to lock because of the inertia of the prosthesis. To understand the process by which this occurs and how the thigh motion affects the leg motion, the kinetics (forces and moments acting on the segments) must be examined.

Kinetic Data

Period I. - During this period the knee continued to flex, but the leg was being accelerated (forward). Therefore moments contributing to the leg acceleration, reflected by a positive L-NET term, resisted knee flexion. The gravitational moment (L-G) was the principle contributor to the leg acceleration. The interactive moments L-VT, L-AT and L-AH assisted in accelerating the leg. The prosthetic knee RJM was negligible except at the slow walking speeds (Figure 8) . Changes in the walking speed did not alter this contribution order.

The gravitational moment and interactive moment due to the linear acceleration of the hip joint (L-AH) correlated well with walking speed (r = .7817 and .7519, respectively). The strong correlation of the term L-AH suggests that the hip joint was being moved at a greater rate forward and/or upward as walking speed increased and this was particularly important at the fast walking speed (Figure 8) . How the hip joint gains in acceleration is unclear, but may be related to what is happening on the contra-lateral side just before this stage of the gait cycle, or the result of the hip flexor moment.

The presence of the term L-AT contributing to knee flexion during the first section, reflected one compensation the AK amputee made to overcome the loss of the plantarflexors. In normal gait, the plantarflexors were the principal initiators of the swing, with the hip flexors assisting them (2,21). The AK amputee relied upon the hip flexors (H1, Figure 9b ) to lift the lower limb off the ground and begin the forward rotation of the thigh. The hip flexor moment increased the thigh angular velocity and acceleration. The increased thigh velocity increased the positive contribution of L-VT moment, which assisted in preventing excessive knee flexion. The increase in thigh acceleration, on the other hand, increased the contribution of the negative L-AT moment, which attempted to flex the knee. The reported lower knee extensor RJM coupled with the contribution of the negative L-AT, led to the continued knee flexion and hence a larger than normal peak knee flexion, particularly at the fast speed.

To reduce the knee flexion, moments contributing to leg acceleration should be increased. An increase of any of the four moments would assist in reducing the knee flexion. The two most obvious moments to increase would be the RJM_k and/or L-G. The disadvantage of increasing the interactive terms, L-AT or L-VT, would be the expected need for an increased hip RJM and hence an increased muscular effort. An increase in the L-G term may not cause a similar increase because of its contribution to the thigh motion through the interactive terms T-AL and T-VL. To increase the knee RJM would require an adjustment to the swing phase controls or re-designing the swing phase control mechanisms. Period 2: In period 2 the knee passes through peak knee flexion and the leg continues to be accelerated as reflected by the positive L-NET term. For all three subjects, regardless of the walking speed, the gravitational moment was the main contributor to the leg acceleration. The secondary contributor was L-AT followed by L-VT.

Only L-VT correlated with walking speed (R= .7710). This was related to the increased thigh velocity and changes in the relative knee angle. The prosthetic knee torque continued to contribute little to the leg acceleration.

During period 2, the hip extensors were active (H2, Figure 9b ). This moment performed a dual function. Most obvious was to extend the hip joint and, second, to assist in limiting knee flexion. The hip extensor torque contributed to decelerating the thigh. The thigh interacted with the leg such that thigh deceleration resulted in the leg being accelerated forward via the term L-AT. Hence, the hip extensors assisted in accelerating the leg forward and resisted excessive knee flexion (the peak flexion angle occurred during this period) and initiated knee extension. Since the prosthetic knee devices could not generate enough force to control the amount of knee flexion, it appeared that the AK amputee increased the hip extensor RJM, as walking speed increased, to alter the thigh kinematics, to influence the leg and knee motion.

Period 3: During this period the knee was extending and the leg was being decelerated, while the thigh was being accelerated. The primary contributor to the leg deceleration was the term L-AT for the slow speed. The knee RJM was the primary contributor for the free and fast speeds. The knee RJM and L-AT were the secondary contributors for slow, free and fast speeds, respectively. The term L-G was the tertiary contributor for all subjects and speeds.

As walking speed increased, the net moment (L-NET) increased (r = - .7382), reflecting the increase in leg acceleration. The knee torque and L-VT term correlated well with walking speed (r= - .7317 and .7713), while L-AH exhibited a weaker correlation (r = - .5937). Because of the low magnitude of the contribution made by L-VT, the correlation was clinically insignificant.

Generally, as leg negative acceleration (deceleration) increased, the negative (flexor) knee torque increased. The prosthetic knee assistive devices, such as the friction control, finally played an important role in the motion of the leg acceleration and the knee joint motion.

In period 3 a hip flexor torque existed. This torque contributed to a positive thigh acceleration. Moments accelerating the thigh forward, and providing a positive thigh velocity and ultimately assisted in decelerating the leg, through the terms L-AT and L-VT. The hip flexor torque contributed to thigh acceleration. Hence, this torque indirectly assisted in contributing to the slowing down of the leg through the interactive term L-AT.

Period 4: The leg and thigh were both decelerating and the knee was being held in full extension. The primary contributor to the leg deceleration was the knee flexor torque, which was assisted by the gravitational (L-G) and L-AH terms.

Both the L-NET and the knee torque terms exhibited very strong correlations with walking speed (r = -.8010 and - .8555). The L-G term also exhibited a weak correlation (r = - .6320).

The fact that the knee torque was the primary contributor for all speeds and had a strong correlation with walking speed, demonstrated the importance of the RJM in decelerating the leg in preparation for the subsequent footstrike.

Terminal swing was characterized by a hip extensor torque (H4, Figure 9b ). This torque not only extended the hip, but indirectly played a vital role in maintaining a stable knee before footstrike. The term L-AT acted in the opposite direction to the other moments, including L-NET. Hence, L-AT attempted to accelerate the leg (or maintain knee extension or resist knee flexion). This moment was critical in maintaining an extended knee. Therefore, moments decelerating the thigh (negative terms) were important in maintaining an extended knee.

Conclusions

• As walking speed increased, the prosthetic leg swing time decreased, and these times were longer than normal subjects walking at comparable speeds.
• Significant changes occurred in the hip displacement pattern during the swing phase.
• The principle moment accelerating the leg forward (resisting knee flexion) was the gravitational moment, L-G; it was assisted by the interactive moments due to thigh angular acceleration and velocity. The residual hip flexors attempt to overcome the loss of the plantarflexors, by accelerating the thigh upward and forward; the greater thigh acceleration, for the faster walking speeds, increased the negative term L-AT, which contributed to negative leg acceleration and excessive knee flexion. The negligible contribution of the prosthetic RJM plus the increased negative L-AT term resulted in the larger knee flexion.
• During the periods of leg deceleration the prosthetic knee RJM became an important contributor. The interactive moment L-AT continued to play a vital role in swing leg motion and exhibited how the AK amputee utilized the hip musculature to control the prosthetic knee.
• Hip extensor torque during terminal swing slowed the thigh down and maintained the knee in extension in preparation for the subsequent weight-bearing stage.

Appendix

X - cross product
* - multiplication
k - knee joint
h - hip joint
l - leg segment
t - thigh segment
R - distance between proximal joint to segment mass center
L - length of segment (l or t)
M - mass of the segment
W - weight (mass * gravitational acceleration) of segment l or t
I - moment of inertia of segment measured about a transverse axis passing through segment's mass center
cm - mass center of the leg or thigh
g - acceleration due to gravity
[theta] - joint (k or h) or segment (1 or t) angle
A - linear acceleration of the segment 1 or t
AV - angular velocity of segment 1 or t
AA - angular acceleration of the segment 1 or t
Ax, Ay - linear acceleration of the hip joint in x and y direction
RJF - resultant joint force about joint h or k
RJM - resultant joint moment about joint h or k

Acknowledgements

Thanks to Dr. C. Putnam for unlimited guidance and support.

S.A. Hale, M, Sc., C.O. (Canada) is with Kawartha Orthopaedic Services, St. Joseph's Hospital, 384 Rogers St., Peterborough, Ontario, Canada K9H 7B6. Work was carried out at the School of Physical Education, Dalhousie University, Halifax, Nova Scotia, Canada.

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