Decreased Knee Extensor Torque During Single-Limb Stance: A Computer Simulation Study of Compensations and Consequences

Abstract

Background/Objectives: For over 50 years, it has been suggested that the plantar flexors and hip extensors can compensate for weak knee extensors and prevent collapse of the leg during a single-limb stance. However, the effects of these compensations have not been studied thoroughly. The purpose of this computer simulation study was to determine, for a given posture, the hip and ankle net joint torque (NJT) required to prevent leg collapse due to systematic decreases in knee NJT and to determine the effect of these compensations on the horizontal ground reaction force. Methods: Single-limb stance was simulated using a static, multisegmented model in eight different postures. For each posture, the knee NJT was systematically decreased. The ankle and knee NJT necessary to prevent lower extremity collapse, along with any net horizontal ground reaction forces, were then calculated. Results: Decreases in knee NJT required linear increases in ankle and hip NJT to prevent the limb from collapsing. There were greater increases in ankle NJT compared to hip NJT, resulting in posteriorly-directed horizontal ground reaction forces. While the magnitudes were different, these findings applied to all postures simulated. Conclusions: For a given posture, ankle and hip NJTs can compensate for a decrease in knee NJT. However, this resulted in a horizontal ground reaction force, which was in the posterior direction for all the postures examined. This horizontal ground reaction force would induce an acceleration on the body’s center of mass that, if not accounted for, could have deleterious effects on achieving a task objective.

1. Introduction

One of the tasks during single-limb stance is to support the bodyweight by preventing a “collapse” (hip flexion, knee flexion, and ankle dorsiflexion) of the lower extremity. The principle of motor abundance [1] suggests that this can be accomplished in many ways. Both Sutherland [2] and Murray et al. [3] have suggested, over 50 years ago, that the plantar flexors could stabilize the knee during single-limb stance if the knee extensors were weak. This idea was extended by Winter’s concept of a support moment [4], which is the algebraic sum of the ankle, knee and hip torque (moment). He proposed that, as long as the overall support torque was sufficiently large, the lower limb would not collapse, despite decreased torque at any one individual joint. However, he never determined exactly how large “sufficiently large” had to be, nor what (if any) “side effects” may result from increasing torque at one joint while decreasing torque at another. Despite these shortcomings, textbooks still describe the ability of the plantar flexors and hip extensors to extend the knee [5].

However, for any given state of the musculoskeletal system, there is one unique solution for each net joint torque (NJT) that satisfies the equations of motion. This would suggest that if one or more joint torques compensated for insufficient torque at another joint in the chain, then either the posture or the forces acting on the system would have to be different. This paper is concerned with the latter.

Simple models have been used in biomechanics research to elucidate important principles of human movement. Hof [6] demonstrated, through simple models, that each net joint torque about a medial–lateral axis will produce both vertical and horizontal forces in the sagittal plane. A hip NJT will produce a force into the ground that has an orientation parallel to a vector from the knee-joint center to the ankle-joint center. Similarly, a knee NJT will produce a force into the ground that is parallel to a vector from the hip-joint center to the ankle-joint center, and an ankle NJT will produce a force into the ground that is parallel to a vector from the hip-joint center to the knee-joint center [6]. This suggests that, for a given posture, any increase in one NJT to compensate for a deficiency in another NJT will create a net horizontal ground reaction force (GRF). However, this idea has not been thoroughly explored.

There is some experimental evidence to support this assertion, however. In a study of hopping, variance in ankle, knee, and hip NJT minimized vertical ground reaction force variance, but increased horizontal ground reaction force variance [7]. However, that study did not link specific ground reaction forces to specific NJTs, nor did it examine how these links are affected by posture. Both are important to further our understanding of compensations and their consequences.

The primary purpose of this computer simulation study was to examine a model’s responses to a decrease in NJT at the knee while a particular posture is maintained and collapse of the leg (operationally defined as creating a vertical ground reaction force equal to the model’s body weight) is prevented. I hypothesized that a decrease in knee NJT would be compensated for by increases in ankle and hip NJT, and that these compensations would create a nonzero horizontal ground reaction force. A secondary purpose of this study was to test the model’s performance when ankle and hip NJTs were reduced.

2. Materials and Methods

The model consisted of 4 rigid bodies representing a foot, shank, thigh, and HAT (head, arms, and trunk). The rigid bodies were connected together by frictionless revolute joints. The generalized coordinates were sagittal-plane angles for the ankle, knee, and hip (q_A, q_K, and q_H, respectively). There was a torque actuator about each joint, representing the net joint torque at the ankle (T_A), knee (T_K), and hip (T_H). Segmental lengths and masses were based on published data as a function of height and body mass [8,9,10] for a man who is 1.8 m tall, with a body mass of 85 kg. See Figure 1.

For a given static posture, initial T_A, T_K, and T_H were solved for using Kane’s Method in the MotionGenesis Kane 6.3 software environment (MotionGenesis LLC, Portola Valley, CA, USA). I verified that these joint torques were identical to those obtained using standard inverse dynamics equations [10]. The contributions of each NJT to the horizontal and vertical GRFs in the sagittal plane were calculated using Jacobian matrices, as outlined by Valero-Cuevas [11]. I compared the sum of these contributions to the original GRF values to validate the calculations. Each NJTs’ resultant ground reaction force vector was calculated using the square root of the sum of the squares, and the angle between it and positive horizontal was calculated using trigonometric functions. The location of the center of pressure (COP) was calculated as described in [12] and the location of the system’s center of mass in the Y- and Z-direction was calculated as the weighted average of the position of each segment in each direction. I confirmed that the COP location was directly under the system’s center of mass. Additionally, I calculated the orientation of the position vectors from joint center to joint center relative to the positive Y-axis (horizontal), and compared them to the orientation of the ground reaction force created by each NJT. See Figure 2 for an example.

To simulate knee extensor weakness, T_K was decreased by 30% in 1% increments, while keeping the vertical GRF equal to body weight. Kane’s Method in the MotionGenesis Kane 6.3 software environment (MotionGenesis LLC, Portola Valley, CA, USA) was used to solve for T_A, T_H, the horizontal GRF, and the location of the COP. I discarded any trial in which the COP was outside the base of support (length of the foot). For each decrement in T_K, the contributions of each NJT to the horizontal and vertical GRF were once again calculated using the methods outlined by Valero-Cuevas [11] and the orientation of the vectors as described above. The sum of the contributions equaled the vertical and horizontal ground reaction forces, validating the calculations.

Only conditions that required a knee extensor NJT were considered. Conditions that were not considered either required no knee NJT (q_A = q_K; q_H = 0) or required a knee flexor NJT (q_A = q_K; q_H > 0 and q_A > q_K). The following conditions were modeled:

• Hip-Joint Center Anterior to Ankle-Joint Center: To meet this condition while still requiring a knee extensor NJT, q_K had to be greater than q_A but less than 2 × q_A, and q_H had to be less than q_K. I chose to model these conditions as q_A = q_H and q_K = 1.5 × q_A.
• Hip-Joint Center Vertically Aligned with the Ankle-Joint Center: For this condition, q_K ≈ 2 × q_A. I chose to model these conditions with q_H ≈ q_A and with q_H ≈ q_K. Because the shank and thigh are not the same length [8], the exact angles inputted into the model for q_K were 20.04124° and 40.08514° when the q_A values were 10° and 20°, respectively.
• Hip-Joint Center Posterior to Ankle-Joint Center: For this condition, q_K had to be greater than 2 × q_A, and q_H had to be larger than q_K. I chose to model these conditions as q_K = 3 × q_A and q_H = 4 × q_A.

Each condition was simulated in an initial configuration (less flexed) and then with all of the angles doubled (more flexed). A complete list of all joint angles is listed in

Table 1. The joint angles used for each simulation condition.

Condition	Less Flexed				More Flexed				HJC Relative to AJC
		qA (°)	qK (°)	qH (°)		qA (°)	qK (°)	qH (°)
qK = 1.5 × qA; qH = qA	A *	10	15	10	B	20	30	20	Anterior
qK ≈ 2 × qA; qH ≈ qA;	C	10	20.04	10.04	D	20	40.08	20.08	Aligned
qK ≈ 2 × qA; qH ≈ qK;	E	10	20.04	20	F	20	40.08	40	Aligned
qK = 3 × qA; qH = 4 × qA	G	10	30	40	H	20	60	80	Posterior

* Letters correspond to postures in Figure 3.

and the angles are represented with stick figures in Figure 3. To test the model’s performance, the process was repeated for each of these conditions with decreases in ankle NJT and then decreases in hip NJT.

3. Results

The same linear relationship existed between changes in T_K and changes in T_A, whether T_K was decreased or T_A was decreased. Similarly, the same linear relationship existed between changes in T_K and changes in T_H, regardless of which of these two values was decreased. In other words, decreases in T_A or T_H could be modeled as an increase in T_K. The results testing the model’s performance will therefore be presented as increases in T_K from 100% to 130%. For all simulations that decreased T_K, the COP remained within the base of support; this was not the case for all conditions in which T_K was increased.

A decrease in T_K led to linear increases in T_A (Figure 4) and T_H (Figure 5), although the slopes of the lines differed between postures. Increases in T_K led to linear decreases in T_A and T_H. For each posture, the slope was the same for increases and decreases in T_K. Similarly, a change in T_K led to a linear increase in the horizontal GRF (Figure 6), with the slopes also being dependent upon the posture. In all cases, a decrease in T_K resulted in a horizontal GRF—one oriented in the negative (posterior) direction, and one oriented in the positive (anterior) direction was seen with an increase in T_K.

Each torque created vertical and horizontal components of the GRF. The sum of all the components met the task objective of creating a vertical GRF equal to body weight (833.85 N). The sum of the horizontal components of the GRF only equaled zero when the T_K was 100% (Figure 7, Figure 8 and Figure 9). The resultant angles from the positive horizontal axis are presented in

Table 2. The angle between the NJT contribution to the ground reaction force and the horizontal.

Condition		Less Flexed				More Flexed
		θFA (°)	θFK * (°)	θFH (°)		θFA (°)	θFK * (°)	θFH (°)
qK = 1.5 ×* qA; qH = qA	A **	95	87.48/86.76	80	B	100	84.99/83.49	70
qK ≈ 2 × qA; qH ≈ qA;	C	100.041	90/89.02	80	D	110.85	90/87.98	70
qK ≈ 2 × qA; qH ≈ qK;	E	100.041	90/89.02	80	F	110.85	90/87.98	70
qK = 3 × qA; qH = 4 × qA	G	110	94.97/93.49	80	H	130	99.93/96.77	70

* The orientation of the position vectors, from joint center to joint center, relative to the positive Y-axis (horizontal) and the orientation of the ground reaction force created by each NJT were identical for the ankle and hip. For the knee, the first number represents the orientation of the position vector from the ankle-joint center to the hip-joint center, while the second number is the orientation of the knee force calculated by NJT. ** Letters correspond to postures in Figure 3.

. This angle was equal to the angle created by the position vector from the respective joint centers, except for the knee. A complete description of the results for each condition is presented below.

3.1. Hip-Joint Center Anterior to Ankle-Joint Center (A and B)

T_A produced a GRF vector that had an orientation parallel to the position vector from the KJC to the HJC, and T_H produced a GRF vector that had an orientation parallel to the position vector from the AJC to the KJC. The difference in orientations between the vector T_K-produced GRF and the position vector from the AJC to the HJC was less than 1° for the less-flexed position and less than 2° for the more-flexed position. Increased flexion increased the angle for the GRF produced by T_A and decreased the angle produced by T_H and T_K. T_A produced a posteriorly-directed GRF, while T_K and T_H produced an anteriorly-directed GRF for both the less-flexed and more-flexed conditions. Because T_A had a greater increase in response to a decrease in T_K than did T_H, there is a net negative horizontal GRF that increased from the less-flexed to the more-flexed position.

3.2. Hip-Joint Center Vertically Aligned with the Ankle-Joint Center (C, D, E, and F)

Two different trunk inclination angles were simulated in this position. The first was when q_H ≈ q_A (Conditions C and D). Initially, T_H was zero. T_A produced a GRF vector that was parallel to the position vector from the KJC to the HJC, and T_H (when present) produced a GRF vector that was parallel to the position vector from the AJC to the HJC. The difference in orientation between the vector T_K-produced GRF and the position vector from the AJC to the HJC was less than 1° for the less-flexed position and approximately 2° for the more-flexed position. Increased flexion increased the angle for the GRF produced by T_A and decreased the angle produced by T_H and T_K. T_A produced a posteriorly-directed GRF, while T_K and T_H produced an anteriorly-directed GRF for both the less-flexed and more-flexed conditions. Because T_A had a greater increase in response to a decrease in T_K than did T_H, there is a net negative horizontal GRF that increased from the less-flexed to the more-flexed position. For the second condition, q_H = q_K (Conditions E and F). This increased T_H. The directions of the force vectors were the same (q_H does not affect the direction of the force vectors) as when q_H ≈ q_A, but the greater T_H meant that there was less of a negative horizontal ground reaction force.

3.3. Hip-Joint Center Posterior to Ankle-Joint Center (G and H)

T_A produced a GRF vector that was parallel to the position vector from the KJC to the HJC, and T_H produced a GRF vector that was parallel to the position vector from the AJC to the HJC. The difference in orientation between the vector T_K-produced GRF and the position vector from the AJC to the HJC was less than 2° for the less-flexed position and approximately 3° for the more-flexed position. Increased flexion increased the angle for the GRF produced by T_A and T_K and decreased the angle for the GRF produced by T_H. T_A and T_K produced a posteriorly-directed GRF, while T_H produced an anteriorly-directed GRF for both the less-flexed and more-flexed conditions. Because T_A had a greater increase in response to a decrease in T_K than did T_H, there was a net negative horizontal GRF that increased from the less-flexed to the more-flexed position.

3.4. Model Performance

Increasing T_K had the opposite effect on the model, compared to decreasing T_K. Whereas decreasing T_K below 100% led to increases in both T_A and T_H, increases in T_K led to decreases in both T_A and T_H. In four conditions (C, D, G, and H), increasing T_K above 100% gradually led T_A to a dorsiflexor torque, and in two conditions (C and D) T_H gradually became a flexor torque. In all cases, increasing T_K above 100% led to a net positive (anterior) horizontal GRF.

4. Discussion

This computer simulation study examined the effects of decreased knee NJT on a planar, multi-joint model emulating a single-limb stance. The findings support the initial hypotheses, in that (1) decreased torque about the knee could be compensated for by increased torque and the hip and ankle, preventing collapse, and (2) the compensatory torques created a net-posteriorly-directed ground reaction force. These findings were consistent across all eight conditions simulated in this study and could be attributed to a greater increase in compensatory torque about the ankle compared to the compensatory torque created about the hip. An increased flexed posture led to greater torques and, thus, greater horizontal ground reaction forces.

The ability of the ankle and hip NJT to compensate for a decrease in knee NJT was originally proposed over 50 years ago [2,3], and was demonstrated more recently during hopping tasks [13]. While Fox and colleagues [14] demonstrated how each NJT can contribute to the vertical force on a barbell during isometric squats, they did not examine the effects of the NJTs on the horizontal forces. Yen et al. [7] studied NJT fluctuations during in-place hopping. They found that the variances in ankle, knee, and hip NJT minimized the variance in the vertical GRF, but also that these NJT variances caused the horizontal GRF to vary from hop to hop. Subjects alternated between forward and backward hops, which kept them in the same horizontal position only when averaged over the 160 hops analyzed for the study. However, the above-cited studies did not explain why NJT variances led to the horizontal GRFs. These computer simulations help to do just that.

Two factors explain these findings. First, posture determines the direction of the ground reaction force produced by each net joint torque [6]. Interestingly, the ankle angle determines the orientation of the AJC to the KJC and is thus responsible for the direction of the GRF produced by the hip NJT. The ankle and knee angles combined determine the orientation of the GRF produced by both the knee and ankle NJT. While the hip angle will influence the amount of torque necessary about the hip, knee, and ankle [15], it does not influence the direction of the ground reaction force produced by each individual NJT. Second, while decreasing the amount of knee NJT led to compensatory increases in both the ankle and hip NJT, the increase in NJT was greater at the ankle in all postures simulated. In all postures studied, the ankle NJT produced a posteriorly-directed GRF while the hip NJT produced an anteriorly-directed GRF. Together, these factors explain why the net effect was a posteriorly-directed GRF for the postures studied.

The results are consistent with inverse dynamics calculations, in which an increase in ankle NJT will lead to a decrease in knee NJT and a subsequent increase in hip NJT as the NJTs propagate up the limb [10]. Experimentally, this propagation has been demonstrated in sit-to-stand [16] and landing tasks [17]. Another study found fatigue of the thigh musculature also decreased the NJT about the knee and increased NJT about the ankle during a single-leg hop [13], but also that this led to an increased knee flexion angle as well. In addition to not being able to control for landing posture, that study also used a step-up exercise to fatigue the knee muscles, which has been shown to put a large demand on the hip as well as the knee extensors [18]. These simulations were able to overcome these limitations by keeping the posture the same and only decreasing the NJT about the knee.

These findings mostly agree with the calculations of Hof [6] in that the ankle NJT produced a GRF vector with the same orientation as a position vector from the KJC to the HJC, and the hip NJT produced a GRF vector with the same orientation as a position vector from the AJC to the KJC. The GRF vector produced by the knee NJT did not have the same orientation as the position vector from the AJC to the HJC. Disagreement between the two vectors ranged from less than 1° to a little over 3°. While it may be tempting to attribute these differences to rounding errors, the differences may be important. An intriguing case occurred when the HJC was directly over the AJC and the hip angle was approximately equal to the ankle angle. In the initial condition, when knee NJT was equal to 100%, the hip NJT was zero. In both the more-flexed and less-flexed positions, the angle between the positive horizontal axis and the position vector from the AJC to HJC was 90°. This would indicate that the knee NJT would create only a vertical GRF. But in this position, the ankle NJT produced a negative horizontal GRF. Since the hip NJT was zero, the only other torque able to counteract the effects of the ankle and create a horizontal GRF equal to zero is the knee. This would not be possible if the knee NJT’s force vector was 90° from the horizontal. The slight angle away from the vertical (89° in the less flexed and 88° in the more flexed conditions) was necessary to maintain the condition of static equilibrium and may be attributed to the calculation of the force vectors relative to the center of mass rather than the “endpoint” (i.e., acromion), as was performed in a previous study [14].

The model performed as expected when ankle and hip weaknesses were simulated. First, the same linear relationships existed between the ankle and knee NJT and between the knee and hip NJT, regardless of whether the hip or ankle NJT were decreased, or the knee NJT was increased. Second, as stated previously and consistent with inverse dynamics calculations, the NJTs propagate up the chain: a decrease in ankle NJT would lead to an increase in knee NJT and a decrease in hip NJT [10]. Third, the effect of increasing knee NJT was the creation of an anteriorly-directed horizontal ground reaction force, which is in the opposite direction of decreases in the knee NJT. Taken together, these results suggest that the model performs well, irrespective of which joint weakness it simulates. One limitation of this study was that the lower extremity was modeled as a single leg. The position of the non-stance leg has been shown to affect both the body’s center-of-mass location during gait [19] and the kinetics of the stance limb during single-leg squats [20,21]. Future studies should include a non-stance limb to determine how that may affect the results.

Another limitation of this study is that it was limited to the sagittal plane. The ability of frontal and transverse plane NJTs to prevent lower extremity collapse is not well understood. However, in previous work, most of the three-dimensional support moment was produced by NJTs about an axis perpendicular to the leg plane during both pirouette turns [22] and golf swings [23]. While it was reasonable for this study to only include sagittal-plane NJTs, further work should evaluate the contributions of frontal and transverse plane NJTs to the vertical and horizontal ground reaction forces.

This study was also limited to a static analysis of eight different postures. While these results appear robust across the different postures representing the hip-joint center as anterior to the ankle-joint center, vertically aligned with the ankle-joint center, and posterior to the ankle-joint center, other postures should also be simulated. For example, the hip and knee NJT torque may demonstrate increased contributions with more-flexed postures [14]. Additionally, the Jacobian matrix has been used to evaluate NJTs’ contribution to endpoint forces, primarily during static tasks [11]. Other methods (with their own set of assumptions and limitations), such as induced acceleration [24] or induced power analysis [25], may need to be employed during dynamic tasks. Finally, the effects of the posteriorly-directed GRF will be different for different tasks. For example, a posteriorly-directed GRF may cause one to lose balance when standing on a single leg, but may slow the horizontal velocity of the center of mass during gait, and may actually be necessary to initiate a stop [26].

Validating the model on human subjects would be problematic. First, it could be difficult to find a sufficient number of subjects who have isolated knee-extensor weakness in varying degrees without any other impairment. Second, those subjects may not be able to hold all of the same postures as the model. Third, each subject may compensate for their knee weakness in different ways. However, this simple model does provide insights into the ability of the ankle plantar flexors and hip extensors to prevent collapse of the leg in the presence of knee extensor weakness, and the “side effects” of doing so. One benefit of this study is that it tests the isolated effects of diminished knee torque to varying degrees with different postures, providing a mechanical rationale to explain experimental findings and to generate hypotheses for further testing.

5. Conclusions

These computer simulations demonstrated that a decrease in torque about the knee joint may be compensated for by increasing the torque about the other joints in a multi-segmented model. However, these compensations come at a cost. Namely, there will be a horizontal ground reaction force that, at least for the postures simulated, will induce an acceleration in the posterior direction on the model’s center of mass. The magnitude of this effect will depend on the location of the hip-joint center relative to the ankle-joint center and the amount of flexion at each joint. Furthermore, this effect is a result of a greater increase in the ankle net joint torque with decreased knee NJT, a phenomenon which created a posteriorly-directed ground reaction force in each posture studied. These findings do not suggest that other compensations are not possible, but those would most likely require a change in posture. The results of these computer simulations should be used in conjunction with experimental evidence to further our understanding of compensatory mechanics and their consequences.