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J Appl Physiol 99: 1093-1103, 2005. First published April 28, 2005; doi:10.1152/japplphysiol.00365.2005
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Uncertainty of knee joint muscle activity during knee joint torque exertion: the significance of controlling adjacent joint torque

Daichi Nozaki, Kimitaka Nakazawa, and Masami Akai

Department of Rehabilitation for Movement Functions, Research Institute of National Rehabilitation Center for Persons with Disabilities, Tokorozawa, Japan

Submitted 31 March 2005 ; accepted in final form 22 April 2005


   ABSTRACT
TOP
 ABSTRACT
METHODS
 RESULTS
 DISCUSSION
 GRANTS
 ACKNOWLEDGMENTS
 REFERENCES
 
In the single-joint torque exertion task, which has been widely used to control muscle activity, only the relevant joint torque is specified. However, the neglect of the neighboring joint could make the procedure unreliable, considering our previous result that even monoarticular muscle activity level is indefinite without specifying the adjacent joint torque. Here we examined the amount of hip joint torque generated with knee joint torque and its influence on the activity of the knee joint muscles. Twelve healthy subjects were requested to exert various levels of isometric knee joint torque. The knee and hip joint torques were obtained by using a custom-made device. Because no information about hip joint torque was provided to the subjects, the hip joint torque measured here was a secondary one associated with the task. The amount of hip joint torque varied among subjects, indicating that they adopted various strategies to achieve the task. In some subjects, there was a considerable internal variability in the hip joint torque. Such variability was not negligible, because the knee joint muscle activity level with respect to the knee joint torque, as quantified by surface electromyography (EMG), changed significantly when the subjects were requested to change the strategy. This change occurred in a very systematic manner: in the case of the knee extension, as the hip flexion torque was larger, the activity of mono- and biarticular knee extensors decreased and increased, respectively. These results indicate that the conventional single knee joint torque exertion has the drawback that the intersubject and/or intertrial variability is inevitable in the relative contribution among mono- and biarticular muscles because of the uncertainty of the hip joint torque. We discuss that the viewpoint that both joint torques need to be considered will bring insights into various controversial problems such as the shape of the EMG-force relationship, neural factors that help determine the effect of muscle strength training, and so on.

monoarticular muscle; biarticular muscle; isometric force exertion; muscle strength training; cosine tuning

A NUMBER OF RESEARCHERS SEEM to believe that an isometric torque exertion at a single joint, such as a knee extension, is the simplest task that is least influenced by a subject's skill. Because of the simplicity of the task, it is considered a suitable and efficient task to perform when muscular force is being measured or must be controlled. Indeed, countless previous studies in the fields of motor control, sports science, and rehabilitation have adopted this type of task so as to control the output force of muscles that span corresponding joints. Besides, in more practical situations like muscle strength training, this type of torque exertion might be the first choice for those who want to increase their muscular strength. In this situation, the output of the muscular force is usually controlled by specifying the magnitude of torque around the relevant joint (21, 29). For example, when one wants to enhance the strength of knee extensor muscles, knee extension torque is specified as a training intensity. On the contrary, quite naturally, the hip joint torque is ignored.

Such neglect of the adjacent joint is based on a tacit assumption that the activity level of the muscles is uniquely determined as long as the magnitude of torque of the joint that they span is provided. That is, the muscle activity is assumed to be a function of the torque of the relevant joint alone. However, the actual situation seems more complicated: biarticular muscles generate interaction between adjacent joint torques. For example, the activity of the rectus femoris (RF) (biarticular muscle) during the exertion of a certain level of knee extension torque is likely to depend on hip joint torque. Hence, the hip joint torque could indirectly affect the activity of monoarticular muscles. The problem to be considered here is how the activities of these two types of muscle are determined in accordance with the knee and hip joint torques. Neglect of the hip joint torque may be justified if there exists a range of hip joint torque within which the activity levels of both mono- and biarticular muscles are determined by knee joint torque alone, and if the subjects conduct their knee joint torque exertion within this range.

Various approaches have been taken to elucidate this problem (see Ref. 27 for a review). For example, it has been proposed that the biarticular muscles contribute to controlling the direction of the limb endpoint (14, 34), that the muscle activity is modulated with the force direction in a cosine-function-like fashion (10), and that the muscle activity is determined so as to minimize a certain cost function (4, 8, 35, 30). In a previous study (25), we proposed a very simple principle by which the muscle activity might be determined and demonstrated how the principle relates to the results of the previous approaches. According to our previous results, the activity level of knee joint muscles (M) during isometric torque exertion by lower limbs is a function of not only the knee joint torque (Tk: extension torque is defined to be positive) but also the hip joint torque (Th) as M {propto} {lfloor}T PT{rfloor} where T = (Tk,Th), P = (cos {phi}, sin {phi}), {lfloor}x{rfloor} = max (x,0) and T indicates transposition. This relation indicates that the M obeys a (half) cosine tuning (10) function whose preferred direction (PD) is the {phi} on the joint torque plane (Tk,Th). A notable point is that monoarticular muscles have the PD that deviates from their mechanical pulling direction (MD) (3, 33). For instance, the MD of monoarticular knee extensors is 0° (i.e., along Tk axis) because they generate only knee extension torque, whereas their PD shifted in an anticlockwise (i.e., hip extension) direction by ~15°. This implies that the activity level of the monoarticular knee joint muscles explicitly depends on the torque of the hip joint, which they do not span. In other words, there does not exist a range of hip joint torque where the knee joint muscle activities are determined by knee joint torque alone. The misalignment between the PD and MD is inevitable in our musculoskeletal system, which equips biarticular muscles as long as the central nervous system (CNS) adopts cosine tuning (25).

One more notable point is that the PD of biarticular muscles considerably deviates from that of monoarticular muscles. For example, the PD of RF, which is a knee extensor as well as a hip flexor, was approximately –45°, and it was different from the PD of the monoarticular knee extensors such as vastus lateralis (VL) and vastus medialis (VM) (~15°). Therefore, the relative contribution between these mono- and biarticular knee joint muscles to the production of a certain level of knee extension torque is not fixed but could depend on which of the PDs is closer to the direction of the torque vector T the subject is actually exerting.

As can be easily understood from the description above, the conventional single-joint torque exertion task in which only the relevant joint torque is specified has a drawback in that the muscle activity level is indefinite. The problem is to what extent the uncertainty of the neighboring joint torque can actually affect the muscle activity level. To our knowledge, no one has examined how much neighboring joint torque accompanies the torque exertion of the relevant joint. Therefore, the first aim of this study was to measure the amount of hip joint torque exerted when a subject was asked to exert just knee joint torque. We examined whether there was intertrial variability or intersubject difference in the relationship between the knee and hip joint torques. The next question was whether the difference in the relationship between both joint torques (if any) virtually affected the activity of the knee joint muscles. The second aim of this study was to investigate this issue by examining the change in the muscle activity when the subjects were asked to change the relationship between the knee and hip joint torques. Some of these results have been published in abstract form (23).


   METHODS
TOP
 ABSTRACT
 METHODS
RESULTS
 DISCUSSION
 GRANTS
 ACKNOWLEDGMENTS
 REFERENCES
 
Measurement device and experimental procedure.   Twelve healthy subjects [age, 23–42 yr old; height, 173 (SD 6.8) cm; mass, 65.3 (SD 8.6) kg] participated in this study. Each subject gave written, informed consent to the experimental procedures, which were conducted in accord with the Helsinki Declaration and approved by the ethics committee of the National Rehabilitation Center for Persons with Disabilities, Tokorozawa, Japan. They were seated on a chair with the right foot placed on a custom-made force measurement device (Fig. 1). Only the left buttock was placed on the seat surface so as not to interfere with the right hip joint torque generation. The line between the knee and ankle joint (the line from lateral epicondyle to lateral malleolus) was set to be vertical [i.e., the ankle joint angle (between the foot sole and this line) was 90°]. The height of the chair and the angle of the backrest were respectively fixed at 50 cm and 100°, thus setting the knee and hip joint angles at a ~70 and 90°, respectively. The pelvis was fixed to the seat by use of a strap so that the hip angle and positions would not change throughout the experiment. The force measurement device can measure the force generated at the ankle position in every direction by use of a force sensor (Kistler 9067) (Fig. 1B). The foot plate can rotate frictionlessly around the bearing, confining the site of the application of the force to just under the force sensor and minimizing the torque generated around the ankle joint.



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  		Fig. 1. Force measurement system. A: subject seated on a chair with the right foot placed on a force measurement device. The lower leg was set to be vertical. Subjects were asked to place only the left buttock on the surface of the chair. The right foot was firmly fixed to the device by a strap. Top (B), side (C), and front (D) views of the force measurement device. The foot plate can rotate around the axis without any friction with the help of the bearing. The triaxial force sensor mounted just below the rotation axis can measure the force that the subject exerts at the ankle joint position.

 
Subjects were asked to exert various levels of isometric knee extension and flexion torque for at least 3 s. Before each force exertion, the subjects relaxed their right leg and then an experimenter reset the amplifier of the force sensor. This procedure enabled us to exclude the weight of the leg segments from the force measured by the force sensor. Only the force component that the subjects voluntarily exerted was measured. The magnitude of the anterior-posterior force component only was displayed to the subjects in real time by a bar chart on a computer display. It should be noted that this force component was proportional to only the knee joint torque because the lower leg was set to be vertical (Fig. 1A). Subjects were instructed to adjust this force component to a value which was varied from 30 and 20 N to the maximal level with an increment of 30 and 20 N for the knee extension and flexion, respectively. No instruction or information on hip joint torque was provided to the subjects ("Free" condition). Therefore, the hip joint torque, even if its amount was substantial, was not volitional but unintentionally generated. Three trials were conducted for each force level. Sufficient rest was taken between trials to reduce the effect of fatigue.

Surface electromyography (EMG) signals were obtained from six major knee joint muscles: VL, VM, RF, biceps femoris short head (BFS), semitendinosus (ST), and biceps femoris long head (BFL). The EMG signal was amplified with band-pass filtering between 20 Hz and 500 Hz (The Bagnoli 8 EMG System, DELSYS). The EMG signals and the output of the force sensor were digitized at 1 kHz (WE7000 system, Yokogawa Electric) using custom-made software written in Visual Basic (Microsoft).

Data analysis.   First, the force vector F = (Fx,Fy) was constructed from the output of the force sensor (Fig. 2A). Note that this force excluded the component originating from the weight of the leg. Then, the net joint torque vector T = (Tk,Th) generated by muscles was estimated by the following equation:

(1)
where

(2)
where the parameters are defined in Fig. 2A. It should be noted that a mapping from F to T is linear. Figure 2 also demonstrates the correspondence of the T to the F. For example, when the knee extension torque accompanies the hip extension and flexion torques (directions 1 and 3 in Fig. 2B), the force is directed downward and upward, respectively (Fig. 2C).



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  		Fig. 2. Transformation from the force coordinate to the torque coordinate. A: definition of parameters. Th, hip joint torque; Tk, knee joint torque; F, force vector. The torque vectors shown in B correspond to the force vectors shown in C.

 
A 1-s time window was chosen where the fluctuation of the force magnitude was smallest and the value of F and T averaged over this 1-s period was obtained. The root mean square (RMS) of the EMG signals during this period was calculated to evaluate the muscle activation level. Our results were not virtually influenced when the integrated value of the rectified EMG signal was used.

Force-direction control trial.   As will be shown later (e.g., Fig. 4), the pattern of the relationship between the knee and hip joint torques was different from subject to subject. However, two patterns were typically observed: for example, in the knee extension torque exertion, there was one pattern in which it was accompanied by relatively large hip flexion torque, and another pattern in which there was almost no hip joint torque. After the Free condition trial had been finished, we asked the subjects to change the strategy from one to another. Specifically, the subjects who applied the force in the forward (or downward) direction against the device tended to generate the larger (or smaller) hip flexion torque (Fig. 2), so we instructed these subjects to change the force direction in a more downward (or forward) direction. We call this trial the "Control" trial. In the knee flexion torque exertion, similarly, we requested that the subjects who used hip flexion (or extension) torque in the Free trial use hip extension (or flexion) torque in the Control trial.



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  		Fig. 4. Relationship between the knee and hip joint torque in all 12 subjects (A–L). This relationship was obtained when the subjects were asked to exert just knee joint torque (Free condition).

 
In our previous study, the PD of the RF was shown to be approximately –45° on the joint torque plane (25). Hence, when the knee extension torque accompanies more hip flexion torque, the relative contribution of the RF to a certain amount of knee extension torque should become larger than that of monoarticular knee extensors such as VL and VM. In contrast, when the knee extension torque accompanies hip extension torque or less hip flexion torque, the activity of the monoarticular knee extensors should be higher. Therefore, the muscle activity is likely to be affected by the strategy the subjects took. Because which strategies corresponded to the Free or Control condition depends on the subjects, we categorized the data not in accordance with Free vs. Control but with the actual strategy (i.e., the profile of the hip joint torque).

The effect of the strategy on the EMG activity was tested by the following three different methods. In the first method, the EMG activity level with respect to a certain value of knee joint torque was evaluated. To do this, for each subject, the knee extension and flexion torque was normalized by the maximal value of knee extension and flexion torque, separately. Because the maximal knee joint torque was usually different between the Free and Control conditions, the smaller value was taken as a normalized factor. Then, the RMS of EMG with respect to a knee joint torque ranging from 10 to 100% in increments of 10% was obtained by a linear interpolation. Then the difference in the EMG activity level between the two categorized strategies was tested for each joint torque level by the paired t-test.

In the first method described above, the subjects' strategies were forcibly categorized into two strategies. However, the amount of hip joint torque accompanied by the knee joint torque was different from subject to subject. To take the amount of hip joint torque into consideration more explicitly, in the second method, a regression analysis was applied to the whole data set (i.e., both Free and Control conditions). The function used for the regression was M = {lfloor}aTk + bTh{rfloor} and the parameters a and b were estimated by the least squares method for each subject. The Tk and Th data during knee extension (or flexion) were used for the regression of the activity of the knee extensors (or flexors). In other words, we did not use the data when the muscle was silent (e.g., the activity of knee flexor during the knee extension) for the regression, implying that the actual regression function was equivalent to a linear model as M = aTk + bTh (i.e., the term {lfloor} {rfloor} was not required). To evaluate the goodness of fit of this model to the data, the coefficient of determination R2 (38) was calculated as the ratio of the variance explained by the model in the total variance. We also investigated to what extent the RMS of EMG was dependent not only on the knee joint torque but also on the hip joint torque by calculating a confidence interval of the regression coefficient b. If the RMS of EMG depended on only knee joint torque, the regression coefficient b would not differ significantly from zero.

In the third method, using the regression coefficients obtained above, the PD of each muscle was calculated as the direction of (a, b) in degrees. The PDs are characterized as data on a circular scale, so circular statistics (e.g., see chapter 26 in Ref. 38) were used. Specifically, the PD vector was constructed for each subject as (cos {phi}, sin {phi}), and then the mean vector was calculated as ({sum} cos {phi}/n, {sum} sin {phi}/n) where n represents the number of subjects. The mean value of the PD () was obtained as the direction of this mean vector. The value analogous to the standard deviation on a linear scale was calculated as s = (s2 is called the angular variance), where r represents the length of the mean vector. For r ≥ 0.9 (in our case, as shown later), the confidence interval of the mean PD can be expressed as ± d where

(3)
where {chi} is a critical value of the {chi}2 distribution with a degrees of freedom of 1 (the value of {alpha} is 0.05 when a 95% confidence interval is needed). The deviation of the PD from the Tk axis was statistically tested by examining whether the confidence interval contained 0° (knee extensors) or 180° (knee flexors). In all statistical tests, the level of significance was set to P < 0.05.

Transform of the PD from torque to force plane.   For a reference, we transformed the PD on the joint torque plane into that on the force plane. Let the PD vector on the torque plane be pt = (cos {phi}, sin {phi}) where {phi} is the PD of each muscle. If the muscle activity M can be represented as M {propto} ptTT, substituting Eq. 1 into this equation gives M {propto} ptAFT. Hence, the PD vector pf on the force plane is given by

(4)
Note that the pt is not directly mapped into the pf by Eq. 1 as pfT = A–1pfT. In general, the A is not an orthogonal matrix, so A–1 = AT does not hold (32). More intuitively, what is transformed by Eq. 1 is the function of the muscle activity (i.e., M {propto} {lfloor}aTk + bTh{rfloor}), and the PD on the force plane is secondarily determined from this mapped function (Fig. 3).



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  		Fig. 3. Transformation of the preferred direction (PD) from torque to force plane. A: contour plot indicates the activity level of a muscle that obeys cosine tuning on the torque plane. Darker shading indicates higher muscle activity. The PD is defined as the direction in which the muscle activity level increases most steeply. B: the contour plot on the force plane can be obtained by transforming the contour plot in A by using Eq. 1. The PD on the force plane can be defined again as the direction in which the muscle activity level increases most steeply. It should be noted that the PD itself is not transformed by Eq. 1. Mathematically, the PD on the torque plane is transformed into the PD on the force plane by Eq. 4.

 

   RESULTS
TOP
 ABSTRACT
 METHODS
 RESULTS
DISCUSSION
 GRANTS
 ACKNOWLEDGMENTS
 REFERENCES
 
The subjects were able to keep the force output stable. This was ascertained by the small value of the coefficient of variation (below 3%). Furthermore, using a different period (within 3 s) for calculating force magnitude and the RMS of EMG activity did not affect our results substantially, indicating that our results shown below were robust.

Relationship between both joint torques.   The relationship between the knee and hip joint torques obtained from all subjects is shown in Fig. 4. It is apparent that the subjects adopted various strategies to exert the knee joint torque, especially during the knee extension torque exertion. For instance, in a subject shown in Fig. 4E, the knee extension torque accompanied almost the same amount of hip flexion torque. In contrast, in a subject shown in Fig. 4B, there was no substantial amount of hip joint torque accompanying the knee joint torque. One more notable point is that the trajectory between the knee and hip joint torques was not necessarily a straight line but was curved or variable as typically shown in the knee extension torque case of Fig. 4, A, C, F, H, and K. In the knee flexion torque exertion, the intersubject variability was also observed in the amount of hip joint torque although it was smaller than that in the knee extension torque exertion.

Effect of change in strategy on muscle activity.   On the basis of the results of the Free trial shown in Fig. 4, we divided the subjects into two groups. In the knee extension, the five subjects shown in Fig. 4, D, E, F, H, and K were categorized into a group that used relatively large hip flexion torque, and the other seven subjects were categorized into the other group whose hip joint torque was relatively small (Fig. 4, B, C, G, I, J, L) or in the extension range (Fig. 4A). In the Control trial, subjects were requested to use the strategy of the other group. Figure 5 demonstrates the relationship between the knee and hip joint torques in the Control condition. The subjects were able to change the strategy in accordance with our instruction both in the knee extension and flexion tasks.



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  		Fig. 5. Relationship between the knee and hip joint torque in Free ({circ}) and Control ({bullet}) trials.

 
The top two panels of Fig. 6 indicate the relationship between the normalized knee and hip joint torques (A, knee flexion; E, knee extension) averaged separately for each strategy taken by the subjects. Needless to say, there was a significant (P < 0.001) difference in the amount of hip joint torque with respect to each level of the knee joint torque between the two strategies. Such a difference in the amount of hip joint torque affected the muscle activity level not only in the biarticular muscles BFL, ST, and RF (Fig. 6, B, D, F) but also in the monoarticular muscles BFS, VL, and VM (Fig. 6, C, G, H). For example, in the range of knee flexion torque from 40 to 100%, the RMS of the EMG of the BFS muscle was significantly larger when the knee flexion torque accompanied hip flexion torque than when it accompanied hip extension torque (Fig. 6C). In contrast, the RMS of EMG of the BFL was more enhanced when the knee flexion torque accompanied the hip flexion torque (Fig. 6B). The influence of the strategy seems to be small in the ST (Fig. 6D). In the case of knee extension, the activity level of the VL and VM was larger when the knee extension torque was accompanied by hip extension torque (Fig. 6, G and H) whereas that of the RF was larger when accompanied by hip flexion torque (Fig. 6F). In the ST, BFL, VL, and VM, the effect of the strategy was not significant when the amount of the knee joint torque was large.



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  		Fig. 6. Effect of the change in the strategy on the muscle activity. A and E: relationship between the hip and knee joint torque in the knee flexion and knee extension tasks, respectively. Each of the closed and open markers corresponds to each of the 2 strategies that the subjects used. The smaller one of the maximal knee joint torques in the Free and Control conditions is used as a normalization factor for the knee and hip joint torques (%Max). B–D and F–H: muscle activity with respect to each knee joint torque level averaged over all subjects. BFL, biceps femoris long head; BFS, biceps femoris short head; ST, semitendinosus; RF; rectus femoris; VL, vastus lateralis; VM, vastus medialis; RMS, root mean square; EMG, electromyography. The RMS of EMG is normalized (%Max) to the maximal value obtained in the range of knee joint torque from 0 to 100%. **P < 0.01; *P < 0.05. Error bars indicate the standard deviation.

 
In the above analysis, we forcibly categorized the strategies taken by the subjects into two types. However, the amount of hip joint torque was different from subject to subject. To take the effect of the hip joint torque into consideration more explicitly, regression analysis was applied to the data: the assumed regression function was M = {lfloor}aTk + bTh{rfloor}. In the knee extensors VL, VM, and RF, the value of a was significantly (P < 0.001) larger than 0. That is, quite naturally, these muscles were more activated as the knee extension torque became larger. Notably, the value of b for these muscles cannot be ignored. Table 1 demonstrates the number of subjects whose b was statistically judged to be larger or smaller than or not different from 0. When the whole data set was used for regression, for example, the b of the VL was significantly different from 0 in nine subjects, and in seven of these b was significantly larger than 0. This tendency was the same in the VM. Hence, in general, the activity level of these monoarticular knee extensors increases with the increase in the hip extension torque. When the data used for regression were limited up to 60% maximum of the knee joint torque, the dependence of the activation level of the knee extensors on the hip joint torque was more apparent (Table 1). In the RF, the value of b was significantly smaller than 0 in 11 subjects, indicating that its activity level increases with the hip flexion torque.


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  		Table 1. Number of subjects whose regression coefficient b was statistically judged to be b > 0, b = 0, or b < 0

 
In the knee flexors BFS, ST, and BFL, quite naturally again, the value of a was significantly (P < 0.001) smaller than 0. The overall characteristics understood from Table 1 were that the value of b was smaller than 0 for the BFS, larger than 0 for the BFL, and not different from 0 for the ST. Hence, the BFS and BFL activity levels seemed to increase with the hip flexion and extension torque, respectively. In contrast, the dependence of ST activity on the hip joint torque was relatively weak.

The overall value of R2, which is a measure of "goodness of fit" for the model, was larger than 0.9 in almost all cases. Therefore, the degree of the fit of a single plane to the data distribution was very high. A typical example of this situation is shown in Fig. 7. The data for the two different conditions occupy the different spaces of the three-dimensional (3D) plot (Fig. 7A). The relationship between the activity level of the VL and knee extension torque (front surface of the 3D plot in Fig. 7B) indicates that the VL activity was increased in the Control trial in which the hip extension torque was generated (top surface of the 3D plot in Fig. 7B). However, rotating the view point of the 3D plot reveals that there is a best direction in which the variability of the data is minimal (Fig. 7C). This indicates that the data distribution can be approximated by a single plane (25) as shown in Fig. 7D. The high value of goodness of fit provides a rationale for calculating the PD from the results of the multiple regression (an arrow in Fig. 7D).



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  		Fig. 7. Typical example of the 3-dimensional (3D) plot of the muscle activity vs. the knee and hip joint torque. A: spatial location of the data plots obtained in the Free ({circ}) and Control ({bullet}) conditions. The muscle analyzed was the vastus lateralis. The data were obtained from the subject shown in Fig. 5E. B: projection of the data plot to the surface of the cube gives the relationship between the muscle activity and the knee joint torque (front surface) and that between the knee and hip joint torque (upper surface). C: changing the viewpoint of the 3D plot dramatically reduces the variability of the data distribution. D: a single plane was identified by a multiple regression analysis. The arrow indicates the PD in which the muscle is maximally activated on the joint torque plane.

 
The result of the PD is shown in Table 2 and Fig. 8A. The PDs of monoarticular (VL and VM) and biarticular (RF) knee extensors were directed to the first and fourth quadrants of the joint torque plane, respectively. The PDs of monoarticular (BFS) and biarticular (BFL) knee flexor muscle were directed to the third and second quadrants, respectively. The PD of ST did not seem different from the Tk axis (i.e., 180°). When the whole data set was used for the regression, the deviation from PD to the Tk axis was significant only in RF and BFS. However, when the range of regression was limited up to 60% of the maximum knee joint torque, the deviation was significant in all muscles except ST (Table 2). The arrangement of the PDs was quite similar to the results of our previous study (25). Taking all these results together, we concluded that the activity level of the knee joint muscles depends not only on the knee joint torque but also on the hip joint torque even in monoarticular knee joint muscles that span only the knee joint.


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  		Table 2. Preferred direction in degrees of each muscle on the torque plane

 


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  		Fig. 8. PD of each muscle. A: PD on the joint torque plane averaged over all 12 subjects. B: PD on the force plane that is a transformation of the PD of A using Eq. 2 with l1 = 0.4, l2 = 0.4, and {phi} = 70°. C: orthogonal projection from the force vector to each muscle's PD can give an estimation of how the force direction affects the muscle activity. The contribution of the vastus lateralis should be larger for the force direction shown by the thick gray line (1) than that shown by the thin gray line (2).

 

   DISCUSSION
TOP
 ABSTRACT
 METHODS
 RESULTS
 DISCUSSION
GRANTS
 ACKNOWLEDGMENTS
 REFERENCES
 
Hip joint torque secondary to knee joint torque exertion.   The first finding of this study was that when some subjects tried to generate only knee joint torques, hip joint torque was also observed, and the patterns of the relationship between the knee and hip joint torque differed from subject to subject (Fig. 4). In some subjects, the ratio of hip joint torque to the knee joint torque was not even constant, but it depended on the amount of knee joint torque (e.g., knee extension in Fig. 4, A and C) or varied from trial to trial (e.g., knee extension in Fig. 4, F and K). Our experimental setting was different from the setting usually used in knee joint torque exertion in terms of the fixation of the thigh. In an ordinary knee extension-flexion task, the thigh is firmly fixed. Therefore, a possible naive counterargument against our results is that the various patterns of the knee and hip joint torque relationship observed here resulted from the thigh not being sufficiently fixed. However, fixing the thigh would not guarantee that the hip joint torque would be controlled, and the problem of the ambiguity of the hip joint torque would still remain.

Effect of change in strategy on muscle activity.   In our previous study (25), we examined the activity of the knee and hip joint muscles when subjects exerted various combinations of knee and hip joint torques. According to the results, the muscle activity was dependent on both the knee and hip joint torques, and this was true even for the monoarticular muscles that span only one of the two joints, indicating that the hip joint torque has to be specified so that the activity level of the knee joint muscles is uniquely determined. This notion is quite contrary to a naive assumption that the activity level of the knee joint muscles is proportional to the knee joint torque (20).

Therefore, the variability of the relationship between the knee and hip joint torques during the knee joint torque exertion (Fig. 4) could lead to a bias of the muscles being activated. The second finding was that such bias really existed because the muscle activity with respect to knee joint torque was significantly changed by the strategy that the subject took. Even in the monoarticular knee joint muscles, the activity level was influenced by the torque of the hip joint, which the muscles do not span. The dependence of muscle activity on the hip joint torque was more apparent when the knee joint torque was relatively small (Figs. 6 and 7, Tables 1 and 2). One of the possible reasons for this finding may be the activity of the gastrocnemius muscle. In the previous study (25), the activity of the gastrocnemius for some subjects was variable (i.e., it did not show a clear tuning pattern) possibly because of the ambiguity of the ankle joint torque. This variability in the gastrocnemius activity could affect the knee joint torque and contaminate the intrinsic cosine tuning. A second possible reason for this finding is the methodological limit to using the EMG level as a measure of the muscle activity level. For example, the EMG level was shown to increase more steeply than the increase in the force when the force level was high (7).

One more important aspect of this study is that the muscle activity levels under both the Free and Control conditions were likely to be expressed by a single model: M = {lfloor}aTk + bTh{rfloor} (Fig. 7, C and D). This model can be written as the inner product between T = (Tk,Th) and (a,b). Further transformation gives M {propto} T cos ({theta}–{phi}) where {phi} and {theta} are, respectively, the direction of (a,b) and (Tk,Th), indicating that the cosine tuning with its preferred direction being {phi} works as a muscle recruitment strategy. Such a simple muscle recruitment pattern gives a rationale for characterizing the muscle activation pattern using just one parameter, PD. The value of PD obtained in this study was almost same as that in the previous study (25). Specifically, the PD of the VL, VM, and BFS slightly shifted from the Tk axis in an anticlockwise direction. In addition, the PD of the biarticular muscles was located in the same quadrants as their own MD. The PD of the BFL was significantly smaller than that of the ST, which had also been observed previously. Therefore, the present study confirms that the cosine tuning with the PD deviated from the MD works as a general muscle recruitment principle. In the previous study (25), we showed that the misalignment between the MD and PD is inevitable in our musculoskeletal system containing biarticular muscles, as long as the CNS adopts cosine tuning as a strategy for recruiting muscles.

The previous mathematical analysis (25) also demonstrated that cosine tuning with a PD shifted from an MD could be explained by a process of minimizing the sum of the squared muscular force, stress, or activation (8, 35). Recently, it has been proposed that this minimization is related to the CNS process by which the variability of the endpoint force vector is minimized (33, 35) under the existence of signal-dependent noise in the muscular force output (9). However, we would like to emphasize that this cost function cannot explain all aspects of the muscle activity. Another cost function describing the maintenance of the joint stability (6) may be needed to explain the slight cocontraction among monoarticular antagonists that was observed in the present (data not shown) and previous studies (10, 35, 37). Similarly, the cost function for minimizing the sum of the cubed values of muscle stress demonstrated better agreement with the EMG activity during cycling movement (28) and locomotion (4), indicating that the cost functions required may differ from task to task.

Significance of control of hip joint torque.   In conventional studies in the field of motor control or sports science, the single joint torque exertion task has been very commonly used to control a muscle activity or to conduct a muscle strength training. In almost all cases, only the relevant joint torque was specified, and the torque of the adjacent joint was totally ignored. In the ordinary muscle strength training using isometric muscle contraction, the training intensity was often given by the amount of joint torque at only one joint (21, 29). For instance, the training intensity at more than 40% of the maximum was reported to be useful for improving the muscular strength (11). However, our findings raise the question of what such a specification of training intensity means. The 40% knee extension torque exertion does not necessarily guarantee that both mono- and biarticular muscles are each activated to 40% of their capacity. For example, for a subject who exerts little hip joint torque as shown by the "1" in Fig. 8C, the force direction should be closer to the PD of the VM or VL, and far from that of the RF. Hence, the 40% of knee extension torque is unlikely to give 40% activity in the biarticular muscles. Indeed, it was reported that the effect of knee extension isometric training on the muscle hypertrophy could be different from muscle to muscle within the quadriceps femoris muscles (21, 29).

The present results imply that the relative contribution of each muscle to the achievement of a certain level of knee joint torque can be different from subject to subject and/or from trial to trial. This fact would mean that the conventional method is not necessarily efficient and reliable. In other words, even this seemingly simplest task requires a factor of "skill" so as to conduct the muscle strength training efficiently and reliably. A possible method to overcome this drawback is to control the torque of the adjacent joint. This can be easily achieved by monitoring the force direction. Keeping force direction constant throughout the training session might increase the consistency of the muscles to be activated. Another merit of controlling the force direction is that it can activate the muscles in a specific manner. The differential activation of mono- and biarticular muscles has been reported in a variety of tasks including cycling (28, 36), force exertion in various directions under an isometric (14, 35, 37) or dynamic condition (34), walking and running (22), and jumping (2). Such previous findings indicate that the types of muscles required differ from movement to movement. The method of controlling the force direction enables us to design the torque exertion task freely to meet the needs of the movement. For example, subjects who need to train the RF (or VL and VM) selectively can choose an upward or downward force direction in accordance with the PD of each muscle (Fig. 8C). This aspect may be important in practical situations such as rehabilitation and sports training. From a practical viewpoint, the finding that the PD can be estimated by using data for only two force directions (Fig. 7D) is important because it suggests that the whole landscape of the muscle activation pattern can be easily estimated.

Maximal voluntary contraction.   One more remarkable point shown implicitly in this study is that the maximal knee joint torque in several subjects varied according to the strategy (however, note that the maximal joint torque in this study is not "maximal" in a strict sense, because the subjects had to keep the torque stable for at least 3 s). In the subjects shown in Fig. 5, D, H, and K, the maximal knee extension torque was even larger for the Control than the Free conditions. Therefore, these subjects did not choose an optimal strategy to achieve the maximal knee joint torque. Their maximal knee joint torque was increased just after the simple instruction of changing the direction of the endpoint force, indicating that conducting the maximal torque exertion around a single joint requires a factor of skill. It is well known that the gain in the maximal muscle strength is not necessarily proportional to the increase in the cross-sectional area of the muscles, especially at the beginning stage of training (15, 17, 31). In addition, the training of one limb has been shown to increase the muscle strength in the contralateral limb muscles (40). The increase was ascribed to several neural factors (5). The present finding can partly explain the increase in the muscle strength without the increase in the CSA of the muscles.

As for the maximal force exertion, an interesting idea has been proposed by Ohshima et al. (26). They theoretically and experimentally demonstrated that the trajectory of maximal force output at the endpoint (wrist or ankle) in every direction on the force plane can be approximated by a hexagon. By evaluating the shape of the hexagon, they tried to estimate the strength of the mono- and biarticular muscles separately. This finding also demonstrates another example that monitoring the adjacent joint torque can bring additional information.

EMG-torque relationship.   There is still controversy about whether the relationship between the EMG level and the joint torque is linear or nonlinear. One can find a thorough review of this issue in Basmajian and De Luca (1). It should be noted that previous studies have tried to examine the relationship between the EMG level and the torque of a single joint. However, as discussed above, the ambiguity of the adjacent joint torque might have affected the relationship. From the standpoint of the cosine tuning on the joint torque plane (Fig. 7), the linear relationship between the EMG level of the knee joint muscles and the knee joint torque is expected only when the hip joint torque is proportional to the knee joint torque. This can be easily understood by substituting M = cTk (c: constant) into M = aTk + bTh. This substitution leads to (c – a)Tk = bTh. However, in some subjects examined here, the hip joint torque was not proportional to the knee joint torque but was curved (Fig. 4).

We investigated what the EMG activity should be like when the knee-and-hip joint torque relationship took each of the two patterns shown in Fig. 6E. We assumed that the muscle activity M was proportional to the orthogonal projection from a data point to the PD of each muscle (Fig. 9A): i.e., the assumed model was M {propto} cos {phi}Tk + sin {phi}Th where {phi} was the PD. In this estimation, the PD was set to be 15° and –40° for the VL and RF, respectively. The actual EMG activity (Fig. 9, D and H) as well as the estimated EMG activity (Fig. 9, E and I) are shown in Fig. 9. To evaluate the curvature of the knee joint torque and EMG activity, we first drew a straight line connecting the origin and the last data point (Fig. 9B) and then calculated the deviation between the line and each data point (Fig. 9C). That is, the deviation was positive if the relationship was convex and negative if the relationship was concave. For the RF, the actual relationship was concave or almost linear for each strategy (Fig. 9F). The estimated relationship realized the profile of the relative curvature between the two strategies, although the degree of curvature was too small (Fig. 9G). On the other hand, for the VL, the type of curvature (i.e., convex or concave) observed in the actual relationship depended on strategy employed (Fig. 9J). This characteristic was accounted for in the estimated relationship, although the degree of curvature was small (Fig. 9K). Therefore, although the nonlinear EMG-joint torque (or force) relation-ship has been accounted for by various factors (7, 39), it can be partly explained by the curved relationship between both joint torques.



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  		Fig. 9. Evaluation of the nonlinear EMG vs. knee joint torque relationship. A: relationship between the knee and hip joint torque (this is the same as Fig. 6E). Each symbol represents the different strategy that the subject used. Muscle activity was estimated as being proportional to the orthogonal projection from a data point to each preferred direction. For data point a, the VL and RF activity level is proportional to the length of oa1 and oa2, respectively. To evaluate the curvature of the relationship between the EMG level and knee joint torque, a line was drawn connecting the origin with the last data point (B), and then the deviation between the line and each data point was calculated (C). Actual (D, H) and estimated (E, I) EMG vs. knee joint torque relationship. The {circ} and {bullet} correspond to the {square} and {blacksquare}, respectively. F, G, J, and K: degree of convexity or concavity for the EMG-knee joint torque relationship.

 
Considering that the dependence of muscle activity on the neighboring joint torque comes from the existence of biarticular muscles, we can understand why the first dorsal interosseous has a linear EMG-force relationship (16), because there is no biarticular muscle associated with the abduction of the index finger around the metacarpophalangeal joint.

Reconsideration of single joint torque exertion.   In summary, we have shown that the variability of hip joint torque during isometric knee joint torque exertion virtually affects the activity of the knee joint muscles including monoarticular muscles that span only the knee joint. The belief that the muscle activity is proportional to the torque of the joint that the muscles span still seems tenable, but we would like to point out that this is not necessarily correct given the conditions under which our musculoskeletal system equips biarticular muscles (25). Taking the adjacent joint torque into consideration not only is a new method of controlling a muscle's output but also will provide an insight into unsolved problems such as determining the neural factors involved in the acute effect of the muscle strength training and the EMG-torque relationship discussed above. In addition, using this approach, we have succeeded in explaining the different muscle activity patterns between soleus and gastrocnemius muscles during human quiet standing without introducing their differences in motoneuron pool properties such as gain and threshold (24). Similarly, examining the effect of adjacent joint torque on various tasks might be fruitful. Possible future studies might focus on the task-dependent control of muscles in the force vs. position task (13) and in muscle contraction modes (18, 19), bilateral deficit (12), and so on.


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 DISCUSSION
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ACKNOWLEDGMENTS
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This work was partly supported by the Descente and Ishimoto Memorial Foundation, by the Combi Wellness Academy, by the Japanese Ministry of Health, Labour and Welfare, and by the Japanese Ministry of Education, Culture, Sports, Science and Technology.


   ACKNOWLEDGMENTS
TOP
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 METHODS
 RESULTS
 DISCUSSION
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REFERENCES
 
We thank Masashi Tanizaki for developing the measurement device.


   FOOTNOTES
 

Address for reprint requests and other correspondence: D. Nozaki, Dept. of Rehabilitation for Movement Functions, Research Institute NRCD, 4-1 Namiki, Tokorozawa, Saitama 359-8555, Japan (e-mail: dnozaki{at}rehab.go.jp)

The costs of publication of this article were defrayed in part by the payment of page charges. The article must therefore be hereby marked "advertisement" in accordance with 18 U.S.C. Section 1734 solely to indicate this fact.


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