Biarticular Hamstring Muscles Can Act as Knee Extensors – A Computer Simulation Study

Purpose: It is unclear whether biarticular hamstring muscles (HAM) can act as knee extensors or not. The purpose of this study is to identify the conditions that HAM can act as a knee extensor by using a computational simulation approach. Methods: The modied Gait2392 musculoskeletal model was used in this study. The posture was determined with a hip exion angle that ranged from -30° to 90° and a knee exion angle that ranged from -10 ° to 90 °. The simulations were executed under two conditions: all segments are free to move, non-contact with the ground (nCG), and the foot is constrained on the ground, contact with the ground (CG). Induced acceleration analysis was applied to determine the contribution of the HAM to the knee angular acceleration. Results: Three key ndings were discovered. 1) HAM can act as knee extensors that have CG condition as well as nCG condition. 2) The HAM function changes depending on the posture. 3) The range of the hip joint that HAM was able to act as a knee extensor was expanded for the CG condition from the nCG condition. Conclusions: We identied the situations in which HAM can act as knee extensors and demonstrated that the HAM function on the knee joint changes depending on the posture and the foot contact condition. Our ndings suggest that HAM can be used as compensatory movement strategy for patients with a reduced capacity to generate knee extension if the patients have enough HAM strength.


Introduction
The capacity to generate knee extension during movement is one of the key factors in predicting movement, which includes locomotion velocity [1][2][3]. When knee extension is impaired because of some conditions, such as stroke patients and patients who have underwent partial or total quadriceps resection, decreasing the capacity to generate knee extension occurred. This causes abnormal knee motion during movement, genu recurvatum, and giving way. Weakness in the quadriceps muscles can cause abnormal knee joint movement during walking [4,5]. The level of activity of daily living decreases when patients show abnormal knee motion during movement [3]. In almost all cases, it is impossible to fully recover the patient's capacity to generate knee extension [6][7][8]. Therefore, this is an important matter in terms of clinical rehabilitation in terms of improving the patient's capacity to generate knee extension.
There is a probability that biarticular hamstring muscles (HAM) can compensate for the decreased capacity to generate knee extension. Some researchers have provided the theoretical perspective that HAM can act as knee extensors [9,10]. This theoretical perspective was supported by the ndings of previous studies that showed the counterintuitive function of the biarticular muscles [11][12][13][14]. Meanwhile, other studies that investigated the HAM function in the knee joint have shown that HAM can act as knee exors [15,16]. Even though this point has been discussed since the 20th century (Lombard and Abbott, 1907), it is not clear whether HAM can act as knee extensors or only act as knee exors.
To break such a stalemate, it is necessary to establish the theoretical framework of how and when HAM can act as knee extensors by considering the main contributing factors to determine the HAM function.
Previous studies have identi ed two main factors that have contributed to this phenomenon. The rst contributing factor is posture because the moment arm ratio and the intersegmental dynamics change depending on the posture. Many researchers have claimed that the biarticular muscle moment arm ratio of the proximal joint to the distal joint is the largest contributor to the counterintuitive muscle function [9,17,19,21]. In addition, some studies have demonstrated that changes in the muscle function are caused by posture-dependent intersegmental dynamics [14,15,18,22]. The second contributing factor is the foot contact condition. A number of previous studies have reported that whether the foot is in contact with the ground or it does not change the exerted muscle function [15,17,23]. Even though these contributing factors have already been identi ed, to the best of our knowledge, there have been no comprehensive studies to investigate the mechanics of how the counterintuitive biarticular muscle function is produced.
The computational simulation approach was a prospective candidate to establish the theoretical framework of how and when the counterintuitive HAM function is exerted. This is because this approach can clarify the causal relationship between the above two contributing factors while exerting a counterintuitive HAM function. When the muscle function was investigated, there are two main types of approaches. One is the experimental approach, and the other is the computational simulation approach.
Electrical stimulation is often used to evaluate the muscle function in an experimental approach [16, 19,24]. This approach is believed to be a powerful method for evaluating muscle function. However, there is a large limitation in which the correct evaluation of the muscle function with electrical stimulation is very di cult when the number of experimental conditions is large. This is because previous studies have shown that repeated contractions with electrical stimulation can cause a decrease in the produced muscle forces [25,26]. Given that all conditions must be con rmed to achieve a comprehensive understanding, the results of those studies indicate that it seems to be impossible to con rm all the conditions. On the other hand, computational approaches have the potential to con rm all conditions without the effect of decreasing the produced muscle forces by repeated muscle contraction.
Biomechanists believe that the advantage of computational simulations is that it demonstrates a causal relationship [27][28][29]. For this reason, computational simulation approaches have often been used in a wide range of elds, such as sports and rehabilitation.
The purpose of this study is to identify the situations under which HAM can act as knee extensors by using a computational simulation approach. We developed the following hypotheses for this investigation. 1) HAM can act as knee extensors regardless of the foot contact conditions. 2) The HAM function changes depending on the posture. Finally, the situation where the foot contacts the ground facilitates a counterintuitive HAM function. We tested these hypotheses by performing a musculoskeletal modeling simulation that takes into account the posture dependent moment arm ratio, intersegmental dynamics, and the contact conditions.

Model implementation
The Gait2392 model is a musculoskeletal model from OpenSim [30] and it was modi ed to conform to our computational simulation study. The Gait2392 model was de ned in a three-dimensional space and it has 23 degrees of freedom and 92 muscles. We aimed to simplify this by restricting the model motion in the frontal and transverse planes. Therefore, the total number of degrees of freedom was 10; there were two translational factors (the point of the pelvis fore-aft and the vertical direction) and eight rotational factors (pelvis tilt, trunk bending, hip exion, knee exion, and ankle dorsal exion). The biceps femoris long head was chosen as the delegation of the HAM.

Simulation
As stated earlier, the purpose of this study was to identify the situations under which HAM can act as knee extensors. In order to achieve this purpose, the simulation in this study was designed to quantify the HAM function on the knee joint for the two main contributing factors: the effect of changing the posture and the effect of the foot contact condition.
To evaluate the changing HAM function on the knee joint, this depends on the posture. The posture was determined by combining the right hip exion angle that ranged from 90° to -30° and the right knee exion angle, which ranged from 90° to -10°. This setting can cover a wide range of the knee and hip angles during movement such as walking [31], running [32], and vertical jumping [33]. In order for the simulation's setting to be simpler, we assumed that all the joint angles, except for the right hip and knee joint angles, were set to 0°.
To examine the effect of the contact condition on determining the HAM function, the simulations were executed under two conditions: non-contact with the ground (nCG) conditions and contact with the ground (CG) conditions. The simulation under the nCG condition was executed without the effects of the ground reaction force. In other words, all segments were able to move freely. In contrast, under the CG condition, the point constraint [34] was applied at the point of the right ankle joint. The point constraint does not allow the ankle joint to translate in any direction (i.e., fore-aft and vertical), although it allows the shank segment to rotate. The simulation under the CG condition was executed while considering the effect of the ground reaction force that is induced by HAM. The biggest difference between the two conditions is whether there is an effect of the ground reaction force that is induced by HAM. The constraint force was determined by using a technique that was previously reported [35].
The induced acceleration analysis [21,22,36] was applied to determine the contribution of HAM to the knee angular acceleration. In the simulation, we quanti ed the HAM-induced knee acceleration when a 1 N HAM force and the corresponding ground reaction force were applied to the model. These accelerations represent the capacity of the HAM per unit force. All calculations were performed by using OpenSim version 3.3 [30].
Three evaluation terms were used in this study to explain the results of our simulation. The rst one is a "moment arm ratio". The moment arm ratio was calculated by dividing the hip extension moment arm of the HAM by the knee exion moment arm. If the moment arm ratio of HAM was larger than 1.0, this means that the effect of the hip extension that is induced by HAM is larger than the effect of the knee exion, and vice versa. The second one is "knee angular acceleration induced by HAM". A positive value indicates that the knee is extended, and vice versa. When the knee angular acceleration that is induced by HAM is positive, this means that the hamstring acts as the knee extensor. The third is "the range of the hip joint that HAM was able to act as a knee extensor (RANGE HAM )". RANGE HAM was de ned as the value of the maximum hip exion angle that HAM was able to act as a knee extensor minus its minimum. This evaluation term was used to quantify the impact of the contact conditions while determining the HAM function on the knee joint. If RANGE HAM is expanded or contracted, this implies that the contact condition has a positive or negative effect to exert a counterintuitive HAM function. RANGE HAM was quanti ed with knee exion angles of 0 °, 20 °, 40 °, 60 °, and 80 °.

Results
The moment arm of HAM changed according to the displacement of the hip and knee angles [37,38]. In other words, the moment arm ratio of HAM was determined by the hip and knee angles. Figure 1 shows the alternation of the moment arm ratio, which depends on the posture. The moment arm ratio of HAM varied from 0.8 to 3.9 depending on the posture. The moment arm ratio was lower than 1.0, which ranged from − 30° to -24° for the hip exion angle and 9° to 74° for the knee exion angle. This means that in almost all cases, the effect of the hip extension that is induced by HAM is larger than the effect of the knee exion that is induced by it.
The HAM can act as knee extensors under both conditions. However, when comparing the nCG condition to the CG condition, this indicates that the HAM function changed from the nCG condition to the CG condition. Figure 2 Figure 3 shows the alternation of RANGE HAM from the nCG condition to the CG condition.
RANGE HAM was expanded from the nCG condition to the CG condition in all individual knee exion situations.

Discussion And Conclusions
The purpose of this study was to identify the situations under which HAM can act as knee extensors. To achieve this purpose, a musculoskeletal model-based computational simulation was designed to quantify the HAM function on the knee joint for the two main contributing factors: the effect of changing the posture and the effect of the foot contact condition. This study revealed three main ndings. 1) HAM can act as knee extensors that have contact with the ground (CG) condition but also non-contact with the ground (nCG) condition.
2) The HAM function changes depending on the posture. 3) Finally, the range of the hip joint that HAM can act as knee extensors (RANGE HAM ) was expanded to the CG condition from the nCG condition.
The rst point to be discussed is the posture dependent HAM function. As mentioned earlier, posture is an important contributing factor. This is because the moment arm ratio and the intersegmental dynamics change depending on the posture. Many researchers have claimed that the biarticular muscle moment arm ratio of the proximal joint to the distal joint is the largest contributor to the counterintuitive function [9,17,19,21]. In addition, some previous studies demonstrated that changing the muscle function was coursed by the posture-induced changes in the effect of the intersegmental dynamics [14,15,18,22]. Despite the situation described above, to the best of our best knowledge, a comprehensive study currently does not exist in terms of how the HAM function changes depending on the posture. This is the rst study to show in detail the change in the HAM function on the knee joint depending on the posture.
The next discussion deals with the interaction between the moment arm ratio and the posture-dependent intersegment dynamics. Some researchers believe that the moment arm ratio has the largest impact on the counterintuitive biarticular muscle function [9,17,19,21]. However, the results of the present study denied this notion. A typical example is shown in Fig. 4. Figure 4 indicates that the two different postures have the same moment arm ratio. Even though both postures have the same moment arm, the HAM function is different. This is evidence that the HAM function is determined by the interaction between the moment arm ratio and the intersegmental dynamics. We can conclude that the counterintuitive HAM function arose from the interaction between the moment arm ratio and the intersegmental dynamics, and that the interaction between these factors dynamically changes depending on the posture.
Thirdly, the effect of the foot contact conditions is discussed. In the present study, we con rmed that RANGE HAM was expanded to the CG condition from the nCG condition (Fig. 3). This nding indicates that whether the foot is in contact with the ground has a large impact on determining the HAM function.
Previous studies have reported that the ground reaction force has a large effect in terms of determining the muscle function during movement [15,17,23]. Frigo et al. [15] investigated the alternation of the HAM function under different CG conditions by using a computational simulation approach. As a result, they clearly showed that the effect of the knee exion that is induced by HAM is dramatically reduced while in contact with the ground conditions in comparison to the nCG condition. These results are consistent with our results. When considering these observations, we concluded that the exertion of the counterintuitive function of HAM is facilitated by the foot contacts on the ground. This nding suggests that the effect of the ground reaction force must be considered when investigating the muscle function.
We discussed the clinical implications of our ndings. This study identi ed a situation in which HAM was able to act as the knee extensor. Previous studies have reported that the capacity to generate knee extension is one of the most important factors in predicting movement performance, such as gait velocity [1][2][3]. In addition, it is well known that weakness of the quadriceps muscle causes abnormal knee joint movement during walking, especially in the stance phase of walking [4,5]. Hence, in the rehabilitation region, therapists often try to improve this matter by enhancing the patients' quadriceps muscle. However, it is impossible for some patients, such as stroke patients with severe motor paralysis on the quadriceps muscles and patients who have undergone partial or total quadriceps resection to enhance the strength of the quadriceps muscle [6-8]. Figure 5 shows the knee angular acceleration that is induced by HAM under the CG condition with the hip and knee joint angles during the stance phase of walking, which was divided from Winter's data [31]. As demonstrated, the knee angular extension acceleration was exerted by HAM in all stance phases of walking. This indicates that HAM can act as knee extensors during the stance phase of walking. Therefore, our ndings suggest that HAM can be used as compensatory movement strategy for patients with a reduced capacity to generate knee extension if the patients have enough HAM strength. We believe that our ndings are meaningful in the biomechanics region and the rehabilitation region.
This study successfully provided a theoretical framework for how and when the counterintuitive HAM function is produced. Indeed, the results of the present study can explain the wide range of ndings in previous studies [11,12,14,15,[17][18][19][20]. However, there are some ndings from the present study that we were not able to explain well. Thelen  should act as the knee extensor during the loading response while walking. There is a potential explanation why this gap occurred. We ignored the HAM function in the frontal and transverse planes, even though it has a three-dimensional moment arm [39]. The muscle action in the frontal and transverse plane has been demonstrated to contribute to the motion in the sagittal plane [40]. Additional studies are necessary to obtain a better understanding of this matter. However, it was determined that an appreciation of the mapping between the interpretations that are drawn from simple and complex models is needed to advance the understanding of movement mechanics [41]. Therefore, we believe our study provides useful information about determining the HAM function because we can clearly show the relationship between the main contributing factors while exerting the counterintuitive HAM function in the sagittal plane with a simple model.
In the present study, we investigated the independent action of HAM on the knee joint. This study did not consider the co-contraction effect of HAM and the quadriceps. Some studies have suggested that the cocontraction effect of HAM and quadriceps muscles is necessary to determine the HAM function [9,10,42]. However, the present study revealed that HAM can act as knee extensors without co-contraction. Frigo et al. investigated the co-contraction effect of HAM and the quadriceps muscle and they showed that it enhances the hip extensor effect of HAM and it reduces the knee exor effect of HAM [15]. Unfortunately, only a few studies have investigated how much impact the co-contraction of HAM and the quadriceps muscles has in determining the HAM function. Therefore, further studies are necessary to identify the co-contraction between HAM and other muscles.
In conclusion, we have demonstrated that HAM can act as knee extensors. Our ndings suggest that HAM might compensate for the capacity to generate knee extension during movement as knee extensors. In addition, our ndings contribute to establishing the theoretical framework of how and when the counterintuitive function of HAM is exerted.  Figure 1 Moment arm ratio of the biarticular hamstring muscles (HAM). The colors for this gure represent the magnitude of the moment arm ratio of HAM. The black solid line indicates the postures that the moment arm ratio of HAM was 1.0. If the moment arm ratio is larger than 1.0, this means that the effect of the hip extension that is induced by HAM is larger than the effect of the knee exion. Therefore, the black solid line implies the boundary to change the predominance of the effect of the hip extension that is induced by HAM over the effect of the knee exion.

Figure 2
Knee angular acceleration that is induced by the biarticular hamstring muscles (HAM) for the conditions of (a) non-contact with the ground (nCG) and (b) contact with the ground (CG). The shift to the color white indicates that the knee extension angular acceleration is induced by HAM. Conversely, when the color is shifted to black, this indicates that the knee exion angular acceleration is induced by HAM. The black solid line means that the postures for the knee angular acceleration is zero. If the knee angular acceleration is larger than 0, this means that HAM generated knee extension angular acceleration.
Therefore, a black solid line implies the boundary to change the HAM function to the knee extensor from the knee exor.