Biomechanics of Forearm Rotation: Force and Efficiency of Pronator Teres

Biomechanical models are useful to assess the effect of muscular forces on bone structure. Using skeletal remains, we analyze pronator teres rotational efficiency and its force components throughout the entire flexion-extension and pronation-supination ranges by means of a new biomechanical model and 3D imaging techniques, and we explore the relationship between these parameters and skeletal structure. The results show that maximal efficiency is the highest in full elbow flexion and is close to forearm neutral position for each elbow angle. The vertical component of pronator teres force is the highest among all components and is greater in pronation and elbow extension. The radial component becomes negative in pronation and reaches lower values as the elbow flexes. Both components could enhance radial curvature, especially in pronation. The model also enables to calculate efficiency and force components simulating changes in osteometric parameters. An increase of radial curvature improves efficiency and displaces the position where the radial component becomes negative towards the end of pronation. A more proximal location of pronator teres radial enthesis and a larger humeral medial epicondyle increase efficiency and displace the position where this component becomes negative towards forearm neutral position, which enhances radial curvature. Efficiency is also affected by medial epicondylar orientation and carrying angle. Moreover, reaching an object and bringing it close to the face in a close-to-neutral position improve efficiency and entail an equilibrium between the forces affecting the elbow joint stability. When the upper-limb skeleton is used in positions of low efficiency, implying unbalanced force components, it undergoes plastic changes, which improve these parameters. These findings are useful for studies on ergonomics and orthopaedics, and the model could also be applied to fossil primates in order to infer their locomotor form. Moreover, activity patterns in human ancient populations could be deduced from parameters reported here.


Introduction
The effect of muscular forces on bone structure can be predicted from osteometric parameters using biomechanical models if the line of action and the origins of the tendon are known [1,2]. The rotational efficiency (E rot ) of the pronator teres (PT) is a measure of its rotational capacity as a function of the applied force [3]. The PT E rot can be calculated from several osteometric parameters of the upper-limb skeleton, basically related to the curvature of the diaphysis of the radius bone, the size and shape of the humeral medial epicondyle, and the proximo-distal location of PT radial enthesis [3][4][5].
During resisted pronation, PT acts as the primary agonist, displaying relatively higher activity levels than pronator quadratus [6]. In contrast to the latter, PT is largely affected by elbow and forearm position [6,7]. The positioning of the forearm may also influence the components of PT force, which in turn can entail structural changes with different functional implications. There-fore, it can be hypothesized that each component influences the skeletal structure in a different way, e.g. a component may enhance the curvature of the radius when it exerts a bending loading on its shaft [3]. However, the biomechanical behavior and structural effects of these components in the pronation-supination and flexion-extension ranges have never been examined.
This study aims to analyze PT E rot for a full range of elbow flexion angles in order to (i) analyze the relationship between the components of the force vector and the skeletal structure, (ii) assess how E rot is modified by changes in several osteometric parameters and (iii) get further knowledge about the functional implications of the variation of this parameter in the flexion-extension range. The results of this biomechanical analysis will lay the foundations for future studies about comparative and evolutionary anatomy, as well as works on applied sciences, such as ergonomics, sports medicine and orthopaedics, providing relevant information about the relationship between the skeletal structure and PT functionality.

Materials
The skeletal remains used in this study are part of an osteological collection housed at Unitat d'Antropologia Biològica (Universitat Autònoma de Barcelona). The skeletons in this collection were ceded to Unitat d'Antropologia Biològica (Universitat Autònoma de Barcelona) by Cementiri de Granollers (Ajuntament de Granollers). This cession was subjected to prior agreement between both institutions. Moreover, this study is part of a project (CGL2008-00800/BOS), which was approved by Ministerio de Ciencia e Innovación (Gobierno de Españ a).
Rotational efficiency was calculated for the right upper-limb skeleton (humerus, radius and ulna) of a 32-year-old male from the abovementioned collection. The E rot of the studied individual was previously shown to be within to the human pattern of variation, calculated for full elbow extension (180u) and intermediate flexion (90u) [5]. The biomechanical model used in the current study enables to calculate E rot as function of the elbow flexion angle by using 3D imaging techniques. NextEngine's 3D Scanner was used to obtain a three-dimensional image of the humerus, which was processed with ScanStudio HD software [8]. The image was exported to the modeling software Rhinoceros 4.0 SR1 [9], where planes and axes were defined and measurements were taken.

Calculation of pronator teres rotational efficiency
Galtés et al. [3] developed the biomechanical model to calculate the rotational efficiency (E rot ) of the pronator teres (PT), which was initially based on the use of CT scan images of the upper-limb. In accordance with this work, E rot is defined by the expression: Erot~cos a| cos b|AO 0 The biomechanical model was later adapted to calculate E rot using skeletal remains by means of photographs of the distal epiphysis of humerus [4]. Although this methodology was simple and practical, the technique was limited since the spatial representation of the humeral planes was not possible. Therefore, some assumptions about the arm and forearm mechanics and the representation of the geometrical points used to calculate E rot had to be made. The use of 3D imaging, basically to define the humeral planes and axes and to determine important details of the medial epicondyle, enabled to obtain more accurate calculations of E rot [5]. As a matter of fact, the calculation of E rot at any elbow position described in the current paper is based on the approach developed by Galtés et al. [4] and Ibáñ ez-Gimeno et al. [5]. In order to calculate angles a and b at a given forearm and elbow position, several parameters are required (Table 1) [5]. Two of these parameters, l 1 distance and O 0 B 0 distance, depend on the value of carrying angle (l) and thus vary throughout the flexionextension range (Fig. 1) [10]. The carrying angle is the angle between the arm and the forearm long axes (Fig. 1). Assuming a linear variation of this angle between a maximum value (l m ) in full elbow extension (180u), which is measured on dry bones [11], and l = 0u in full elbow flexion (40u) [10,[12][13][14], and using XB and XC values (Fig. 1B) obtained from the three-dimensional image of the humerus, we can obtain:  Fig. 1 and Eq. 1) [10]. Then, the distance l 1 in maximum elbow extension can be calculated as follows: Figure 1. Representation of the influence of carrying angle (l) on several parameters used to calculate rotational efficiency. A: Anterior view of right distal arm and forearm bones in supination position. Humeral and forearm axes, the flexion-extension axis (FE axis), as well as the force exerted by pronator teres muscle ( F ! , from point A to point B), are represented. F ! p is the vertical component of pronator teres force. Point A indicates pronator teres distal enthesis just at the apex of radial curvature. Point B indicates its proximal attachment site, just at the apex of the medial epicondyle, and point B9 is the projection of point B on plane P 3 . Planes P 1 , P 2 and P 3 are parallel to each other and perpendicular to the forearm axis. P 1 passes through point B, P 2 passes through the most proximal point of the radial head and P 3 passes through point A. Point X is the intersection between humeral axis and a plane perpendicular to the humeral axis that passes through point B. Point C is the most distal humeral point of the humeral axis. Point O is the intersection between plane P 3 and the forearm axis. The dashed circle indicates the zoomed-in area of the right image. B: Detail of the left image. Plane P 2 is represented for a position of full elbow extension (P 2 (180u)) and for a position of full elbow flexion (P 2 (40u)). The variation of this plane is caused by the variation of the carrying angle (l). This causes a variation in l 1 and in OB 0 (Dl 1 (l) and DOB 0 (l)), which depends on the elbow angle. Angle v is the angle between plane P 2 (40u) and CB segment. where Dl 1 (l 180 ) is obtained from Eq. 1. Distance AO is the curvature of the radius, measured on dry bone [4,15]. Distance OB 0 (projection of CB segment on plane P 3 ), is dependent on l value and can be obtained for each position of the elbow using Eq. 2. Then, AO 0 and O 0 B 0 can be respectively calculated from AO and OB 0 by determining distance OO 0 [4]. Knowing the position of point B and its projection on plane P 3 (Fig. 3), we can calculate, at any flexion angle of the elbow, distance AB 0 , angle b, F ! 00 and F ! r and, therefore, the value of E rot . Figure 4 shows PT E rot throughout the flexion-extension and the pronation-supination ranges. Rotational efficiency for each elbow position is maximal when the forearm is close to the neutral position (position attained when the thumb points superiorly and the palm faces medially, at 0u of pronation-supination [10]) and is minimal in the extremes of the pronation-supination range. Maximum E rot for each elbow position undergoes a nonlinear growth of 18% from full extension (180u) to full flexion (40u). The position of maximum E rot displaces from 211u of pronation at 180u of elbow extension towards 6u of supination at 40u of elbow flexion (40u) (Fig. 4B). At 120u of elbow flexion, E rot reaches its maximum when the forearm is in the neutral position.

Results
The force that PT exerts on the radius is represented by F ! (Fig. 1). The projection of this vector on the axial plane P 3 is  represented by F ! 0 (Fig. 3), whereas the projection on the vertical axis is named F ! p (Fig. 1). F ! 0 can be decomposed into two vectors: F ! 00 , which is tangent to the trajectory of the radial rotational movement, and F ! r, which is directed to the rotation center (Fig. 3). Figure 5 shows the relative values for these components at three elbow angles (40u, 90u and 180u). F ! 0 is greater in supination than in pronation, regardless of the elbow angle. The modulus for the perpendicular component of F ! (F p ) displays values that vary between the 86% and 99% of F value, depending on the pronation-supination and flexion-extension angles. This component increases as the forearm pronates, regardless of the elbow position. It is also slightly greater in extension than in flexion of the elbow.
Concerning the radial component ( F ! r), its modulus values are approximately the 50% of F value in a full supination position for the entire flexion-extension range. These values decrease as the forearm pronates, reaching negatives values in pronation. The forearm position where F r = 0 depends on the elbow angle. At 180u of elbow extension, the modulus for F ! r reaches 0 at 246u of pronation, whereas in full elbow flexion this occurs at 225u of pronation. Therefore, this component reaches lower negative values, i.e. higher values for its modulus being the vector negative, as the elbow flexes (up to 15% in full flexion). Regarding F ! 00 , which modulus does not reach the 40% of F, its values change in a similar way to the values of E rot . The pronation-supination angle where F0 = F r is dependent on the elbow position. At 180u, it is close to the neutral position of the forearm (at 11u of supination). As the elbow flexes, it displaces towards a more supinated position of the forearm, reaching 31u of supination in full flexion.
Several simulations were carried out to assess how E rot and F ! components are modified by changes in some osteometric parameters. From the mathematic expression to calculate E rot (see Materials and Methods and Figures 1, 2 and 3), it is easily inferable that a greater curvature of the radius (AO), and so a greater rotational radius (AO 0 ), leads to a proportional increase in the E rot values (Fig. 6A). Moreover, any change in one or several osteometrical parameters that causes an increase of a and/or b angles will consequently enhance E rot (see Materials and Methods). Hence, the moment of force can be improved by rising either the tangential force ( F ! 00 ) or the curvature of the radius (AO).
Although both scenarios would cause an increase in E rot , they are related to different structural characteristics. A 10% rise of AO has a minor effect on the components of F ! , being F ! 00 modulus constant (Fig. 6A) and slightly reducing F ! r modulus at 270u of pronation, from 8-15% to 6-13% of F value, depending on elbow position. When a similar increase in E rot is simulated by a 10%  decrease of l pr , a completely different behavior of F ! vectors can be observed: F ! 00 modulus increases proportionally to E rot during the entire flexion range (Fig. 6B), whereas F ! r modulus rises from 46-48% to 49-51% of F value at 70u of supination and from 8-15% to 10-19% at 270u of pronation. A 10% increase of XB has broadly the same effect on E rot and F ! 00 and F ! r moduli than the reported 10% decrease of l pr . F ! p is not meaningfully modified by changes in the abovementioned parameters, whereas the forearm positions where F ! r modulus reaches 0, where F ! 00 and F ! r become equal and where E rot is maximal are considerably affected by these changes. All this information is summarized in Table 2.
Conversely to the abovementioned simulations, the direction of the change in E rot and F ! 00 modulus (increase or decrease) caused by a change in the position of point B depends on the elbow angle ( Fig. 6C). When coordinate x increases, maximal E rot and F ! 00 modulus rise in flexion of the elbow and decrease in extension. When this coordinate is lower, the values for these parameters decrease in flexion and rise in extension. When coordinate y is closer to 0, E rot and F ! 00 modulus increase in flexion and fall in extension, whereas when this coordinate is lower, these parameters decrease in flexion and rise in extension. Efficiency is also lower when the carrying angle (l) is greater. A change of 5u in this angle leads to a variation of about 6.5% of the maximum E rot value at 180u of elbow extension. This effect is lesser as the elbow flexes and it is null in full flexion (40u).

Rotational efficiency, force components and structural implications
In the current study, PT E rot has been assessed throughout the entire flexion-extension range using three-dimensional technology. The variation of E rot obtained from our innovative biomechanical model is in agreement with the results of kinematic studies using cadaveric specimens [16,17] and virtual and resin models of the upper-limb skeleton [18][19][20], as well as with analysis on forearm discomfort [21] and on electromyographic signals of the forearm pronators [7].
Pronator teres E rot is dependent on the skeletal structure of the arm, elbow and forearm, which in turn can be modified by the usage of this muscle. In this regard, the analysis of E rot has enabled to study the effect of the components of PT force vector on the upper-limb skeleton. The perpendicular component of this vector ( F ! p) shows high relative values, which indicates that an important part of PT force is employed in a direction parallel to the rotational axis of the forearm, i.e. compressing the radius lengthwise towards the distal epiphysis of the humerus. This compression is also produced during the contraction of other upper-limb muscles, such as biceps brachii and the wrist and fingers flexors. As regards PT, the compressive effect on the radius is higher in pronation than in supination and slightly decreases from elbow extension to maximum flexion. Although the differences are slight, this indicates that pronated positions of the forearm enhance the curvature of the radius through radial compression in a greater degree than supinated positions. This effect is independent of the skeletal structure, as F ! p is broadly unaffected by changes in the parameters used to calculate E rot .
When the forearm is pronated, biceps brachii is reflexly inhibited and plays little if any role in flexion, because it would supinate the forearm during contraction [22][23][24]. As mentioned, the perpendicular component of PT force is greater on the prone forearm. This increase may partially compensate the lack of action of biceps brachiii, enhancing the assisting role of PT as elbow flexor and joint stabilizer in this position.
The component F ! r, which directs to the rotation center, reaches negative values in forearm pronation. When this component is negative, the vector directs opposite to the rotational center, and so it enhances the curvature of the radius. Negative values are reached in a position of the forearm that gets closer to the neutral position as the elbow flexes. Therefore, the curvature of the radius (AO) is enhanced during pronation in flexion rather than in extension of the elbow. In any case, the relative values for F ! r responsible for the curvature of the radius are low when compared to values for F ! p, which suggests that radii with marked curvatures are more probably associated to compression forces from PT, among other muscles, than to forces applied perpendicularly to the forearm axis. These findings are in agreement with a previous empirical study that revealed that the pattern of muscular loading exerted on the apex of the radial shaft curvature by the PT muscle plays an important role as a mechanical stimulus involved in diaphyseal bowing [15].  The effect that the skeletal structure and form have on E rot and on the force vectors has also been assessed. The results show that a greater bowing of the radius entails an increase of E rot . This is consistent with previous studies suggesting an enhancement of PT action and forearm rotational power by a markedly bowed radius [3,[25][26][27][28]. An increase of the radial curvature also causes that F ! r becomes negative in a more pronated position of the forearm, and so its modulus reach lower values in full pronation. Therefore, the radius is more easily bowed when its curvature is low. The radial location of PT muscle also affects E rot : at any elbow angle, E rot increases when this enthesis is more proximally located, which is consistent with previous observations in full elbow extension [3]. Moreover, radii with a more proximal enthesis for PT muscle lead to the reach of negative values of F ! r closer to the neutral position of the forearm. Therefore, these radii entail lower negative values for this component, i.e. higher values for its modulus being the vector negative, which stimulates radial curvature.
Concerning the humeral medial epicondyle, E rot also tends to increase in all elbow positions as this structure enlarges. Even though the current analysis uses a different approach to quantify humeral medial epincondylar projection (XB), the results are consistent with previous studies [3,5,29]. Moreover, an enlargement of the medial epicondyle has the same effect on F ! r than a more proximally located radial enthesis for PT, and therefore a more medially projected epicondyle enhances radial curvature.
The orientation of the medial epicondyle is also relevant for the determination of E rot . The alteration of this orientation causes changes in F ! 00 that lead to changes in E rot . A more posteriorly oriented epicondyle, i.e. more retroflexed [5,30], is associated with a greater E rot in full extension of the elbow, whereas an epicondyle with a lower degree of retroflexion has greater values of E rot in full flexion. Moreover, a more proximally oriented epicondyle enhances E rot in full flexion, whereas when it is more distally oriented, E rot increases in extension. The orientation of the medial epicondyle was reported to present a partially activity-dependent plasticity, and so it was hypothesized that it can be modified to enhance certain abilities [30]. In this regard, and in agreement with the abovementioned relationship between E rot and epicondylar orientation, a simple observation of the upper-limb positioning shows that a habitual and continued contraction of PT in full elbow flexion would reorient the epicondyle towards a more proximal position. Conversely, this reorientation would occur distally in full elbow extension. The reported association between the orientation of the medial epicondyle and E rot may have important implications on the evolutionary pathway of the upper-limb skeleton, as this structure plays a central role in the locomotor diversity of primates [5,[31][32][33].
Regarding the carrying angle of the elbow [11], this is the first quantitative insight into its biomechanical implications on forearm rotation. An increase in this angle leads to lower values of E rot . This effect is the greatest in full elbow extension and becomes lower as it is flexed, given that carrying angle is maximal in full extension, decreases during flexion and reaches 0u when the elbow is completely flexed [10,[12][13][14].

Functional implications of forearm rotation
The biomechanical model described here enables to make a thorough quantitative approach to clarify some mechanical aspects about the relationship between PT and forearm and elbow motion. For instance, the results indicate that maximal E rot for each elbow angle is close to the neutral position of the forearm. This position has been commonly associated to the functional position, which minimizes the expenditure of muscular energy as it implies a natural equilibrium between antagonist muscles [10]. Moreover, it entails an enhancement of the precision of the grip, because the forearm axis is in line with the pronation-supination axis [10,34].
Although skeletal structures with different morphologies can have the same E rot , they do not have to imply the same abilities in terms of precision. For instance, if the same force is exerted by PT on two upper-limbs with the same E rot , the one with the greatest curvature will be able to reach a lower rotational angle (File S1 and Fig. S1), and thus a greater rotational precision.
The stability of the elbow joint has relevant implications for sports medicine and other pathological conditions [14,35,36].
During pronation, F ! 0 plays an important role in the stability of the radio-ulnar joint (see File S1 and Fig. 3), whereas F ! p participates in the radio-humeral stability (see Fig. 1A). The global stability of the elbow is thus enhanced by the combination of both components. Although F ! p is slightly lower when the forearm is supinated, the great F ! 0 value in this position leads to a better fitting between the radius and the ulna than in pronation [37]. Rotational efficiency and the forces that act during pronation may be associated to the estimations of the discomfort levels done by Mukhopadhyay et al. [38], as the movements implying a high level of discomfort correspond to positions where E rot is low. Moreover, the concept ''efficiency'' provides information about the rotational stability of a given position, as the work needed to modify the forearm rotational angle depends directly on the value of E rot in each position (see File S1). For a given pronation movement, the model can be used to determine the trajectories of the hand that entail the minimal expenditure of energy for PT. In humans, pronation is usually performed when an object has been reached with the hand supinated and the elbow extended and is then brought closer to the body. The final position of this movement will entail a full flexion of the elbow and a close-toneutral position of the forearm. This movement implies an increase of PT E rot , which indicates that the energy expenditure of this muscle diminishes when the elbow is flexed (see Fig. 4B). The results show that a great part of this trajectory entails upper-limb positions where F ! 00 and F ! r have a similar value, i.e. around the forearm position where F0 = F r . This indicates that during this trajectory the forces that the elbow joint is submitted to are quite equilibrated.
These two conditions (equilibrium between forces and increase of E rot ) are not observed for other trajectories, such as those where the end point is close to full pronation with the elbow flexed. In this case, the final part of the trajectory entails a decrease of E rot and very different values of these two force components. If the forearm was very regularly used to perform this movement with important mechanical loads, the disequilibrium could have pathological consequences, as a result of heavy tensions in the radio-ulnar proximal joint. In order to minimize the expenditure of energy and to rebalance the forces, the radial curvature could increase. As reported by our simulations, a rise of the curvature of the radius causes an increase of E rot and a displacement of the position where F r = 0 towards full pronation, as well as a displacement of the position where F ! r and F ! 00 become equal from supination towards a position closer to the neutral. Therefore, the results of the current study also indicate that the upper-limb skeleton may experience plastic changes as a result of PT activity when it is used in positions where E rot is low and forces are not equilibrated. These changes entail an improvement of these parameters, in order to adjust to the unfavorable conditions. Conversely, if the upper-limb is mainly used in positions with high E rot values, PT will not trigger changes in the skeletal structure. Overall, E rot and its relationship with energy expenditure and rotational stability, as well as the relative values of the force components, which are linked to the joint stability, provide a global insight into pronation and its association to the forearm skeletal structure. These findings will be useful for future analyses on ergonomics and orthopaedics of the upper-limb skeleton, as they provide relevant information about the relationship between PT functionality and the skeletal structure. Applying this biomechanical model on studies about comparative anatomy of primate taxa would provide valuable information about the functional meaning of the upper-limb skeletal interspecific differences, which will probably be associated to the locomotor abilities of each taxon. Moreover, an analysis of PT E rot and force components on a fossil primate would very useful for inferring its upper-limb involvement in locomotion.
Some skeletal parameters essential for the determination of PT functionality, such as the radial curvature and the medial epicondylar form, are reported to be plastic and related to the usage of PT muscle [15,30]. Therefore, this biomechanical model would also be of great interest if applied to ancient human populations. Differences in parameters derived or related to E rot between sexes, groups or populations would account for differences in occupational and habitual activities. Figure S1 Diagrams showing the displacement caused by a force in different scenarios. A: Linear displacement (Ds) and rotational angle (Dh) caused by a force F ! on a rotating body, being the center of rotation O and the rotational radius R. B:

Supporting Information
Representation of the differences in the force ( F ! 1 and F ! 2 ) that are needed to obtain the same rotational angle when the rotation performed by pronator teres muscle starts at a different point (A 1 and A 2 ) (see Fig. 3). C: Representation of the differences in the rotational angle (Dh 1 and Dh 2 ) obtained when the same force ( F ! ) is applied by pronator teres muscle on two radii with different rotational radius (r 1 and r 2 ) (see Fig. 3). Note that the same linear displacement (Ds) is obtained in both cases. (TIF) File S1 Rotating work during forearm pronation, stability and muscular energy expenditure. (DOCX)