Computer Methods in Biomechanics and Biomedical Engineering

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1 Neck Muscle Paths and Moment Arms are Significantly Affected by Wrapping Surface Parameters Journal: Manuscript ID: GCMB-0-0.R Manuscript Type: Original Article Date Submitted by the Author: -May- Complete List of Authors: Suderman, Bethany; Washington State University, Mechanical and Materials Engineering Krishnamoorthy, Bala; Washington State University, Mathematics Vasavada, Anita; Washington State University, Bioengineering Keywords: neck muscles, moment arms, wrapping surface, muscle path

2 Page of Neck Muscle Paths and Moment Arms are Significantly Affected by Wrapping Surface Parameters a Bethany L. Suderman, b Bala Krishnamoorthy, a,c,d Anita N. Vasavada* a School of Mechanical and Materials Engineering, Washington State University, Pullman, WA,, U.S.A.; b Department of Mathematics, Washington State University, Pullman, WA,, U.S.A.; c Gene and Linda Voiland School of Chemical and Bioengineering, Washington State University, Pullman, WA,, U.S.A.; d Department of Veterinary and Comparative Anatomy, Pharmacology and Physiology, Washington State University, Pullman, WA,, U.S.A. *Corresponding author. vasavada@wsu.edu

3 Page of We studied the effects of wrapping surfaces on muscle paths and moment arms of the neck muscle, semispinalis capitis. Sensitivities to wrapping surface size and the kinematic linkage to vertebral segments were evaluated. Kinematic linkage, but not radius, significantly affected the accuracy of model muscle paths compared to centroid paths from images. Both radius and linkage affected the moment arm significantly. Wrapping surfaces that provided the best match to centroid paths over a range of postures had consistent moment arms. For some wrapping surfaces with poor matches to the centroid path, a kinematic method (tendon excursion) predicted flexion moment arms in certain postures, whereas geometrically they should be extension. This occurred because the muscle lengthened as it wrapped around the surface. This study highlights the sensitivity of moment arms to wrapping surface parameters and the importance of including multiple postures when evaluating muscle paths and moment arm. Keywords: neck muscles, moment arms, wrapping surface, muscle path. Introduction Studies using three-dimensional musculoskeletal models have shown that accurate representation of muscle paths improves model estimates of moment arm, an important determinant of a muscle s mechanical action at a joint. In lower limb (Arnold et al. 00) and upper limb (Murray et al., Garner and Pandy 00) models, muscle paths which curved around geometric objects ( wrapping surfaces or obstacle sets ) were defined from magnetic resonance imaging (MRI) or cryosection photographs. Compared to straight line paths, model moment arm estimates using these curved paths more closely matched moment arms measured experimentally. Muscle paths are curved in the human neck as well, but defining the anatomical constraints on muscle paths has been more difficult. Existing neck musculoskeletal models have used via points (intermediate muscle points fixed with respect to a bone segment) to constrain neck muscles around bones (Kruidhof and Pandy 0, Vasavada et al. ). This alteration of muscle paths significantly affected model estimates of neck muscle moment arm, force, and moment; and model estimates of neck strength with the altered muscle paths more closely matched experimentally measured neck strength compared to straight path models (Kruidhof and Pandy 0). However, wrapping muscles over bone does not account for the

4 Page of B. Suderman et al. superficial anatomical constraints provided by soft tissue, and thus other methods are required to model neck muscle paths. A previous study used MRI to define neck muscle centroid paths (the locus of cross-sectional area centroids (Jensen and Davy )) and introduced an objective method for defining and evaluating neck muscle paths using wrapping surfaces in a musculoskeletal model (Vasavada et al. 0) In that study, modelled muscle paths were validated in static postures by comparison to centroid paths. However, the analysis of neck muscle mechanics ultimately depends on the moment arm estimates, which involves modelling neck kinematics. The effect of wrapping surfaces on neck muscle moment arms has not been studied. Evaluating model predictions of neck muscle moment arms is difficult because experimentally measured neck muscle moment arms have not been reported. The most common method to determine moment arm is the tendon excursion method (where moment arm is the change in muscle length over a change in joint angle); but these experiments are challenging for neck muscles because of the multi-joint kinematics of the spine and muscle attachment sites on multiple vertebrae. However, non-invasive methods have also been used to determine moment arm from images. Geometrically, moment arm of a straight path is defined as the perpendicular distance from the muscle line of action to the instantaneous axis of rotation (IAR) of a joint (An et al. ). Moment arm of a curved path has been defined as the shortest distance from the IAR to the centroid path (Wilson et al. ) Using this geometric definition, moment arms can be determined from either a series of MRI scans or from the muscle path geometry in a model, as long as the IAR is also calculated. In the current study, our goal was to evaluate the effect of wrapping surfaces on model-predicted moment arms of neck muscles. In a multi-joint system, especially when the superficial soft tissues constraining the muscle may change shape

5 Page of throughout the range of motion, the choice of wrapping surface parameters are not clear from the anatomy. Further, the wrapping surface must be kinematically linked to one rigid body segment (i.e., vertebral body) in the model, and the choice of this linkage affects the position of the wrapping surface in different postures. Therefore, the specific objective of this study was to analyze the sensitivity of neck muscle paths and moment arms to size (radius) of a cylindrical wrapping surface and the vertebral segment to which it is linked. We assumed that accurate representation of muscle paths will lead to improved estimation of moment arm. We evaluated the modelled muscle path by comparison with the centroid path from MRI scans using subjectspecific static models in different postures. Model-predicted moment arms were evaluated using two different methods: () a virtual tendon excursion method and () a geometric method (i.e., shortest distance from the muscle path to the IAR). We hypothesized that both wrapping surface radius and vertebral segment linkage significantly affect neck muscle paths and estimates of moment arm. Methods In this study, we describe two types of models, three types of muscle paths and two methods of estimating muscle moment arms. The terminology is explained briefly below: Subject-specific (static) model A three-dimensional (D) model of the cervical spine created from MRI scans of one subject in seven different static postures. For every static posture this model includes anatomic (centroid) muscle paths, modelled straight muscle paths and modelled wrapped muscle paths (described below). This model was used to calculate error metric, the difference between a modelled path and the anatomic (centroid) path.

6 Page of B. Suderman et al. Kinematic (generic) model A D model of the cervical spine that represents the geometry of a 0 th percentile male. The model kinematics are defined according to data in the literature. This model was used to estimate moment arm of straight and wrapped muscle paths. Anatomic (centroid) muscle path A muscle path created by connecting the centroids of muscle cross-sectional areas from MRI scans, used in the subject-specific model. Modelled straight muscle path A muscle path created by connecting the first and last centroid path points in the subject-specific model, or by attachments on bony landmarks in the generic model. Modelled wrapped muscle path A muscle path created by wrapping the straight muscle path over a geometric object, used in both the subject-specific and generic models. Kinematic moment arm A continuous estimate of moment arm throughout a range of motion using the tendon excursion method. Geometric moment arm An estimate of moment arm in one plane defined as the shortest distance from the muscle path to the instantaneous axis of rotation (IAR) of the skull with respect to the torso... Muscle paths.. Anatomic paths Anatomic muscle paths were obtained from axial MRI scans of a male subject with th percentile neck circumference (Gordon et al. ), as described in a previous study (Vasavada et al. 0). Axial proton density-weighted MR images (TR=0 ms; TE= ms; slice thickness.0 mm; gap.0 mm) were obtained from the base of the skull to the second thoracic vertebra to identify muscle boundaries. T-weighted

7 Page of images (TR=00 ms; TE= ms; slice thickness.0 mm; gap 0. mm) were obtained in the sagittal and coronal planes to define vertebral position and orientation. Scans were obtained with the subject in seven different head postures - neutral, 0 flexion, 0 extension, 0 left axial rotation, left lateral bending, cm protraction (anterior translation of the head), and cm retraction (posterior translation of the head). The protocol was approved by the Institutional Review Board of Washington State University, and the subject provided informed consent. The anatomic path was approximated by the centroid path (Jensen and Davy ) in this study. Neck muscle boundaries were traced on each MRI slice, and the centroid path was defined as a series of straight lines connecting the centroids of cross-sectional areas (CSAs) from consecutive axial slices. A straight muscle path was also modelled between the first and the last centroid points. Results are reported here for the semispinalis capitis muscle (Figure ), one of the major extensors of the head and neck... Modelled paths and wrapping surface definition Subject-specific static models were created using Software for Interactive Musculoskeletal Modeling (SIMM; Musculographics, Santa Rosa, CA). These models were designed to reproduce the subject s musculoskeletal geometry in each of the seven head postures from the MRI scans, and were used () to define and apply wrapping surfaces and () to evaluate the accuracy of the modelled muscle paths compared to the centroid path from MRI (Figure ). For visualization purposes (e.g., Figures and ), generic model bones were implemented in the model. To recreate the subject s posture from the MRI scans, the model s vertebrae and skull were placed so that they had the same position and orientation with respect to the sternal notch as the bones in the sagittal MR images. The vertebral position and orientation were

8 Page of B. Suderman et al. defined in terms of a local coordinate system with its origin at the centroid of the four corners of the vertebral body in the sagittal plane. The midpoints of the upper and lower vertebral endplates were identified, and the y-axis was aligned with the two midpoints, positive in the superior (cranial) direction. The x-axis was oriented perpendicular to the y-axis in the sagittal plane and was positive anteriorly; and the z- axis was the cross product of the y-axis and x-axis vectors, positive to the right. Cylindrical wrapping surfaces were defined using the neutral posture data, such that the straight path was constrained to be closer to the centroid path (Figure and Figure A). First, the point along the centroid path which was furthest away from the straight path was identified; we assumed that this was the location where the straight path needed greatest adjustment. The x-axis of the cylindrical wrapping surface was defined as the perpendicular direction from the straight path to the furthest point on the centroid path; the y-axis was parallel to the straight path; and the z-axis (long axis of the cylinder) was mutually perpendicular. Although radius was varied in the sensitivity analysis (below), the target radius of the cylinder was defined as the distance from the straight path to the furthest centroid point. The centre of the cylinder was placed such that the surface of the cylinder touched the furthest centroid point, i.e., at a distance equal to the cylinder radius from the furthest centroid point to the straight path along the x-axis (Figure ). The wrapping surface parameters - cylinder orientation, radius, and centre, were defined separately for the left and right muscles. The values for left and right were averaged and mirrored to create symmetric wrapping surfaces about the midsagittal plane, which were applied to both the left and right muscles for analysis of the resulting muscle path and moment arm estimates. This step followed from our assumption that the model should be symmetric; averaging the wrapping surface

9 Page of parameters would account for any asymmetries in the musculoskeletal geometry and possible tracing errors... Wrapping surface parameter evaluation We assessed the effect of varying two properties of the wrapping surface on muscle path and moment arm: () the radius and centre location of the cylinder (which were varied together), and () the vertebral segment to which the wrapping surface is linked, which defines the wrapping surface kinematics. Radius. We tested three different values of cylinder radius (the target radius (r t ), 0.*r t, and.*r t ; Figure B); each radius had a different centre translated along the cylinder s x-axis so the outer surface of the cylinder was touching the same point (furthest centroid from the straight path). A large radius often provides a better fit to the centroid path (as measured by the error metric defined below); this phenomenon is illustrated schematically (in dimensions) in Figure B. However, if a wrapping surface is too large, the endpoint of the muscle may fall within the surface in extreme postures, and the path will not wrap around the surface (seen in some moment arm curves in the Results). Kinematic linkage. We also assessed the effect of linking wrapping surfaces to different vertebral segments. The wrapping surface was initially defined in a global coordinate system with the model in the subject s neutral posture. We then defined the wrapping surface relative to each of the vertebral local coordinate systems (C- C; one at a time) and rigidly fixed it to that vertebra as the model was placed in other postures. The vertebral segment chosen for linkage does not change the muscle path in the neutral posture (Figure B; compare top and bottom images), but changes how the cylinder rotates in other postures, potentially affecting the muscle path and moment arm over the range of motion (Figures A & C).

10 Page of B. Suderman et al... Muscle path evaluation The quality of the fit between the model wrapped path and MRI centroid path is measured by the error metric. In our previous study, we defined the error metric (EM) as the average deviation of the centroid path from the wrapped path, measured at each MRI slice (Vasavada et al. 0). Error metric was determined for all combinations of three radii and seven vertebral linkages over all seven postures. For a given combination of radius and linkage, the average error metric over all postures was evaluated. We chose the wrapping surface that minimized average error metric to represent the most accurate muscle path in multiple postures. The effect of radius and vertebral linkage on error metric was analyzed using a two-way ANOVA... Moment arm.. Generic kinematic model Information about intervertebral kinematics between postures, which is necessary for estimating moment arm, was not available from the static MRI scans. Thus, a generic kinematic model of a 0 th percentile male head and neck musculoskeletal system (Vasavada et al. ) was utilized to evaluate the effect of muscle wrapping on moment arms. At each of the eight intervertebral joints between the skull and T, the axis of rotation for flexion-extension motions was defined according to radiographic studies (Amevo et al. ). The amount of motion occurring at each intervertebral joint was constrained to be a function of one angle (generalized coordinate) the angle of the head relative to the trunk. In the generic model, the straight line path of the semispinalis capitis muscle was defined by its attachments relative to bony landmarks on the skull (midway between the superior and inferior nuchal lines) and vertebra (transverse process of

11 Page of T). Although the semispinalis capitis has attachments to other vertebrae, its path was simplified for this analysis as the position of the other vertebral attachments (or muscle subvolumes) could not be determined from the MRI scans in the subjectspecific model. Wrapping surfaces defined relative to the vertebral local coordinate system in the subject-specific static model were applied to the kinematic model... Definitions of moment arm In the musculoskeletal modelling software, the moment arm was calculated kinematically using the tendon excursion method, equivalent to the change in musculotendon length with respect to head angle:. () We also calculated the moment arm of the modelled path using a geometric method. The instantaneous axis of rotation (IAR) of the skull with respect to the torso was calculated using the Rouleaux method (Panjabi ). Coordinates of the infraorbital socket and the external occipital protuberance on the skull of the kinematic model were recorded at ±. of the desired position, and the perpendicular bisectors of the lines connecting those points were generated. The IAR was then calculated from 0 flexion to 0 extension at increments as the intersection of the perpendicular bisectors (Figure ). At each angle, the extension moment arm was determined as the shortest distance in the sagittal plane from the straight or wrapped path to the IAR (Wilson et al. )... Moment arm evaluation Both kinematic and geometric moment arms were calculated for the straight and wrapped paths for sagittal plane (flexion-extension) motion, where semispinalis capitis has its largest moment arms (Vasavada et al. ). Moment arm was

12 Page of B. Suderman et al. evaluated over a range of 0 flexion to 0 extension because the MRI data (subjectspecific model) covered that range of motion, even though the generic kinematic model has a larger range of motion. As with the evaluation of muscle path, we varied the radius and vertebral linkage. All seven vertebral linkages (C to C) were examined for each of the three cylinder radii, resulting in moment arm curves. Similar to the error metric analysis, we performed two-way ANOVA tests on kinematic moment arm data to investigate sensitivities to radius/centre position and vertebral linkage. Geometric moment arms are only reported for the target radius with linkages to each vertebra; one-way ANOVA tests were performed on geometric moment arm data to investigate sensitivities to vertebral linkage. Paired t-tests were performed to compare kinematic and geometric moment arms at each vertebral linkage (for the target radius only). Results. Error Metric All implementations of wrapping surface radius/centre and vertebral linkage improved the error metric from the straight path (Table ); the average improvement for all wrapping surface radii, linkages, and postures was % (from an average straight path error metric of. mm to an average wrapped path error metric of. mm). Averaged over all postures, the lowest error metric was. mm, occurring with the wrapping surface linked to C with 0% of the target radius. With this combination (C linkage and 0% target radius), the error metric was smallest (. mm) in the flexion posture and largest (. mm) in the lateral bending posture for the contralateral muscle. All six of the lowest error metrics occurred for kinematic linkages to C or C.

13 Page of For a given wrapping surface radius, the error metric varied significantly depending on the vertebral segment to which the wrapping surface was linked (Figure A; p = 0). However, for a given vertebral linkage, varying the wrapping surface radius and centre did not significantly affect the error metric (Figure B; p = 0.0). Error metric varied significantly with posture, and this variation was influenced by linkage. For all linkages, the error metric was generally larger for axial rotation and lateral bending (both contralateral and ipsilateral muscles). However, for linkages to C or C, the error metric was larger for all postures compared to linkages to other vertebrae. To separate the outlier values for linkages to C and C, we calculated the standard deviations of the error metric for individual postures using different sets of vertebral linkages; all vertebrae (C-C), C-C (leaving out C), C-C plus C (leaving out C), and C-C (Figure ). If we examine the standard deviation for C-C only (i.e., discount the large EM values for C and C), axial rotation had the largest variation over all linkages, followed by lateral bending for both contralateral and ipsilateral muscles.. Moment Arm Moment arms for wrapped paths were smaller than those for straight paths (Figures and ), whether calculated kinematically or geometrically. Moment arms calculated using the kinematic method varied significantly with both radius and kinematic linkage (p = 0 for both factors, two-way ANOVA). As radius increases, moment arm values decrease (or become more negative at some angles with wrapping surfaces linked to C-C).(Figure ). Moment arms also have a wider range of values with linkages to lower cervical vertebrae (Figure, Table ). Over the range of parameters we tested, the standard deviation of kinematic moment arm is larger for the variation of vertebral linkage than for the variation of radius (Table ). The

14 Page of B. Suderman et al. geometric moment arm values were calculated for only one radius value, but these moment arms also decrease significantly with linkages to lower cervical vertebrae (Figure ; p = 0, one-way ANOVA). For each vertebral linkage, with the radius set to the target radius, kinematic and geometric moment arms were significantly different (paired t-tests, p < 0.0). For linkages to C-C, geometric moment arms have lower values compared to kinematic moment arms (Figure ). Kinematic moment arms for wrapping surfaces linked to C, C or C become negative when the head goes into extension, whereas the geometric moment arms remain positive throughout the entire range of motion. The muscle paths resulting in the smallest six error metrics (Figure ) had small standard deviation of the moment arm curves (. mm). In contrast, muscle paths resulting in the largest six error metrics had larger moment arm curve standard deviation (. mm). Discussion. Muscle paths The accuracy of neck muscle paths over a range of postures was not significantly affected by the size and centre position of the wrapping surface. Altering the vertebral segment to which the wrapping surface was linked significantly altered the muscle path over a range of postures because each vertebra moved by a different amount. For the semispinalis capitis muscle, the best muscle paths (lowest error metric) occurred with linkages to upper cervical vertebrae (C or C), whereas the worst paths (largest error metric) occurred with linkages to lower cervical vertebrae (C or C). The motion of the upper cervical vertebrae (in a global coordinate system) are a cumulative sum of the motion of all vertebrae below it. Therefore, wrapping surfaces

15 Page of linked to upper cervical vertebrae move a greater amount than those linked to lower vertebrae (c.f., Figure ). For C and C, the vertebral kinematics appeared to be appropriate to match centroid paths, but wrapping surfaces linked to C had higher error metrics. This work shows that, at least for the neck, evaluating muscle paths in multiple postures is important to accurately represent the musculoskeletal anatomy. In fact, the error metric varied very little in the neutral posture; the standard deviation over all sets of wrapping surface parameters (radius and linkage) was 0. mm, making it difficult to choose any set of wrapping surface parameters over the other. However, the standard deviation of error metric over all postures was. mm, and it is clear that some wrapping surface parameters provide a superior fit in non-neutral postures (Figure ). As we found in our earlier study (Vasavada et al. 0), modelled muscle paths were least accurate in out-of-sagittal plane postures (i.e, lateral bending and axial rotation). However, the wrapping surface parameters that resulted in the lowest average error metric also resulted in the lowest maximum error metric (Table ). In this study, wrapping surfaces were defined in the neutral posture and evaluated in other postures. For applications in which more accurate representation of muscle paths in non-sagittal plane postures is important, it may be necessary to use these other postures to define different wrapping surface parameters.. Moment arm Moment arm varied significantly with both linkage and radius, but was more sensitive to linkage. A surprising result of this study was the prediction of flexion moment arms in extended postures by the kinematic method. This occurred for wrapping

16 Page of B. Suderman et al. surfaces linked to lower cervical vertebrae, C-C (Figure ). Because the lower cervical vertebrae (and associated wrapping surface) did not move as much as upper cervical vertebrae (cf. Figure ), the muscle lengthened as the head extended. According to the kinematic definition of moment arm (change in muscle-tendon length vs. change in joint angle), this is a negative (flexion) moment arm. However, the wrapping surface parameters which led to negative moment arms are clearly not the appropriate geometry, as evidenced by their lack of correspondence to the muscle path (Table and Figure ). The semispinalis capitis may be an extreme example of moment arm sensitivity to wrapping surface linkage. In other highly curved neck muscles, we also see large variation in moment arm with different kinematic linkages. However, other neck muscles do not appear to have the extreme changes in sign of moment arm as seen in semispinalis capitis with linkages to C-C. It is important to recognize that inappropriate wrapping surface kinematics may lead to incorrect muscle paths and moment arms over a range of motion, even though the muscle paths are accurate in the neutral posture. We found differences in the moment arms calculated by a kinematic method compared to those with a geometric method. Other studies have found that moment arms calculated kinematically are larger than those calculated geometrically (Wilson et al. ), but to our knowledge none have shown that the kinematic method results in moment arms of opposite sign to the geometric method. Geometric estimates of semispinalis capitis moment arm are positive (i.e., extension moment arms) for all linkage and radius combinations. We believe the negative moment arm predicted in some cases by the kinematic method is physically incorrect, but this may be due to poor muscle path definition, rather than an error in the kinematic method. We also

17 Page of found that moment arm estimates from the kinematic and geometric method agree for straight paths and do not agree for curved paths. This may indicate that alternate methods of calculating moment arms may need to be explored for curved paths. The model-predicted moment arms have not been compared to experimentally measured neck muscle moment arms, because to our knowledge these data are not available in the literature. This study suggests that it is critical to measure neck muscle moment arms by both the tendon excursion (kinematic) method and imaging (geometric) method in cadavers, as neither method has been validated with curved muscle paths experimentally. Although the tendon excursion method can only be used in cadavers, imaging may also be used in vivo in humans. We found that the wrapping surface parameters which produced the best fit to muscle centroid path (i.e., had the lowest error metrics) resulted in more consistent moment arm curves. Other studies have found that modelling the muscle path accurately should result in the most accurate moment arm estimation (Arnold et al. 00). Without experimental measurements of neck muscle moment arms for comparison, we assume that the wrapping surfaces with the best fit to muscle centroid path predict the most realistic moment arm values.. Considerations for selecting wrapping surface parameters In this study, we also modified the method for selecting wrapping surface parameters that was described in an earlier study (Vasavada et al. 0). Previously, cylindrical wrapping surfaces were oriented in the transverse plane when the head and neck were in the neutral posture and linked to the nearest vertebra. In the current study, we oriented the wrapping surface according to the direction of the greatest perpendicular distance between the straight and centroid paths. By considering the threedimensional straight and centroid paths to define the wrapping surface and examining

18 Page of B. Suderman et al. linkages to different vertebrae, we were able to obtain only a slightly better fit to the centroid path of semispinalis capitis (average error metric. mm vs..0 mm). However, the semispinalis capitis muscle is primarily oriented in a vertical plane, so the wrapping surface orientation was similar for either method; therefore, this new method may be more appropriate to define muscle paths for other neck muscles (e.g., trapezius), which have more complex paths that would require wrapping surfaces placed in non-transverse planes. Some of the issues raised in this study may be related to the fact that we used a subject-specific model to define wrapping surfaces and evaluate muscle paths but a generic model to calculate and evaluate moment arms. Moment arm variation with linkage may have been larger because the geometry and kinematics of the generic model were different from the subject-specific models. This may also explain partially why radius did not affect muscle path (evaluated in the subject-specific model) but did affect moment arm (evaluated in the generic model). To compare the generic and subject-specific models, we also calculated geometric moment arms from the subject-specific MRI data in the neutral posture. Because it is known that IAR (and thus the geometric moment arm) can vary with the angular increment used to calculate it, we used an IAR calculation from 0 range of motion (MRI data were only available in 0 extension and 0 flexion postures). The moment arm in neutral from the subject-specific MRI data was mm, whereas the average geometric moment arm for all kinematic linkages in the generic model was mm. Conclusions We found that neck muscle paths and moment arms were significantly affected by the parameters used to define wrapping surfaces. In particular, the vertebral body to

19 Page of which a wrapping surface is linked is a critical parameter in the definition of wrapping surfaces. However, wrapping surface parameters which resulted in a better match to muscle paths also resulted in a smaller range of moment arm estimations. We urge caution in selecting the vertebral body to which the wrapping surface is linked and suggest that a range of postures be used to select appropriate wrapping surface parameters. Acknowledgements: We thank Richard Lasher and Travis Meyer. Supported by NSF (CBET #00) and the Whitaker Foundation.

20 Page of References B. Suderman et al. Arnold A, Salinas S, Asakawa D, Delp S. 00. Accuracy of muscle moment arms estimated from MRI-Based musculoskeletal models of the lower extremity. Computer Aided Surgery. :-. Available Murray WM, Arnold AS, Salinas S, Durbhakula MM, Buchanan TS, Delp SL.. Building Biomechanical models based on medical image data: an assessment of model accuracy. Lecture Notes in Computer Science.-. Available Garner B, Pandy M. 00. The obstacle-set method for representing muscle paths in musculoskeletal modeling. Computer Methods in Biomechanics and Biomedical Engineering. ():-0. Available Kruidhof J, Pandy M. 0. Effect of muscle wrapping on model estimates of neck muscle strength.. ():-. Available Vasavada AN, Li S, Delp SL.. Influence of muscle morphometry and moment arms on the moment-generating capacity of human neck muscles. SPINE. ():-. Available Jensen R, Davy D.. An investigation of muscle lines of action about the hip: a centroid line approach vs the straight line approach. Journal of Biomechanics. ():-. Available Vasavada AN, Lasher RA, Meyer TE, Lin DC. 0. Defining and evaluating wrapping surfaces for MRI-derived spinal muscle paths. Journal of Biomechanics. ():0-. Available An KN, Takakashi K, Harrington TP, Chao EY.. Determination of muscle orientation and moment arms. J Biomech Eng. :-. Available Wilson DL, Zhu Q, Duerk JL, Mansour JM, Kilgore K, Crago PE.. Estimation of tendon moment arms from three-dimensional magnetic resonance images. Annals of Biomedical Engineering. ():-. Available Gordon CC, Churchill T, Clauser CE, Bradtmiller B, McConville JT, Walker RA.. anthropometric survey of U.S. army personnel: methods and summary statistics. Amevo B, Worth D, Bogduk N.. Instantaneaous axes of rotation of the typical cervical motion segments: a study in normal volunteers. Clinical Biomechanics. ():-. Available Panjabi M.. Centers and angles of rotation of body joints: a study of errors and optimization. Journal of Biomechanics. ():-. Available

21 Page of Figure Captions Figure. Subject-specific model displaying semispinalis capitis muscle paths in the neutral posture. A. Straight path (yellow) and centroid path (blue), B: Wrapped path (yellow) and centroid path (blue). Figure. Two-dimensional representation of the muscle path and wrapping surface definition. A. Centroid paths (circles), straight path, and wrapping surface with the target radius displaying cylinder axes orientations and the wrapped path. * indicates the centroid point furthest from the straight path. B. Three wrapping surfaces, each with different radii (0%, 0% and 0% of target radius). The red circles indicate the centres of the wrapping surface, which change position as the radius changes. The wrapped path (not shown for clarity) would be slightly different in each case. Figure. Subject-specific model with wrapped path (yellow) and centroid path (blue). Top row: with 0% target radius wrapping surface linked to the C vertebrae (resulted in smallest error metric over all postures). Bottom row: 0% target radius wrapping surface linked to the C vertebrae (resulted in largest error metric over all postures). A. 0 extension posture. B. Neutral Posture. C. 0 flexion Posture. Note that the wrapped muscle paths are similar in the neutral posture, but very different in flexion and extension because the wrapping surface linked to C moves very little, compared to the wrapping surface linked to C. Figure. Instantaneous axis of rotation (IAR) calculation using the Roleaux method for -0 (flexion) to 0 (extension) at increments. Perpendicular bisectors of the infraorbital socket and external occipital protuberance are shown for ±. about neutral as an examples. IARs are shown for all increments between 0 flexion and extension. For reference, the location of the vertebral bodies and skull in the neutral posture are shown. Figure. Error metric for each posture (N: Neutral, F: Flexion, E-Extension, L-C: Contralateral bending, L-I: Ipsilateral bending, A-C: Contralateral axial rotation, A-I: Ipsilateral axial rotation, P: Protraction, R: Retraction, Ave: Average). A. Error metric for each kinematic linkage, averaged over all radii. B. Error metric for each radius, averaged over all linkages. Error bars reflect the standard deviation. Figure. Standard deviations of error metric over linkages (averaged over radii, presented in Figure B) for each posture (N: Neutral, F: Flexion, E-Extension, L-C: Contralateral bending, L-I: Ipsilateral bending, A-C: Contralateral axial rotation, A-I: Ipsilateral axial rotation, P: Protraction, R: Retraction). The standard deviations are presented for linkages to C-C, C-C, C-C and C, and C-C, to examine the variation over postures while accounting for large standard deviations with linkages to C and C. Figure. Kinematic moment arm estimates throughout 0 flexion to 0 extension for all combinations of linkage and radius. dotted line = 0% of target radius, solid line = 0% target radius and dash line =0% target radius. The curves for linkages to C and C with 0% target radius exhibit discontinuities in the extended posture

22 Page of B. Suderman et al. because the muscle endpoint became inside the wrapping surface and therefore reverted to a straight path (did not wrap). Figure. Kinematic (solid lines) and geometric (square markers) moment arm estimates throughout 0 flexion to 0 extension, for all seven kinematic linkages with 0% target radius. Figure. Kinematic (lines) and geometric (square markers) moment arm estimates for the wrapping surfaces that resulted in the lowest error metrics. Wrapping surface parameters listed in the legend correspond to best to worst error metric, top to bottom.

23 Page of Table. Error Metric (mm) average, with minimum and maximum in parenthesis, over all postures for each combination of radius (rows, given as percentage of target radius) and linkage (columns). The error metric for the straight path averaged over all postures was. mm (.-.0). Linkage C C C C C C C Radius 0% 0% 0% (.-.0) (.-.) (.-.) (.-.) (.-.) (.00-.) (.-.0) (.-.) (.-.0) (.-.) (.-.0) (.-.) (.-.) (.-.) (.-.) (.-.) (.-.) (.-.0) (.-.) (.-.) (.-.)

24 Page of Table. Standard deviation (S.D.; mm) of moment arm curves, averaged over the range of motion for flexion/extension. Linkage data are the standard deviation over all three radii, and radius data are the standard deviation over all seven linkages. Linkage S.D. Radius S.D. C 0. 0%. C 0. 0%. C. 0%.* C. C.* C.* C. * Because of the discontinuity in the curves for linkages to C and C with 0% target radius (see Figure ), the data were only averaged up to the point where the discontinuity occurred ( for analyses containing C and for analyses including C).

25 Page of Figure. Subject-specific model displaying semispinalis capitis muscle paths in the neutral posture. A. Straight path (yellow) and centroid path (blue), B: Wrapped path (yellow) and centroid path (blue). xmm (00 x 00 DPI)

26 Page of Figure. Two-dimensional representation of the muscle path and wrapping surface definition. A. Centroid paths (circles), straight path, and wrapping surface with the target radius displaying cylinder axes orientations and the wrapped path. * indicates the centroid point furthest from the straight path. B. Three wrapping surfaces, each with different radii (0%, 0% and 0% of target radius). The red circles indicate the centres of the wrapping surface, which change position as the radius changes. The wrapped path (not shown for clarity) would be slightly different in each case. 0x0mm (00 x 00 DPI)

27 Page of Figure. Subject-specific model with wrapped path (yellow) and centroid path (blue). Top row: with 0% target radius wrapping surface linked to the C vertebrae (resulted in smallest error metric over all postures). Bottom row: 0% target radius wrapping surface linked to the C vertebrae (resulted in largest error metric over all postures). A. 0 extension posture. B. Neutral Posture. C. 0 flexion Posture. Note that the wrapped muscle paths are similar in the neutral posture, but very different in flexion and extension because the wrapping surface linked to C moves very little, compared to the wrapping surface linked to C. xmm (00 x 00 DPI)

28 Page of Figure. Instantaneous axis of rotation (IAR) calculation using the Roleaux method for -0 (flexion) to 0 (extension) at increments. Perpendicular bisectors of the infraorbital socket and external occipital protuberance are shown for ±. about neutral as an examples. IARs are shown for all increments between 0 flexion and extension. For reference, the location of the vertebral bodies and skull in the neutral posture are shown. xmm (00 x 00 DPI)

29 Page of Figure. Error metric for each posture (N: Neutral, F: Flexion, E-Extension, L-C: Contralateral bending, L-I: Ipsilateral bending, A-C: Contralateral axial rotation, A-I: Ipsilateral axial rotation, P: Protraction, R: Retraction, Ave: Average). A. Error metric for each kinematic linkage, averaged over all radii. B. Error metric for each radius, averaged over all linkages. Error bars reflect the standard deviation. xmm (00 x 00 DPI)

30 Page of Figure. Standard deviations of error metric over linkages (averaged over radii, presented in Figure B) for each posture (N: Neutral, F: Flexion, E-Extension, L-C: Contralateral bending, L-I: Ipsilateral bending, A-C: Contralateral axial rotation, A-I: Ipsilateral axial rotation, P: Protraction, R: Retraction). The standard deviations are presented for linkages to C-C, C-C, C-C and C, and C-C, to examine the variation over postures while accounting for large standard deviations with linkages to C and C. xmm (00 x 00 DPI)

31 Page 0 of Figure. Kinematic moment arm estimates throughout 0 flexion to 0 extension for all combinations of linkage and radius. dotted line = 0% of target radius, solid line = 0% target radius and dash line =0% target radius. The curves for linkages to C and C with 0% target radius exhibit discontinuities in the extended posture because the muscle endpoint became inside the wrapping surface and therefore reverted to a straight path (did not wrap). xmm (00 x 00 DPI)

32 Page of Figure. Kinematic (solid lines) and geometric (square markers) moment arm estimates throughout 0 flexion to 0 extension, for all seven kinematic linkages with 0% target radius. xmm (00 x 00 DPI)

33 Page of Figure. Kinematic (lines) and geometric (square markers) moment arm estimates for the wrapping surfaces that resulted in the lowest error metrics. Wrapping surface parameters listed in the legend correspond to best to worst error metric, top to bottom. xmm (00 x 00 DPI)