ELSEVIER EDITORIAL SYSTEM(TM) FOR JOURNAL OF BIOMECHANICS

Elsevier Editorial System(tm) for Journal of Biomechanics Manuscript Draft

Manuscript Number: BM-D-08-00503 Title: A transformation method to estimate muscle attachments based on three bony landmarks Article Type: Full Length Article (max 3000 words) Section/Category: Keywords: Scaling, linear mapping, musculoskeletal modelling, shoulder. Corresponding Author: PT Ricardo Matias, Corresponding Author's Institution: First Author: Ricardo Matias Order of Authors: Ricardo Matias; Carlos Andrade; António P Veloso Manuscript Region of Origin:

Cover Letter

Cover Letter We hereby declare that there is no duplicate publication elsewhere of any part of this work. There are no commercial relationships which might lead to a conflict of interests. The typescript has been read and agreed by all authors. We hereby declare that all authors were fully involved in the study and preparation of the manuscript and the material within has not been and will not be submitted for publication elsewhere.

Authors: Ricardo Matias Carlos Andrade António Prieto Veloso

Corresponding author: Ricardo Matias [email protected] Escola Superior de Saúde – Instituto Politécnico de Setúbal Campus do IPS, Estefanilha Edifício da ESCE 2914 – 503 Setúbal

Referee Suggestions

Referee Suggestions

Thomas Kepple [email protected]

Scott White [email protected]

* Manuscript

A transformation method to estimate muscle attachments based on three bony landmarks Ricardo Matias 1, 2, 3, Carlos Andrade 1, 2, António Prieto Veloso 1, 2

1

Faculty of Human Kinetics, Technical University of Lisbon, Portugal

2

Neuromechanics of Human Movement Group - Interdisciplinary Centre of Human Performance

3

Physiotherapy Department, School of Health, Polytechnic Institute of Setúbal, Setúbal, PORTUGAL

Submitting for Original Article

Word count: 2941

CORRESPONDING AUTHOR Ricardo Matias Escola Superior de Saúde – Instituto Politécnico de Setúbal Campus do IPS, Estefanilha Edifício da ESCE 2914 – 503 Setúbal [email protected] Phone +351 265 709 391 • Fax 265 709 392

1

ABSTRACT

In order to create musculoskeletal models that can be scalable to different subject specificities the calculation of the exact locations of muscle attachment is required. For this purpose a scaling method is presented that estimates muscle attachment locations in homologous segments using three bony landmarks per segment. A data-set of seventeen muscles' attachment lines from the shoulders of seven cadavers was used to assess the estimation quality of scaling method. By knowing from the cadaver data the measured location of the muscles' attachment lines it is possible to assess the quality of the estimated ones. The scaling results showed an overall mean RMSE for the scapula and humerus muscles of 7.6 and 11.1 mm, respectively. The results presented were considered to be satisfactory. Among other error contributors, the inter and intra-subject variability should be further investigated, along with the sensitivity of a biomechanical model to these error variations.

Keywords: Scaling, linear mapping, musculoskeletal modelling, shoulder.

2

1. INTRODUCTION

Modelling is undoubtedly one of the key features concerning the development of human movement knowledge. Nevertheless, care should be taken when extrapolating the results from a generic model based on mean data to a specific subject; not because the validity of the model is being questioned, but due to the subject specificity. A well validated model is in fact a very strong and powerful tool, not surprisingly: the word validity derives from the Latin word validus that can be translated as "strength" or "power". In order to respond to the subject specificity demand, models that are based in any cadaver data-set studies or other, should take into account subject parameters, thus making the scaling process necessary. Not knowing, yet, what scaling method will produce the most valid results in the shoulder model that we are developing, we believe that by adapting the model data-set to the subject characteristics, we will improve the quality of the inferences. A more "simple" approach to the geometric scaling problem is accepting the oversimplification of the isometric model, that all people are geometrically similar (Zatsiorsky, 2002); other possibility, is using a least squares solution for an affine scaling transformation, extensively described by Sommer et al. (1982). In the second half of the last century a wide variety of publications have approached the anthropometric scaling challenge. For example, Lewis et al. (1980) presented a technique using four or eight-noded finite elements to locate equivalent points in two different human femora sizes, shifting from a previous homogenous scaling method proposed some years before by Lew and Lewis (1977; 1978). Sommer et al. (1982) described a least squares solution for an affine scaling transformation between homologous specimen-subject landmark coordinates. Kepple et al. (1994) used two indirect accuracy tests and a three-dimensional computer graphics program to evaluate the estimation of muscle origin and insertion locations. Recently Kaptein and van der Helm (2004) used three-dimensional models of the scapula, humerus and clavicle to assess the accuracy of some of the cadavers' mapping muscle attachment contours studied by van der Helm et al. (1992). This assessment was conducted by measuring the distance between the muscles' attachment contours and its corresponding bone surfaces given by the three-dimensional geometric models. Along this, the authors demonstrated that it is possible to predict muscle attachment locations by means of the mentioned geometric models. 3

The method presented here uses techniques of linear algebra, in particular linear mappings, to effect the scaling of a specimen (or data-set) to a given subject. The purpose of this study is to assess the predictive quality of a transformation method to correctly locate a set of muscle attachment lines.

2. METHOD AND RESULTS

The transformation method presented in this article used three bony landmarks of the scapula and humerus in order to predict the muscle attachment in the corresponding segment of six other known scapulae and humeri. For landmarks and muscle attachment lines the base dataset of seven cadavers (known as the VU-study 1988-1996) provided by

the

Dutch

shoulder

group

web

page

(www.fbw.vu.nl/research/Lijn_A4/shoulder/overview.htm) was used. For each cadaver the coordinates of trigonum spinae (TS), angulus inferior (AI), scapula angulus acromialis (AA), glenohumeral rotation center (GH), lateral epicondyle (EL), medial epicondyle (EM) and seventeen muscles' attachment locations - for the scapula: biceps caput breve (BC), biceps caput longum (BL), deltoideus (DL), coracobrachialis (CB), triceps (TC), trapezius (TP), rhomboideus (RB), levator scapulae (LS), pectoralis minor (PM), serratus anterior (SA); and for the humerus: subscapularis (SUB), supraspinatus (SUP), infraspinatus (INF), teres minor (TMi), teres major (TMa), latissimus dorsi (LD), pectoralis major (PM) - were gathered. The muscle attachment lines used were only those directly related to the scapula and humerus bones, and their corresponding coordinates were calculated using a t-polynome as described by van der Helm (1992). Thus, the muscle attachment lines presented here are not in fact the real anatomical location of the muscle insertion. These lines are obtained by the least squares fitting technique to a sufficient number of digitized points that after a visual inspection judgement were considered sufficient to cover the anatomical insertion area (van der Helm et al., 1992). All data was processed using Matlab software (version 7.4.0).

2.1 TRANSFORMATION METHOD

2.2 DESCRIPTION OF THE METHOD 4

Let

, and

Euclidean space

be three vectors in

. If these vectors are linearly independent, they constitute a basis of

and thus, given three

vectors

,

, there is a linear map , and

and such that

,

. This map is given by

(1)

where

and

Let through

and ,

and

Then, given a point , we have

Thus, the image of a point in and

in the object plane

passing

and by linearity

is a point in the image plane

passing through

,

.

The idea is to use this type of mapping to effect a scaling of muscular attachment lines of a given specimen to a subject. Three bony landmarks of the specimen are chosen as the vectors vectors

and and

, and the respective bony landmarks of the subject are the

. With these six vectors we construct . Given a specimen point

(for example, a point in the muscular attachment line), the scaling of will be given by and

For example, we can have

. The image points

and

to the subject ,

,

will be the subject bony landmarks,

5

and

The map

defined in (1) is obtained, and,

given the coordinates of a point in the muscular attachment line of the specimen, image,

, its

, will represent the coordinates of the respective point of the muscular

attachment line scaled to the subject scapula. is continuous, that is, given two elements

The mapping

and

of

, we have

(2)

where the norm of ,

, is given by the 2-norm of the matrix

square root of the maximum eigenvalue of orthogonal projection of

in

, which is the

. In particular, if

is the

, then

(3)

which means that points near the object plane should have their image near the image plane. We also see by (3) that the quality of this fit will be dependent on the norm of If

.

is too large, then the image of points near the object plane cannot be guaranteed

to be near the image plane. On the other hand, a small value of

will result in image

points too near the image plane. We have verified experimentally a correlation between values of the norm of

and the distance between the transformed data and its real

location by using the four landmarks of a specimen,

and

in the

following way: three of them were used to fit the fourth one, in the four possible combinations, to the other six possible specimens. This correlation only becomes apparent for larger values of

, as can be seen in Figure 1.

Figure 1 here

The higher values of used, namely,

and

, visible in Figure 1, lie in the fact that two of the landmarks , are very close to each other. The vectors

,

and

should not be too close to each other, since this can increase the condition number of the matrix

, thereby increasing the numerical error associated to the scaling. Plotting

the mapping error against the condition number of the matrix, led us to observe a correlation which is similar to the one presented on Figure 1.

6

Another possible source of error is the distance between the object plane the vectors

and

, containing

, and the origin of the coordinate system. This distance should

not be small, since if this plane intersects the origin, the vectors linearly dependent, and the existence of

and

are

cannot be guaranteed.

2.3 APPLICATION OF THE METHOD

In order to test this transformation method a data set from seven cadavers was used. For each of them three landmarks were assigned to the scapula (AC, TS and AI) and other three to the humerus (GH, EM and EL). In order to obtain the attachment line coordinates, eleven points were calculated between and

and

, using the vectors

for each muscle of each cadaver, except for the zero-order polynome cases, in

which only one point was calculated. After calculating these coordinates for each cadaver, the scaling method was applied, transforming the coordinates to each of the other six corresponding segments. In the following, we will speak of object segment when referring to the segment which we associate with the landmarks

and

,

and of image segment when referring to the segment which we associate with the landmarks

and

.

Knowing the given coordinates of all the attachments of each cadaver, the root mean square error (RMSE) between the transformed data from the object segment and the given coordinates of the corresponding attachment in the image segment was used to assess the predictive quality of the method described above. For each pair of homologous segments, the mean residual errors supplied by the Dutch shoulder group web page and the palpator error (Pronk and van der Helm, 1991) were considered in the following manner: If the respective image

is the error related to an object point will be given by

, then by (2) the error of

. Since to each attachment line is

associated an error of 1.43 mm coming from the use of the palpator and an error , meaning that

which is the mean residual error of this line, the total error is when we apply the scaling procedure to an attachment line transported to the image scapula is

. Thus, if

the attachment line in the image scapula corresponding to

, the error that is

is the error associated to , the total error when

measuring the RMSE between these lines is given by

7

For each pair of homologous segments, this value was subtracted to our measurements of the distance between the transformed attachment line of the object segment and the corresponding attachment line of the image segment, thus producing the error values we present in the tables below.

2.4 TRANSFORMATION RESULTS

The method described above was applied using three scapula and humerus bony landmarks of each of the seven cadavers in order to predict ten and seven muscles' attachment lines of the scapula and humerus, respectively. In Tables 1 and 2 we present the mean error of forty-two possible transformations of each muscle attachment, for scapula and humerus, respectively. The mean RMSE for the scapula (Table 1) was 7.6 mm with a variation from 2 to 12 mm. For the humerus scaled muscle attachments (Table 2) the mean RMSE was 11.1 mm, with a variation from 6 to 17 mm. If we consider the data from BC in Table 1, the value of S1 (9.7 mm) represents the mean error value of the transformations from scapula one to the other six.

Table 1 here

Table 2 here

3. DISCUSSION

The scaling method presented here is, as described above, based on the use of linear mappings which are obtained from the coordinates of three object and three image points. This procedure has an advantage in simplicity, since it does not involve more than the algebraic manipulation of 3x3 matrices and it relies only on three landmarks per segment. However, we have to be careful about the mathematical prerequisites and calculations for application of the procedure. As a prerequisite, the object plane must not be too near the origin of the coordinate system. As to the calculations involved, the 8

points should not be too near to each other, which would result in bad conditioning of the matrices involved. Because we are not able to measure the distance between the transformed muscles' attachment lines and their corresponding bone surfaces, which would give us an estimation of our transformation error, we chose to measure the RMSE between the transformed muscles' attachment lines and their measured location given by a tpolynome. In order to remove the error from the experimental procedure performed by van der Helm et al. (1992), the absolute measurement error of the palpator, mean residual errors and the norm of

were taken in consideration for each scaling pairs,

even so, mean errors of 7.6 and 11.1 mm were observed for the scapula and humerus transformations, respectively. The larger error value means transformation errors were obtained when using the humerus. These results are probably due to the increased proximity of two (EM and EL) of the three landmarks used, in relation to the scapula. As we described and illustrated in Figure 1, the scaling RMSE increases when two of the closest landmarks were used in the set of three that represents the object segment. In the four possible combinations used the increased RMSE values were only verified when AC and AA were used in combination. Apparently this corroborates this assumption. One issue that is still unclear is the identification of the contributors to the error obtained and their relative weight. One of these sources is the transformation method itself, which can in some cases induce some error, as previously explained when we studied the norm of

. A second source could be the inter-subject anatomical

variability of the muscle attachments can bring error to our measurements. One of the assumptions underlying our transformation method is that when the morphologic dimensions of the bones changes, the relative position of the muscle attachments that are related to this bone also change in the same proportion. This can be in fact a misleading assumption. Duda et al. (1996) have described muscle attachment variability and outline the caution that should be taken when considering mean data of different subjects in biomechanical modelling analysis field. Others have also addressed the inter and intra-individual differences found in the attachment shapes and the fact of not having achieved a scaling procedure with sufficient similarity (van der Helm et al., 1992). Further research will be conducted to test the assumption previously described, as well as the possible influence of the inter-subject muscle attachment variability and the norm of

on the transformed error results. 9

Other methods have been described to perform scaling transformations. In the work of Lewis et al. (1980), the mean absolute error, understood as the magnitude of the error vector between the calculated and the directly measured location of equivalent points, was 10.75, 7.13 and 6.31 mm for the four-noded element, eight-noded element and two eight-noded elements, respectively. From these results the authors suggested that by using an element that can describe and be more sensitive to the nonhomogeneity of bone shapes the scaling results would be more accurate. The method described by Sommer et al. (1982) resulted in an accuracy, when scaling beagle dried femur, tibia and humerus, inferior to 2 mm. The results from one of the tests (landmark residual errors) used by Kepple et al. (1994) when mapping the measured location of three landmarks to their anatomically based locations, vary between 6 to 12 mm, with an average value of less than 12 mm for the four segments (foot, shank, thigh and pelvis). Kaptein and van der Helm (2004), using a 5 mm criterion as the maximum acceptable distance between the muscle attachment contours and the bone surface, reported that eighty five per cent of the muscles' attachments were measured and processed correctly, and that fifteen per cent of the analyzed data have probably an experimental error associated. The amplitude of our results was 2-12 mm for the scapula and 6-17 mm for the humerus, with an overall mean of 7.6 and 11.1 mm, respectively. Because different methodologies were used on the studies previously mentioned, a direct numerical comparison of the results should be done with care. Notwithstanding this precaution, and taking into account that we are using only three landmarks per subject, we can consider the amplitudes of the RMSE obtained in this study to be satisfactory. One of our primary motivations is to develop a scaling method that can be easily used and implemented. Another objective is that the method might have the particularity of using only three landmarks per segment, using landmarks that are accessible for direct digitizing on the subject or that can be extrapolated from others (i.e. glenohumeral rotation center) and making possible the direct linkage with the ones proposed by the International Society of Biomechanics recommendation for the reporting of human joint motion (Wu et al., 2005). In this sense, and taking into consideration that one of our current concerns is to make a shoulder model scalable to different subjects' anthropometric specificities, the method here presented could contribute therefore for a less time consuming experimental setup, because it would take as input parameters the landmarks proposed on the above mentioned kinematic proposal. 10

4. CONCLUSION Musculoskeletal modelling faces the huge challenge of subject specificity. With the increased technical solutions (i.e. image-based and open source solutions) available nowadays, the creation of scalable and subject adaptable models can realistically be a further step in the study of human movement. In this study we presented a method that among others can give a small contribution to this major goal. This method uses a muscle attachment lines data-set of seventeen shoulder muscles referent to seven scapulae and humeri, to assess the predictive quality of the estimated muscle attachment locations in homologous segments. Its primary advantages are: a simple and easy to implement method; the use of only three palpable bony landmarks per segment and the use of landmarks that are suggested in a widely used shoulder kinematic protocol. Not knowing what is the amount of error that can significantly influence the output of a biomechanical model, it is a hard task to objectively judge a cut-point for the RMSE values. Further research is therefore required.

CONFLICT OF INTEREST The authors of this paper have no financial or personal relationships with other people or organizations that could inappropriately influence (bias) our work.

ACKNOWLEDGEMENTS We would like to thank the School of Health of the Polytechnic Institute of Setúbal Portugal, for its support, and Prof. Luís Canto de Loura for his mathematical remarks. This work was supported by the Foundation for Science and Technology (FCT) and the European Union under the program POCI/DES/61761/2004 and SFRH/BD/41846/2007.

11

REFERENCES

Dutch Shoulder Group 2003, viewed 5 May, http://http://www.fbw.vu.nl/research/Lijn_A4/shoulder/overview.htm >.

2004,

<

Duda, G. N., Brand, D., Freitag, S., Lierse, W., Schneider, E., 1996. Variability of femoral muscle attachments. J Biomech 29(9), 1185-90. Kaptein, B. L., van der Helm, F. C., 2004. Estimating muscle attachment contours by transforming geometrical bone models. J Biomech 37(3), 263-73. Kepple, T. M., Arnold, A. S., Stanhope, S. J., Siegel, K. L., 1994. Assessment of a method to estimate muscle attachments from surface landmarks: a 3D computer graphics approach. J Biomech 27(3), 365-71. Lew, W. D., Lewis, J. L., 1977. An anthropometric scaling method with application to the knee joint. J Biomech 10(3), 171-81. Lew, W. D., Lewis, J. L., 1978. A technique for calculating in vivo ligament lengths with application to the human knee joint. J Biomech 11(8-9), 365-77. Lewis, J. L., Lew, W. D., Zimmerman, J. R., 1980. A nonhomogeneous anthropometric scaling method based on finite element principles. J Biomech 13(10), 815-24. Pronk, G. M., van der Helm, F. C. T., 1991. The palpator: an instrument for measuring the position of bones in three dimensions. Journal Medical Engeneering Technology 15(1), 15-20. Sommer, H. J., 3rd, Miller, N. R., Pijanowski, G. J., 1982. Three-dimensional osteometric scaling and normative modelling of skeletal segments. J Biomech 15(3), 171-80. van der Helm, F. C. T., Veeger, H. E., Pronk, G. M., Woude, V. D., Rozendal, R. J., 1992. Geometry parameters for musculoskeletal modelling of the shoulder system. Journal Biomechanics 25, 129-144. Wu, G., van der Helm, F. C., Veeger, H. E., Makhsous, M., Van Roy, P., Anglin, C., Nagels, J., Karduna, A. R., McQuade, K., Wang, X., Werner, F. W., Buchholz, B., 2005. ISB recommendation on definitions of joint coordinate systems of various joints for the reporting of human joint motion--Part II: shoulder, elbow, wrist and hand. J Biomech 38(5), 981-992. Zatsiorsky, V. M., 2002. Inertial properties of the human body. In: Kinetics of human motion. Human Kinetics, pp. 265-364.

12

Table and Figures

TABLES and FIGURES Table 1 - Mean error (mm) of each transformed muscle attachment line from each scapula (S) to the other possible six. Mean and standard deviation (SD) of these values are also presented. Muscles BC BL* DL* CB TC* TP RM* LS* PM SA

S1 9.7 8.9 3.4 5.1 6.0 12.7 1.6 20.0 10.4 7.6

S2 S3 S4 S5 S6 S7 Mean 14.1 12.8 7.4 7.0 16.9 11.0 11.3 11.5 NaD 7.1 6.0 8.5 6.2 8.0 8.1 6.6 3.2 5.5 5.3 4.7 5.2 10.0 9.6 4.6 4.9 13.7 9.5 8.2 6.2 7.2 3.2 4.4 5.8 2.9 5.1 4.5 6.4 6.3 7.4 7.6 8.7 7.7 0.1 -0.8 0.7 4.5 1.0 7.2 2.0 10.3 14.3 5.6 6.4 21.6 7.2 12.2 13.3 11.9 7.5 5.3 17.3 12.1 11.1 2.7 4.4 4.1 3.5 8.3 7.5 5.4 NaD – No available muscle data from this cadaver. * - Indicates that the structures being compared are fitted by polynomes of the same order.

SD 3.6 2.1 1.7 3.4 1.6 2.6 2.8 6.6 3.9 2.3

Table 2 - Mean error (mm) of each transformed muscle attachment line from each humerus (H) to the other possible six. Mean and standard deviation (SD) of these values are also presented. Muscles SUB* SUP* INF* TMi* TMa* LD* PM*

H1 NaD NaD NaD 6.7 5.0 10.8 18.4

H2 H3 H4 H5 H6 H7 Mean 17.8 12.0 9.1 12.5 6.6 8.9 11.2 21.7 NaD 15.7 9.8 9.6 8.6 13.1 12.4 19.6 11.5 7.5 9.3 7.3 11.3 5.5 4.3 4.9 5.1 9.0 3.6 5.6 9.7 4.7 6.3 5.7 7.1 5.6 6.3 16.8 16.0 10.2 14.2 14.5 17.6 14.3 17.8 12.9 21.6 14.4 15.5 16.3 16.7 NaD – No available muscle data from this cadaver. * - Indicates that the structures being compared are fitted by polynomes of the same order.

Figure 1 - Distance between the transformed data and its real location (mm) against the norm of title of each plot indicates the landmark that is being mapped.

SD 3.9 5.6 4.6 1.8 1.7 2.8 2.8

. The

Figure 1 Click here to download high resolution image

Conflict of Interest Statement

Conflict of Interest Statement

We declare that we have no financial or personal relationships with other people or organizations that could inappropriately influence (bias) our work.

Ricardo Matias Carlos Andrade António Prieto Veloso