- Email: [email protected]
Response to “Assumption of a ‘gravity only region’ for gravity correction of passive joint moment data may be problematic”
Letters to the Editor / Journal of Biomechanics 43 (2010) 2653–2656
described by Anderson et al. should only be used when the presence of a gravity-only region has been confirmed.
2655
Thompson, R.E., Barker, T.M., Pearcy, M.J., 2003. Defining the neutral zone of sheep intervertebral joints during dynamic motions: an in vitro study. Clinical Biomechanics 18, 89–98.
Conflict of interest statement
E.C. Clarke n Research Fellow, The George Institute for International Health, The University of Sydney, PO Box M201, Missenden Rd., Sydney, NSW 2050, Australia E-mail address: [email protected]
No conflicts of interest to disclose.
Acknowledgement
J.H. Martin Prince of Wales Medical Research Institute, University of New South Wales, Sydney, Australia
The authors wish to thank Mr. Kevin Wang for his assistance in conducting the experiments. References
R. Herbert Research Fellow, The George Institute for International Health, The University of Sydney, PO Box M201, Missenden Rd., Sydney, NSW 2050, Australia
Anderson, D.E., Nussbaum, M.A., Madigan, M.L., 2010. A new method for gravity correction of dynamometer data and determining passive elastic moments at the joint. Journal of Biomechanics. doi:10.1016/j.jbiomech.2009.11.036. Clarke, E.C., Appleyard, R., Bilston, L.E., 2007. Immature sheep spines are more flexible than mature spines: an in vitro biomechanical study. Spine 32, 2970–2979. Hoang, P.D., Gorman, R.B., Todd, G., Gandevia, S.C., Herbert, R.D., 2005. A new method for measuring passive length–tension properties of human gastrocnemius muscle in vivo. Journal of Biomechanics 38, 1333–1341. doi:10.1016/j.jbiomech.2010.03.051
Response to ‘‘Assumption of a ‘gravity only region’ for gravity correction of passive joint moment data may be problematic’’ In our recent paper (Anderson et al., 2010) we describe a method for determining the gravitational moment in dynamometer data that also accounts for the passive elastic moment due to passive tissue deformation. Our method assumes that the passive elastic moment is negligible within a middle region of the joint range of motion, which we dubbed the ‘‘gravity-only region’’. In their letter to the Editor entitled ‘‘Assumption of a ‘gravity only region’ for gravity correction of passive joint moment data may be problematic’’, Clarke et al. present data indicating that this assumption is invalid for the ankle, and as a result the method tends to overestimate the magnitude of the gravitational moment at the ankle. We would like to thank Clark et al. for their scrutiny of our method, as this is a critical component of scientific progress. In this response we describe an addition to our method that accounts for the presence of a non-negligible passive elastic moment in the gravity-only region, and as a result should address the valid concern expressed by Clarke et al. We would also like to point out that, based upon the data presented by Clarke et al. and our own retrospection, the term ‘‘gravity-only region’’ that we used in the original work is somewhat of a misnomer for the midrange region of passive moment data. This is because the passive elastic moment in this region, while small, may still be present. As such, this region will henceforth be referred to as the ‘‘mid-range region’’. Clarke et al. reported a passive elastic moment at the ankle that is not uniformly zero in the mid-range region. Based upon their data, this moment could be reasonably approximated by a straight line. Other studies also report non-zero yet linear passive elastic moments in the mid-range region of the knee (McFaull and Lamontagne, 1998) and hip (Yoon and Mansour, 1982). To account for the presence of a non-zero passive elastic moment in this mid-range region, a linear approximation of the passive elastic moment may be incorporated into our previously published method. Thus, when fitting the passive moment data from
30 March 2010
n
Corresponding author. Tel.: + 61 2 9657 0343; fax: + 61 2 9657 0301.
the mid-range region, a line representing the passive elastic moment should be added to the sinusoidal gravitational model (Eq. (1) in Anderson et al., 2010) giving Mmidrange ¼ A1 sinðyÞ þ A2 cosðyÞ þb1 y þ b2 where b1 and b2 are coefficients for the linear passive elastic component. To estimate these coefficients, an iterative approach can be used. In the first iteration, b1 and b2 are set to zero, and the coefficients for gravitational moment (A1 and A2) are found using a least squares fit as described originally. The passive moment data is then corrected for gravity as described originally, but using the equation above with only the A1 and A2 terms (not including the b1 and b2 terms). The passive elastic moment constants (B1, k1, B2, and k2) are then found as described originally. Coefficients b1 and b2 can then be estimated by evaluating the passive elastic moment (Eq. (2) in Anderson et al., 2010) at the ends of the midrange region and determining the equation of the line between those two points. In subsequent iterations, the estimates of b1 and b2 are included when determining A1 and A2, and updated values for B1, k1, B2, k2, b1 and b2 may be found. Iteration continues until the fit is optimized (i.e. the concordance correlation coefficient reaches a maximum). It should be noted that with this approach the final equation for fitting the measured passive moment (Eq. (3) in Anderson et al., 2010) remains the same and does not include the terms b1 and b2. This is because the linear trend of the passive elastic moment within the mid-range region is accounted for in the passive elastic moment equation (Eq. (2) in Anderson et al., 2010). In testing using data from a single subject, this addition to the method reduced the magnitude of estimated gravitational moment for the ankle, knee, and hip. The concordance correlation coefficient was optimized for all three joints in three iterations. At the ankle, the largest magnitude gravitational moment in the range of motion was reduced by 1.84 N m or 18.6%. Similarly, gravitational moment was reduced by 0.69 N m (4.1%) at the knee and by 5.55 N m (8.1%) at the hip. Thus, this revised method appears to be less likely to overestimate the magnitude of the gravitational moment.
2656
Letters to the Editor / Journal of Biomechanics 43 (2010) 2653–2656
In conclusion, we agree with Clarke et al. that the model as originally published is susceptible to errors due to the assumption of near-zero moments in the mid-range region. The addition to the model described here can account for data that does not conform to this assumption, and as a result should eliminate the overestimation of gravitational moment. As such, this method should offer an improvement over commonly used approaches to gravity correction that do not consider the passive elastic moment.
Yoon, Y.S., Mansour, J.M., 1982. The passive elastic moment at the hip. Journal of Biomechanics 15, 905–910.
Dennis E. Anderson Department of Engineering Science and Mechanics, Virginia Tech, Blacksburg, VA, USA Michael L. Madigan n Department of Engineering Science and Mechanics (MC 0219), Virginia Tech, Blacksburg, VA 24061, USA E-mail address: [email protected]
Conflict of interest statement We have no conflict of interest to report.
Maury A. Nussbaum Department of Industrial and Systems Engineering, Virginia Tech, Blacksburg, VA, USA
References Anderson, D.E., Nussbaum, M.A., Madigan, M.L., 2010. A new method for gravity correction of dynamometer data and determining passive elastic moments at the joint. Journal of Biomechanics 43, 1220–1223. McFaull, S.R., Lamontagne, M., 1998. In vivo measurement of the passive viscoelastic properties of the human knee joint. Human Movement Science 17, 139–165. doi:10.1016/j.jbiomech.2010.05.016
Authors’ response to ‘‘Comments on ‘Validation of a musculoskeletal model of wheelchair propulsion and its application to minimizing shoulder joint forces’’’ A recently published Letter to the Editor by N. Louis (2009) raised the question as to whether the errors in the article, ‘‘Validation of a musculoskeletal model of wheelchair propulsion and its application to minimizing shoulder joint forces’’ (2008 October 20; 41 (14): 2981–8), were data processing or transcription errors. We, the authors of this Original Article publication, wish to respond to the concerns raised by N. Louis. First, we would like to thank N. Louis for his interest in our work in musculoskeletal modeling and wheelchair propulsion biomechanics. Second, we would like to thank him for his thoroughness in reading our article. After reviewing the comments questioning our data processing section (2.4. page 2983), thorough investigation into the suspect matrices have confirmed that there are indeed, two obvious transcription errors. In Eq. (1), the second column of the third row of the RX matrix should indeed be a 1 (corresponding to sin(901)), not a 0. The correct RX matrix should be 0 1 0 1 1 0 0 1 0 0 B C B C RX ¼ @ 0 cosð90Þ sinð90Þ A ¼ @ 0 0 1 A ð1Þ 0 sinð90Þ cosð90Þ 0 1 0
14 May 2010
n
Corresponding author. Tel.: + 1 540 231 1215; fax: + 1 540 231 4574.
system for transformation into the ISB coordinate system, Eq. (2) should correctly be 0 1 1 0 0 B C 0 1A RX RY ¼ @ 0 ð2Þ 0 1 0 Not only have previous versions of the Original Article been verified to have the correct transformation matrix (RX), but the resulting 3-D coordinates in the ISB Global Coordinate System have been confirmed to have been calculated using the correct matrix product (RXRY). In conclusion, the authors would again like to thank N. Louis for his comprehensive review of our paper, and reiterate to the readers of the Journal of Biomechanics that the two errors that were brought to our attention were indeed transcription errors. We would like to restate that the integrity of the data processing itself was never compromised.
Sarah R. Dubowsky University of Medicine and Dentistry of New Jersey, 30 Bergen Street, Newark, NJ 07101, United States E-mail address: [email protected]
Additionally, the matrix product RXRY, applied to the 3-D coordinates of the reflective markers in the Vicon coordinate
6 May 2010
DOI of original articles: 10.1016/j.jbiomech.2009.06.036 doi:10.1016/j.jbiomech.2010.05.007
n
Corresponding author. Tel.: + 1 540 231 1215; fax: + 1 540 231 4574.
described by Anderson et al. should only be used when the presence of a gravity-only region has been confirmed.
2655
Thompson, R.E., Barker, T.M., Pearcy, M.J., 2003. Defining the neutral zone of sheep intervertebral joints during dynamic motions: an in vitro study. Clinical Biomechanics 18, 89–98.
Conflict of interest statement
E.C. Clarke n Research Fellow, The George Institute for International Health, The University of Sydney, PO Box M201, Missenden Rd., Sydney, NSW 2050, Australia E-mail address: [email protected]
No conflicts of interest to disclose.
Acknowledgement
J.H. Martin Prince of Wales Medical Research Institute, University of New South Wales, Sydney, Australia
The authors wish to thank Mr. Kevin Wang for his assistance in conducting the experiments. References
R. Herbert Research Fellow, The George Institute for International Health, The University of Sydney, PO Box M201, Missenden Rd., Sydney, NSW 2050, Australia
Anderson, D.E., Nussbaum, M.A., Madigan, M.L., 2010. A new method for gravity correction of dynamometer data and determining passive elastic moments at the joint. Journal of Biomechanics. doi:10.1016/j.jbiomech.2009.11.036. Clarke, E.C., Appleyard, R., Bilston, L.E., 2007. Immature sheep spines are more flexible than mature spines: an in vitro biomechanical study. Spine 32, 2970–2979. Hoang, P.D., Gorman, R.B., Todd, G., Gandevia, S.C., Herbert, R.D., 2005. A new method for measuring passive length–tension properties of human gastrocnemius muscle in vivo. Journal of Biomechanics 38, 1333–1341. doi:10.1016/j.jbiomech.2010.03.051
Response to ‘‘Assumption of a ‘gravity only region’ for gravity correction of passive joint moment data may be problematic’’ In our recent paper (Anderson et al., 2010) we describe a method for determining the gravitational moment in dynamometer data that also accounts for the passive elastic moment due to passive tissue deformation. Our method assumes that the passive elastic moment is negligible within a middle region of the joint range of motion, which we dubbed the ‘‘gravity-only region’’. In their letter to the Editor entitled ‘‘Assumption of a ‘gravity only region’ for gravity correction of passive joint moment data may be problematic’’, Clarke et al. present data indicating that this assumption is invalid for the ankle, and as a result the method tends to overestimate the magnitude of the gravitational moment at the ankle. We would like to thank Clark et al. for their scrutiny of our method, as this is a critical component of scientific progress. In this response we describe an addition to our method that accounts for the presence of a non-negligible passive elastic moment in the gravity-only region, and as a result should address the valid concern expressed by Clarke et al. We would also like to point out that, based upon the data presented by Clarke et al. and our own retrospection, the term ‘‘gravity-only region’’ that we used in the original work is somewhat of a misnomer for the midrange region of passive moment data. This is because the passive elastic moment in this region, while small, may still be present. As such, this region will henceforth be referred to as the ‘‘mid-range region’’. Clarke et al. reported a passive elastic moment at the ankle that is not uniformly zero in the mid-range region. Based upon their data, this moment could be reasonably approximated by a straight line. Other studies also report non-zero yet linear passive elastic moments in the mid-range region of the knee (McFaull and Lamontagne, 1998) and hip (Yoon and Mansour, 1982). To account for the presence of a non-zero passive elastic moment in this mid-range region, a linear approximation of the passive elastic moment may be incorporated into our previously published method. Thus, when fitting the passive moment data from
30 March 2010
n
Corresponding author. Tel.: + 61 2 9657 0343; fax: + 61 2 9657 0301.
the mid-range region, a line representing the passive elastic moment should be added to the sinusoidal gravitational model (Eq. (1) in Anderson et al., 2010) giving Mmidrange ¼ A1 sinðyÞ þ A2 cosðyÞ þb1 y þ b2 where b1 and b2 are coefficients for the linear passive elastic component. To estimate these coefficients, an iterative approach can be used. In the first iteration, b1 and b2 are set to zero, and the coefficients for gravitational moment (A1 and A2) are found using a least squares fit as described originally. The passive moment data is then corrected for gravity as described originally, but using the equation above with only the A1 and A2 terms (not including the b1 and b2 terms). The passive elastic moment constants (B1, k1, B2, and k2) are then found as described originally. Coefficients b1 and b2 can then be estimated by evaluating the passive elastic moment (Eq. (2) in Anderson et al., 2010) at the ends of the midrange region and determining the equation of the line between those two points. In subsequent iterations, the estimates of b1 and b2 are included when determining A1 and A2, and updated values for B1, k1, B2, k2, b1 and b2 may be found. Iteration continues until the fit is optimized (i.e. the concordance correlation coefficient reaches a maximum). It should be noted that with this approach the final equation for fitting the measured passive moment (Eq. (3) in Anderson et al., 2010) remains the same and does not include the terms b1 and b2. This is because the linear trend of the passive elastic moment within the mid-range region is accounted for in the passive elastic moment equation (Eq. (2) in Anderson et al., 2010). In testing using data from a single subject, this addition to the method reduced the magnitude of estimated gravitational moment for the ankle, knee, and hip. The concordance correlation coefficient was optimized for all three joints in three iterations. At the ankle, the largest magnitude gravitational moment in the range of motion was reduced by 1.84 N m or 18.6%. Similarly, gravitational moment was reduced by 0.69 N m (4.1%) at the knee and by 5.55 N m (8.1%) at the hip. Thus, this revised method appears to be less likely to overestimate the magnitude of the gravitational moment.
2656
Letters to the Editor / Journal of Biomechanics 43 (2010) 2653–2656
In conclusion, we agree with Clarke et al. that the model as originally published is susceptible to errors due to the assumption of near-zero moments in the mid-range region. The addition to the model described here can account for data that does not conform to this assumption, and as a result should eliminate the overestimation of gravitational moment. As such, this method should offer an improvement over commonly used approaches to gravity correction that do not consider the passive elastic moment.
Yoon, Y.S., Mansour, J.M., 1982. The passive elastic moment at the hip. Journal of Biomechanics 15, 905–910.
Dennis E. Anderson Department of Engineering Science and Mechanics, Virginia Tech, Blacksburg, VA, USA Michael L. Madigan n Department of Engineering Science and Mechanics (MC 0219), Virginia Tech, Blacksburg, VA 24061, USA E-mail address: [email protected]
Conflict of interest statement We have no conflict of interest to report.
Maury A. Nussbaum Department of Industrial and Systems Engineering, Virginia Tech, Blacksburg, VA, USA
References Anderson, D.E., Nussbaum, M.A., Madigan, M.L., 2010. A new method for gravity correction of dynamometer data and determining passive elastic moments at the joint. Journal of Biomechanics 43, 1220–1223. McFaull, S.R., Lamontagne, M., 1998. In vivo measurement of the passive viscoelastic properties of the human knee joint. Human Movement Science 17, 139–165. doi:10.1016/j.jbiomech.2010.05.016
Authors’ response to ‘‘Comments on ‘Validation of a musculoskeletal model of wheelchair propulsion and its application to minimizing shoulder joint forces’’’ A recently published Letter to the Editor by N. Louis (2009) raised the question as to whether the errors in the article, ‘‘Validation of a musculoskeletal model of wheelchair propulsion and its application to minimizing shoulder joint forces’’ (2008 October 20; 41 (14): 2981–8), were data processing or transcription errors. We, the authors of this Original Article publication, wish to respond to the concerns raised by N. Louis. First, we would like to thank N. Louis for his interest in our work in musculoskeletal modeling and wheelchair propulsion biomechanics. Second, we would like to thank him for his thoroughness in reading our article. After reviewing the comments questioning our data processing section (2.4. page 2983), thorough investigation into the suspect matrices have confirmed that there are indeed, two obvious transcription errors. In Eq. (1), the second column of the third row of the RX matrix should indeed be a 1 (corresponding to sin(901)), not a 0. The correct RX matrix should be 0 1 0 1 1 0 0 1 0 0 B C B C RX ¼ @ 0 cosð90Þ sinð90Þ A ¼ @ 0 0 1 A ð1Þ 0 sinð90Þ cosð90Þ 0 1 0
14 May 2010
n
Corresponding author. Tel.: + 1 540 231 1215; fax: + 1 540 231 4574.
system for transformation into the ISB coordinate system, Eq. (2) should correctly be 0 1 1 0 0 B C 0 1A RX RY ¼ @ 0 ð2Þ 0 1 0 Not only have previous versions of the Original Article been verified to have the correct transformation matrix (RX), but the resulting 3-D coordinates in the ISB Global Coordinate System have been confirmed to have been calculated using the correct matrix product (RXRY). In conclusion, the authors would again like to thank N. Louis for his comprehensive review of our paper, and reiterate to the readers of the Journal of Biomechanics that the two errors that were brought to our attention were indeed transcription errors. We would like to restate that the integrity of the data processing itself was never compromised.
Sarah R. Dubowsky University of Medicine and Dentistry of New Jersey, 30 Bergen Street, Newark, NJ 07101, United States E-mail address: [email protected]
Additionally, the matrix product RXRY, applied to the 3-D coordinates of the reflective markers in the Vicon coordinate
6 May 2010
DOI of original articles: 10.1016/j.jbiomech.2009.06.036 doi:10.1016/j.jbiomech.2010.05.007
n
Corresponding author. Tel.: + 1 540 231 1215; fax: + 1 540 231 4574.