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  • Summary: Functional method to locate the hip joint center

    Dear colleagues,

    About 10 days ago, I posted a request for algorithms commonly used in the functional method of determining the location of the hip joint center. Here is the original request:

    > We are currently reviewing different methods of locating the hip joint
    > center. It seems to me that most of the indirect methods
    > basically fall into
    > two categories: (1) methods based on the pelvic markers, in some cases in
    > conjunction with anthropometric measurements, and (2) numerical
    > methods s
    uch
    > as the functional method originally reported by Cappozzo (1984).
    >
    > The methods that fall into group 1 include
    >
    > Tylkowsky et al. (1982)
    > Andriacchi et al. (1980)
    > Bell et al. (1990): hybrid of Tylkowsky and Andriacchi
    > Davis et al. (1991)
    >
    > Studies based on the numerical method (rotational method) include
    >
    > Piazza et al. (2001)
    > Leardini et al. (1999)
    > Bell et al. (1990)
    > Cappozzo (1984)
    >
    > I am particularly interested in the functio
    nal method which is
    > based on the
    > least square approximation of
    >
    > L = sum[ ( (xi-xc)^2 + (yi-yc)^2 + (zi-zc)^2 )^0.5 - R ] ==> min
    >
    > where xi, yi, zi = the coordinates of a marker on the thigh (or
    > centroid of
    > several markers) at frame i described in the pelvis reference
    > frame, xc, yc,
    > zc = hip joint center described in the pelvis reference frame, and R =
    > radius of the sphere formed by this marker about the hip joint center.
    >
    > Can anyone suggest a numerical alogori
    thm that will perform the
    > aforementioned least square approximation effectively. I could
    > not find any
    > specifics on this in the published papers. I am in the process of
    > writing a
    > program to do this and I'd like to compare different algorithms including
    > mine. Any pointers or suggestions will be appreciated. As usual,
    > I will post the summary on Biomch-L.

    I am grateful to those who responded to my request: Sian Jenkins, Tomislav Pribanic, Mark Thompson, Xavier Savatier, Arnel Aguinald
    o, Nels Madsen, Kjartan Halvorsen, Alberto Leardini, and James Shippen. Individual responses are compiled at the end of this summary.

    Here are the studies recommended that deal with the algorithms:

    * Gander, W., Golub, G. H., & Strebel, R. (1994). Fitting of circles and ellipses: least squares solution. TechReport Departement Informatik, ETH Zurich. ftp.inf.ethz.ch/doc/tech-reports/2xx/217.ps

    * Halvorsen, K., Lesser, M., & Lundberg, A. (1999). A new method for estimating the axis of rotation and th
    e center of rotation. J Biomechanics 32, 1221-1227.

    * Hiniduma Udugama Gamage, S.S., & Lasenby, J. (2002). New least squares solution for
    estimating the average centre of rotation and the axis of rotation. Journal of Biomechanics, 35, 87-93.

    Traditional algorithms recommended include Newton's method (Newton-Raphson), and Nelder-Mead downhill simplex method.

    Some colleagues sent me MATLAB codes (Mark and Alberto), manuscript under review (Kjartan), and even a detailed rundown of the equations (Nel
    s). Kjartan also offered his doctoral dissertation. I am grateful to you all.

    I compiled details of the methods mentioned above and put them on a webpage. Follow the following links: http://kwon3d.com > Theories > II. Theories and Practices in Motion Analysis > Joint Kinematics > Joint Center: Functional Method. Any feedback regarding this page would be appreciated.

    I am in the process of writing a program for the functional method algorithms. I will report the comparison results soon.

    With gratitu
    de,

    Young-Hoo Kwon
    ------------------------------------------------------
    - Young-Hoo Kwon, Ph.D.
    - Biomechanics Lab, Texas Woman's University
    - ykwon@twu.edu
    - http://kwon3d.com
    ------------------------------------------------------

    Responses to my request:
    =======================================
    Professor Kwon,

    In addition to the literature you've already mentioned, I would suggest that you
    read
    the paper 'New least squares solutions for estimating the average ce
    ntre of
    rotation and the axis of rotation' by Sahan S. Hiniduma Udugama Gamage and Joan
    Lasenby Journal of Biomechanics, 35 (1) : 87-93. 2002;
    and also
    Halvorsen, K., Lesser, M. and Lundberg, A., 1999. A new method for estimating
    the axis of rotation and the center of rotation. Journal of Biomechanics 32 :
    1221-1227.

    There are a number of groups currently working on functional techniques, and so
    I expect that you will get an interesting set of responses.
    I look forward to your summary,

    Sian J
    enkins
    ---------------------------------------------
    Sian E. M. Jenkins
    Orthopaedic Engineering
    University of Oxford

    Oxford Gait Laboratory
    Nuffield Orthopaedic Centre NHS Trust
    tel/ (fax): +44 1865 227946 / (742348)
    ---------------------------------------------

    =======================================
    Dear Sir,

    have you been lately, long time no hear from you?
    I do not know much about finding joint centers and what you need it for (I
    can only guess that your aim is inverse dynamic solutio
    n values?).
    Furthermore, I suspect that expression you are minimizing is of non-linear
    nature. In that case I can recommend you Levenberg-Marquardt method. I have
    used it and I found it very effective for camera calibration. Also, many
    others think of it as standard for the non-linear lest square (i.e. data
    modeling) problems. Here below is a site where you can find out more about
    it and some other numerical methods too (along with codes in C for many of
    the things). If I can help anything more do no
    t hesitated to contact me.

    Regards, Tomislav.

    http://www.ulib.org/webRoot/Books/Numerical_Recipes/bookcpdf.html

    Tomislav Pribanic, M.Sc., EE
    Department for Electronic Systems and Information Processing
    Faculty of Electrical Engineering and Computing
    3 Unska, 10000 Zagreb, Croatia
    tel. ..385 1 612 98 67, fax. ..385 1 612 96 52
    E-mail : tomislav.pribanic@fer.hr

    =======================================
    Dear Young-Hoo

    I am also interested in the problem of locating the hip joint centre, but

    have come across the problem of fitting spheres to point clouds in the
    context of characterising the shape of the human acetabulum.

    In the paper Thompson et al 2000 I used an algorithm from a technical
    report from ETH Zurich, Gander et al 1994. I merely added a third dimension
    to their circle fitting routine.

    In fact the algorithm they use turns the minimisation problem to a root
    finding problem using the Newton Raphson method. There is then the problem
    of giving the algorithm a good initi
    al estimate. I just use the mean of the
    point cloud and the magnitude of its position vector, but Gander et al do
    propose a more sophisticated method in the technical report.

    The code is below.

    I would also like to draw your attention to the paper by Gamage et al
    (2002) which uses a slightly different cost function to the one you have
    quoted. This allows for small relative motions between markers on the same
    body segment. Also they present a much more general numerical solution to
    the prob
    lem.

    I hope that this helps!

    best wishes

    Mark

    references:

    S. S. H. U. Gamage and J. Lasenby. New Least Squares Solution for
    Estimating the Average Centre of Rotation and the Axis of Rotation. Journal
    of Biomechanics 35:87-93, 2002.

    W. Gander, G. H. Golub, and R. Strebel. Fitting of Circles and Ellipses:
    Least Squares Solution. TechReport Departement Informatik, ETH Zurich.
    1994 ftp.inf.ethz.ch/doc/tech-reports/2xx/217.ps

    M. S. Thompson, T. Dawson, J-H. Kuiper, M. D. Northmore-B
    all, and K. E.
    Tanner. Acetabular Morphology and Resurfacing Design. Journal of
    Biomechanics 33 (12):1645-1653, 2000.

    =======================================
    Hi,

    I'm currently working on the same problem. I found a paper which describes a
    least squares algorithm based on the fact that every markers on the thigh
    lie on co-centric spheres. See :

    S.S.H.U. Gamage et al., J. of Biomech, 35 (2002) 87-93

    I tried to implement the algorithm of this article with Mathcad but
    unsuccessfully at thi
    s time. If you find some equivalent solutions, I will
    very pleased if you could share them with me !

    I hope this will help you.
    Best regards,

    Xavier SAVATIER
    Research Engineer
    ESIGELEC
    Mont St Aignan - FRANCE

    Best congratulations for your web site !

    =======================================
    Hi Young-Hoo:

    It's great to hear from you again.

    We're actually finishing up a study on the functional method for hip joint
    center location using MRI-estimated hjc locations as validation.

    Ou
    t of the four articles you mentioned in group 2, only Piazza et al. (2001)
    described in relatively greater detail the algorithm (mixed quadratic/cubic)
    used to minimize the error function. What's unique about their study was
    that the initial values for the minimization process were randomly located
    within a cube centered on the true hjc location (albeit a ball-n-socket
    model not a real hip). Additionally, they increased the max # of iterations
    and function evaluations to 1000 and 10000, respectively,
    but I believe
    these max values simply prevent the minimization from crashing out before an
    adequate fit is achieved.

    Shea et al. (1997) presented a study on current software (Orthotrak) that
    specifies hjc locations as % of pelvic width (ie, Bell's model) using the
    functional method as their validation. The algorithm they used to minimize
    the error function was a downhill simplex method but was not detailed in the
    abstract, but I imagine this was an optimization tool coded in Matlab.

    The algorit
    hm we are employing is based on the Nelder-Meed simplex method,
    which is available in Matlab's function toolbox (use fminsearch function for
    more than one variable). What I am testing out is a process in which the
    minimization results from the trajectory a thigh fixed marker (or centroid)
    are used as initial values for the error minimization based on the
    trajectory of a virtual knee center. The idea is similar to Piazza's study
    in that the initial guesses will "drive" the minimization to a relatively

    better fit (at least that's the idea). The biggest difference with this case
    is that we are evaluating hip joints in-vivo. The optimal motion being
    analyzed is the standard combination of hip flex/ext, abd/add, and
    circumduction. The estimated hip joint locations are then compared to MRI
    based coordinates for validation.

    You can find a very detailed description of the Nelder-Meed algorithm in
    Matlab's function (funfun) toolbox. I can send it to you if you don't have
    it.

    I would be interested in
    testing out your algorithm should it become
    available. Like others in our field, we wish to estimate the hjc location
    using only the hip motion of walking rather than that of a fully 3D range of
    motion. This would be advantageous for patients with very limited hip ROM.
    The challenge, of course, is estimating this point from what ultimately is a
    predominantly planar trajectory during walking.

    Hope this helps!

    Arnel

    ---
    Arnel Aguinaldo
    Biomechanical Engineer
    Motion Analysis Laboratory
    Child
    ren's Hospital San Diego
    San Diego, California USA

    =======================================
    Attached is something I worked up in response to your question (Word file).
    Let me know if you have any difficulty accessing what I have prepared.
    Let me know if you have any questions or concerns about what I have prepared.
    As I didn't have much time to spend on it, I apologize in advance for any
    spelling, grammatical, or typographical errors.
    Let me know if you have any interest in pursuing the proposed id
    ea.

    Enjoy, Nels.

    Nels Madsen
    Associate Dean for Assessment and Special Programs
    Samuel Ginn College of Engineering
    Ramsay Hall
    Auburn University, AL 36849
    334 844 3329
    334 844 3307 FAX
    nmadsen@eng.auburn.edu

    =======================================
    Dear Dr. Kwon,

    I would like to express my appreciation for your biomechanics web pages.
    They cover a nice collection of important methods in biomechanics.

    To your question.
    Look up the recent paper
    Gamage & Lasenby "New least squares sol
    utions for estimating the average
    center of rotation and the axis of rotation". J Biomechanics 35: 87-93,
    2002.

    Gamage & Lasenby provides a closed form solution to the least squares
    criterion you set up in your message.
    An improvement to the method of Gamage & Lasenby has recently been
    submitted to J Biomechanics:
    Halvorsen "A bias compensated least squares estimate of the center of
    rotation"
    I may provide you with a preprint upon request. For a more in depth
    treatment of functional method
    s for determing the joint center you may
    want to have a look at my PhD thesis
    "Model-based methods in Motion Capture"
    which I will defend the coming friday, October 4. I will be glad to send
    you a copy.

    Yours sincerely,

    Kjartan Halvorsen
    PhD student


    The Department of Systems and Control
    Uppsala University
    http://www.syscon.uu.se/
    + 46 18 471 3150

    =======================================
    Dear Young-Hoo, my original work is based on a very simple procedure I
    enclose (in Matlab). It is
    based on the sphere equation. You don't need
    numerical algorithms in a first approximation.

    Do not forget the following in those methods based on pelvic 'geometry'
    (rather than
    'markers'):
    Seidel, G.K., Marchinda, D.M., Dijkers, M., Soutas-Little, R.W., 1995. Hip
    joint center location from palpable bony landmarks - a cadaver study.
    Journal of Biomechanics 28, 995-998.

    Bell et al. (1990) compared in fact his own hybrid (from Tylkowsky and
    Andriacchi), based on regression equations on the pelvic
    'geometry', with
    the functional of Cappozzo: I would not include this Bell et al. (1990) in
    the 'functional' list.

    By the way, you may find a useful review in
    Wu G, Siegler S, Allard P, Kirtley C, Leardini A, Rosenbaum D, Whittle M,
    D'Lima DD, Cristofolini L, Witte H, Schmid O, Stokes I.
    ISB recommendation on definitions of joint coordinate system of various
    joints for the reporting of human joint motion-part I: ankle, hip, and
    spine. J Biomech. 2002 Apr;35(4):543-8

    Regards,

    ***************
    ************************************************** ********
    Alberto Leardini, DPhil
    Movement Analysis Laboratory
    Centro di Ricerca Codivilla-Putti
    Istituti Ortopedici Rizzoli
    Via di Barbiano 1/10, 40136 Bologna ITALY
    tel: +39 051 6366522
    fax: +39 051 6366561
    email: leardini@ior.it
    http://www.ior.it/movlab/

    "Where is the Life we have lost in living,
    Where is the wisdom we have lost in knowledge,
    Where is the knowledge we have lost in information."
    Thomas Stearns Eliot, Choruses from ''The Rock'
    ' (1934)
    ************************************************** ************************

    =======================================
    HI

    If you are (or know) a Matlab user I recommend the function FMINSEARCH which uses the Nelder-Mead simplex (direct search) method. This will reaily eat your problem.

    Regards

    James Shippen
    Marlbrook

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