- Email: [email protected]
Mapping ligament insertion sites onto bone surfaces in knee by co-registration of CT and digitization data
ARTICLE IN PRESS Journal of Biomechanics 42 (2009) 2624–2626
Contents lists available at ScienceDirect
Journal of Biomechanics journal homepage: www.elsevier.com/locate/jbiomech www.JBiomech.com
Short communication
Mapping ligament insertion sites onto bone surfaces in knee by co-registration of CT and digitization data Kang Li a,d, Madelyn O’Farrell a, Daniel Martin a, Sebastian Kopf a, Christopher Harner a, Xudong Zhang a,b,c, a
Department of Orthopaedic Surgery, University of Pittsburgh, USA Department of Mechanical Engineering and Materials Science, University of Pittsburgh, USA c Department of Bioengineering, University of Pittsburgh, USA d Department of Mechanical Science and Engineering, University of Illinois at Urbana-Champaign, USA b
a r t i c l e in f o
a b s t r a c t
Article history: Accepted 25 June 2009
This paper describes a new methodology that enables mapping of the ligament insertion sites onto bone surfaces in the knee joint by co-registration of the data acquired using digitization and computed tomography (CT). Local coordinate systems on the distal femur and proximal tibia were established by three spherical fiducial markers rigidly affixed to each bone. The fiducial marker centroid locations were identified by a least-squares sphere-fitting algorithm. An optimization correction procedure was proposed to mitigate the effect of the target registration error (TRE) on the alignment of coordinate systems for co-registration. A test with four cadaveric specimens demonstrated successful mapping of insertion sites for five ligaments. Fiducial registration error (FRE) as measured by the differences in inter-marker distances between the two modalities was smaller than 2%. The optimization procedure corrected the insertion site invisibility or partial visibility problem and improved the overall mapping quality, as indicated by substantial reduction of the mean and dispersion of distances from digitized insertion site points to the bone surfaces. & 2009 Elsevier Ltd. All rights reserved.
Keywords: Co-registration Ligament insertion site morphology Digitization CT Quantitative anatomy
1. Introduction Accurate knowledge of soft tissue insertion sites in the knee is needed in both basic research and clinical practice. Data on component tissue attachment or insertion morphology are critical, but currently insufficient, for building subject-specific anatomically accurate knee biomechanical models (Kaptein and van der Helm, 2004; Seim et al., 2008). Clinically, while it is easy to appreciate the concept of re-creating the anatomy (and ultimately the function) in ligament reconstruction, it can be difficult to locate the ligament insertion sites intra-operatively. Several studies have attempted to quantify cruciate ligament insertion site morphology, with techniques ranging from direct caliper measurement (Girgis et al., 1975; Morgan et al., 1997) to laser micrometry (Harner et al., 1995; Harner et al., 1999). Others have characterized the cruciate ligament insertions on plain radiographs (Colombet et al., 2006; Zantop et al., 2008), and demonstrated the variability in insertion site morphology (Edwards et al., 2007, 2008). These studies, however, have not
produced quantitative data of insertion sites in three-dimensional (3D) space and in relation to the bony geometry. This void was due mainly to the difficulty of acquiring both insertion and bone morphological data accurately with a single modality or with two modalities combined seamlessly. Imaging modalities have yet to allow accurate identification of the ligament insertions; while digitization may be the most effective method to record insertion morphological data, it is hardly a viable means for creating highfidelity bone surface models. Existing data registration methods for identifying anatomical sites (Kaptein and van der Helm, 2004; Gray et al., 2008; Seim et al., 2008) or estimating kinematics (Fischer et al., 2001) are challenged by the heighten demands in accuracy and practicality. We introduced a new method to precisely map the ligament insertion sites onto bone surfaces in cadaveric knees by coregistration of 3D digitization data of insertion sites and computed tomography (CT) data of the distal femur and proximal tibia surfaces. We tested the method using four specimens.
2. Methods Corresponding author at: Department of Orthopaedic Surgery, University of Pittsburgh, USA. Tel.: +1 412 586 3940; fax: +1 412 586 3979. E-mail address: [email protected] (X. Zhang).
0021-9290/$ - see front matter & 2009 Elsevier Ltd. All rights reserved. doi:10.1016/j.jbiomech.2009.06.042
Soft tissues of cadaveric knee specimens were removed from the femur, tibia, and fibula in areas more than 6 cm away from the joint line, to facilitate the placement and identification (in CT) of fiducial markers. These fiducial markers
ARTICLE IN PRESS K. Li et al. / Journal of Biomechanics 42 (2009) 2624–2626 were precision Nylon spheres (radius 9.525 mm), rigidly affixed through plastic screws to the femur and tibia, three on each bone. They were placed close to the joint line without possibly disrupting the soft tissues of interest and were separated as widely as possible in non-collinear positions (Soderkvist and Wedin, 1993). High-resolution CT scans of the knee were then taken with slice spacing of 0.625 mm. After completion of the imaging, careful dissection of the specimens was performed by an experienced arthroscopic surgeon. A MicroScribe Digitizer (Immersion Corp., CA) with a reported accuracy of 70.2 mm was used to digitize in 3D the surfaces of the spherical markers and the peripheries of the insertion sites for the following soft tissue structures: anterior cruciate ligament (ACL) consisting of both anteromedial (AM) and posterolateral (PL) bundles, posterior cruciate ligament (PCL) consisting of both anterolateral (AL) and posteromedial (PM) bundles, medial collateral ligament (MCL), lateral collateral ligament (LCL), and popliteus tendon. The marker surfaces were digitized such that ‘point-clouds’ of the spherical surfaces were formed. The Mimics software (Materialise, Belgium) was used to segment the bones and spherical markers separately from the CT scans. As a result, 3D surface models of the femur, tibia, and spherical markers were generated. The centroids of spherical markers were obtained by fitting the marker surface data, digitization- or CT-acquired, by a least-squares geometric fitting algorithm. Given m surface points (m ranged about 600–2500 for the digitization data and 1200–15,000 for the CT data), the algorithm determined the four parameters of a sphere, the centroid coordinates (xc, yc, zc) and radius R of the sphere, by minimizing the following residual: vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u 2 ! m qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u1 X : ðxi xc Þ2 þ ðyi yc Þ2 þ ðzi zc Þ2 R Res ¼ t m i¼1
Spherical markers
PCL(AL,PM)
ð2Þ
where (xi, yi, zi) is a digitized insertion site point and (x*i, y*i, z*i) is the nearest point on the corresponding bone surface and n is the total number of the insertion points on each bone (n ranged 117–150). Note that the RMSD, mean distance (MD), and standard deviation of the distances (sD) have the following relation: RMSD2 ¼ MD2+sD2.
3. Results The proposed method was able to map the ligament insertion sites onto the bone surfaces (Fig. 1). The centroids of the spherical fiducial markers were successfully identified using the least-squares fitting algorithm. The average radii of the sphere markers in both modalities were within 72% of the nominal radius (Table 1). The residual errors were all less than 0.25 mm. The differences in inter-marker distances between the two modalities were less than 2%, confirming a small fiducial registration error (FRE)—the difference between homologous fiducial points after registration (Fitzpatrick et al., 1998). The optimization procedure reduced the TRE as measured by RMS distances from the insertion sites to the bone surface from 0.82 (70.21) mm to 0.59 (70.10) mm for the femur, from 0.78 (70.12) mm to 0.63 (70.10) for the tibia. The grand mean (7SD) distances from the insertion sites to the bone surface were reduced from 0.74(70.40) mm to 0.53 (70.29) mm for the femur,
ACL(PL,AM)
LCL Popliteus
MCL
LCL
ACL(PL,AM)
Spherical markers
ð1Þ
The centroids of three markers on each bone established a local coordinate system. Thus, two local coordinate systems were affixed to each bone, one based on digitization data and one on CT data. However, directly applying the transformation matrix between the two coordinate systems may not guarantee the proper positioning of the transformed insertion sites relative to the bone surface in the CT-based coordinate system (e.g., an insertion site could partially or completely lie beneath the bone surface). This is due to the inevitable target registration error (TRE), defined as the distance between the homologous points in both modalities (Fitzpatrick et al., 1998). An optimization correction procedure was formulated to alleviate the effect of TRE on the alignment of coordinate systems. The procedure varied the position of the most distorted marker (whose distance to the other two markers changed most from one modality to the other) within a range of 71 mm in all three dimensions in the digitization-based coordinate system, and through enumeration identified the position resulting in a global minimum root-mean-square distance (RMSD) from a digitized insertion site point to the bone surface. Mathematically, the following objective function was minimized: vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u n u1 X ½ðx xi Þ2 þ ðyi yi Þ2 þ ðzi zi Þ2 RMSD ¼ t n i¼1 i
2625
Fig. 1. The CT-based femur and tibia surface models with six spherical fiducial markers and insertion sites for five ligaments.
Table 1 Statistical summary of the radii of fitted spheres and residual fitting errors. CT
Digitization
Radii (mm) Femur Tibia
9.565 (70.035) 9.544 (70.065)
9.673 (70.117) 9.650 (70.145)
Fitting error (mm) Femur Tibia
0.163 (70.042) 0.139 (70.023)
0.168 (70.036) 0.164 (70.046)
The mean (7SD) values are averaged across all fiducial markers (3 markers per specimen 4 specimens ¼ 12 markers) on each bone within each modality.
Before correction
After correction LCL Popliteus
Popliteus LCL
Fig. 2. Insertion site mapping before and after the optimization-based correction. Note that before the correction, the lateral collateral ligament (LCL) insertion sites on the femur and the tibia were inside the bones.
from 0.69 (70.45) mm to 0.55 (70.37) mm for the tibia. Without the correction, the misalignment (‘‘tilting’’ and ‘‘translating’’) of the coordinate systems caused some insertion footprints to lie completely or partially underneath the bone surfaces and some others to be unrealistically farther away from the surfaces. With the correction, all digitized insertion footprints became visible (Fig. 2).
ARTICLE IN PRESS 2626
K. Li et al. / Journal of Biomechanics 42 (2009) 2624–2626
4. Discussion The technique established in this study will enable systematic investigation of the morphology of soft tissues in relation to 3D bone geometry in the knee, and lead to clinical tool development facilitating more precise, more anatomical, and better-navigated surgeries. A sufficiently accurate yet practical enabling technique has been lacking. The co-registration between an image modality and a physical space based on rigid-body points has been widely used in medical applications, but mostly for intra-operative or radiotherapy applications (cf. Maintz and Viergever, 1998), which required a sophisticated apparatus and a large number of fiducal markers. Practicality concerns impeded the use of these existing techniques for systematic studies of musculoskeletal anatomy. On the other hand, once such concerns are reduced or removed—for instance, as the feasibility to image a patient with fiducial markers on is improved—our technique can be combined with a surgical navigation system to acquire intraoperative data. The accuracy requirement presented another challenge. A few biomechanical studies employed ‘‘inter-modality registration’’ of CT and digitization data (Fischer et al., 2001; Gray et al., 2008). In fact, our exploration of a new method was preceded by testing the method by Fischer et al. (2001), which employed cubic registration blocks. We discovered that the reconstruction of a cubic block from CT, unlike a spherical fiducial marker, was sensitive to its orientation in relation to that of the CT slices; the accuracy or uncertainty, as indicated by the differences in the angles between adjacent planes of the registration block in both modalities, in determining the bone positions might be acceptable for estimating the gross knee joint kinematics, but not for mapping the ligament insertion sites. The screw-based co-registration used by Kaptein and van der Helm (2004) in mapping muscle attachment contours to the bone is not as accurate as the sphericalmarker-based co-registration; the same TRE measures, the attachment-to-bone distances, ranged mostly from 1 to 10 mm. The surfacebased method by Gray et al. (2008) relied on an iterative closest point (ICP) algorithm, and would require meticulous removal of all soft tissues to expose clean bone surfaces to be digitized. Further, the accuracy of ICP-based registration approach would be inferior (Maurer et al., 1998; Lee et al., 2008), while the point-based approach using spherical fiducial markers is considered a ‘‘gold standard’’ (West et al., 1997). Several sources of the mapping error have been effectively reduced by the two optimization procedures. The fiducial location error (FLE) (Fitzpatrick et al., 1998), defined in this study as the distance between the calculated centroid to the actual centroid (unknown), was contained by the sphere fitting procedure as evidenced by the small error in the radius of the fitted sphere. The discrepancy of the intermarker distances indicated the presence of the FRE. Yet, the FRE did not seem to affect the overall rigidity of the fiducial point configuration, which allowed another optimization procedure to alleviate the TRE by perturbing one of the fiducial points. Although in theory using more fiducial markers in the rigid-body point-based registration can decrease the TRE, in practice it may not be feasible to accommodate more markers with radius greater than a certain size.
Conflict of interest statement None.
References Colombet, P., Robinson, J., Christel, P., Franceschi, J.P., Djian, P., Bellier, G., Sbihi, A., 2006. Morphology of anterior cruciate ligament attachments for anatomic reconstruction: a cadaveric dissection and radiographic study. Arthroscopy 22, 984–992. Edwards, A., Bull, A.M., Amis, A.A., 2007. The attachments of the anteromedial and posterolateral fibre bundles of the anterior cruciate ligament: Part 1: tibial attachment. Knee Surgery, Sports Traumatology, Arthroscopy 15, 1414–1421. Edwards, A., Bull, A.M., Amis, A.A., 2008. The attachments of the anteromedial and posterolateral fibre bundles of the anterior cruciate ligament. Part 2: femoral attachment. Knee Surgery, Sports Traumatology, Arthroscopy 16, 29–36. Fischer, K.J., Manson, T.T., Pfaeffle, H.J., Tomaino, M.M., Woo, S.L.Y., 2001. A method for measuring joint kinematics designed for accurate registration of kinematic data to models constructed from CT data. Journal of Biomechanics 34, 377–383. Fitzpatrick, J., West, J., Maurer, C.J., 1998. Predicting error in rigid-body point-based registration. IEEE Transactions on Medical Imaging 17, 694–702. Girgis, F.G., Marshall, J.L., Monajem, A., 1975. The cruciate ligaments of the knee joint, anatomical, functional and experimental analysis. Clinical Orthopaedics and Related Research, 216–231. Gray, H.A., Taddei, F., Zavatsky, A.B., Cristofolini, L., Gill, H.S., 2008. Experimental validation of a finite element model of a human cadaveric tibia. Journal of Biomechanical Engineering 130, 31016. Harner, C.D., Baek, G.H., Vogrin, T.M., Carlin, G.J., Kashiwaguchi, S., Woo, S.L., 1999. Quantitative analysis of human cruciate ligament insertions. Arthroscopy 15, 741–749. Harner, C.D., Livesay, G.A., Kashiwaguchi, S., Fujie, H., Choi, N.Y., Woo, S.L., 1995. Comparative study of the size and shape of human anterior and posterior cruciate ligaments. Journal of Orthopaedic Research 13, 429–434. Kaptein, B.L., van der Helm, F.C.T., 2004. Estimating muscle attachment contours by transforming geometrical bone models. Journal of Biomechanics 37, 263–273. Lee, Y.S., Seon, J.K., Shin, V.I., Kim, G.H., Jeon, M., 2008. Anatomical evaluation of CT-MRI combined femoral model. Biomedical Engineering Online 7, 6. Maintz, J.B., Viergever, M.A., 1998. A survey of medical image registration. Medical Image Analysis 2, 1–36. Maurer, C.R., Maciunas, R.J., Fitzpatrick, J.M., 1998. Registration of head CT images to physical space using a weighted combination of points and surfaces. IEEE Transactions on Medical Imaging 17, 753–761. Morgan, C.D., Kalman, V.R., Grawl, D.M., 1997. The anatomic origin of the posterior cruciate ligament: where is it? Reference landmarks for PCL reconstruction. Arthroscopy 13, 325–331. Seim, H., Lamecker, H., Heller, M., Zachow, S., 2008. Segmentation of bony structures with ligament attachment sites In: Tolxdorff, T., Braun, J., Deserno, T. ¨ die M., Horsch, A., Handels, H., Meinzer, H.-P. (Eds.), Bildverarbeitung fur Medizin Springer, pp. 207–211. Soderkvist, I., Wedin, P.A., 1993. Determining the movements of the skeleton using well-configured markers. Journal of Biomechanics 26, 1473–1477. West, J., Fitzpatrick, J.M., Wang, M.Y., Dawant, B.M., Maurer, C.R., Kessler, R.M., Maciunas, R.J., Barillot, C., Lemoine, D., Collignon, A., Maes, F., Suetens, P., Vandermeulen, D., vandenElsen, P.A., Napel, S., Sumanaweera, T.S., Harkness, B., Hemler, P.F., Hill, D.L.G., Hawkes, D.J., Studholme, C., Maintz, J.B.A., Viergever, M.A., Malandain, G., Pennec, X., Noz, M.E., Maguire, G.Q., Pollack, M., Pelizzari, C.A., Robb, R.A., Hanson, D., Woods, R.P., 1997. Comparison and evaluation of retrospective intermodality brain image registration techniques. Journal of Computer Assisted Tomography 21, 554–566. Zantop, T., Wellmann, M., Fu, F.H., Petersen, W., 2008. Tunnel positioning of anteromedial and posterolateral bundles in anatomic anterior cruciate ligament reconstruction: anatomic and radiographic findings. American Journal of Sports Medicine 36, 65–72.
Contents lists available at ScienceDirect
Journal of Biomechanics journal homepage: www.elsevier.com/locate/jbiomech www.JBiomech.com
Short communication
Mapping ligament insertion sites onto bone surfaces in knee by co-registration of CT and digitization data Kang Li a,d, Madelyn O’Farrell a, Daniel Martin a, Sebastian Kopf a, Christopher Harner a, Xudong Zhang a,b,c, a
Department of Orthopaedic Surgery, University of Pittsburgh, USA Department of Mechanical Engineering and Materials Science, University of Pittsburgh, USA c Department of Bioengineering, University of Pittsburgh, USA d Department of Mechanical Science and Engineering, University of Illinois at Urbana-Champaign, USA b
a r t i c l e in f o
a b s t r a c t
Article history: Accepted 25 June 2009
This paper describes a new methodology that enables mapping of the ligament insertion sites onto bone surfaces in the knee joint by co-registration of the data acquired using digitization and computed tomography (CT). Local coordinate systems on the distal femur and proximal tibia were established by three spherical fiducial markers rigidly affixed to each bone. The fiducial marker centroid locations were identified by a least-squares sphere-fitting algorithm. An optimization correction procedure was proposed to mitigate the effect of the target registration error (TRE) on the alignment of coordinate systems for co-registration. A test with four cadaveric specimens demonstrated successful mapping of insertion sites for five ligaments. Fiducial registration error (FRE) as measured by the differences in inter-marker distances between the two modalities was smaller than 2%. The optimization procedure corrected the insertion site invisibility or partial visibility problem and improved the overall mapping quality, as indicated by substantial reduction of the mean and dispersion of distances from digitized insertion site points to the bone surfaces. & 2009 Elsevier Ltd. All rights reserved.
Keywords: Co-registration Ligament insertion site morphology Digitization CT Quantitative anatomy
1. Introduction Accurate knowledge of soft tissue insertion sites in the knee is needed in both basic research and clinical practice. Data on component tissue attachment or insertion morphology are critical, but currently insufficient, for building subject-specific anatomically accurate knee biomechanical models (Kaptein and van der Helm, 2004; Seim et al., 2008). Clinically, while it is easy to appreciate the concept of re-creating the anatomy (and ultimately the function) in ligament reconstruction, it can be difficult to locate the ligament insertion sites intra-operatively. Several studies have attempted to quantify cruciate ligament insertion site morphology, with techniques ranging from direct caliper measurement (Girgis et al., 1975; Morgan et al., 1997) to laser micrometry (Harner et al., 1995; Harner et al., 1999). Others have characterized the cruciate ligament insertions on plain radiographs (Colombet et al., 2006; Zantop et al., 2008), and demonstrated the variability in insertion site morphology (Edwards et al., 2007, 2008). These studies, however, have not
produced quantitative data of insertion sites in three-dimensional (3D) space and in relation to the bony geometry. This void was due mainly to the difficulty of acquiring both insertion and bone morphological data accurately with a single modality or with two modalities combined seamlessly. Imaging modalities have yet to allow accurate identification of the ligament insertions; while digitization may be the most effective method to record insertion morphological data, it is hardly a viable means for creating highfidelity bone surface models. Existing data registration methods for identifying anatomical sites (Kaptein and van der Helm, 2004; Gray et al., 2008; Seim et al., 2008) or estimating kinematics (Fischer et al., 2001) are challenged by the heighten demands in accuracy and practicality. We introduced a new method to precisely map the ligament insertion sites onto bone surfaces in cadaveric knees by coregistration of 3D digitization data of insertion sites and computed tomography (CT) data of the distal femur and proximal tibia surfaces. We tested the method using four specimens.
2. Methods Corresponding author at: Department of Orthopaedic Surgery, University of Pittsburgh, USA. Tel.: +1 412 586 3940; fax: +1 412 586 3979. E-mail address: [email protected] (X. Zhang).
0021-9290/$ - see front matter & 2009 Elsevier Ltd. All rights reserved. doi:10.1016/j.jbiomech.2009.06.042
Soft tissues of cadaveric knee specimens were removed from the femur, tibia, and fibula in areas more than 6 cm away from the joint line, to facilitate the placement and identification (in CT) of fiducial markers. These fiducial markers
ARTICLE IN PRESS K. Li et al. / Journal of Biomechanics 42 (2009) 2624–2626 were precision Nylon spheres (radius 9.525 mm), rigidly affixed through plastic screws to the femur and tibia, three on each bone. They were placed close to the joint line without possibly disrupting the soft tissues of interest and were separated as widely as possible in non-collinear positions (Soderkvist and Wedin, 1993). High-resolution CT scans of the knee were then taken with slice spacing of 0.625 mm. After completion of the imaging, careful dissection of the specimens was performed by an experienced arthroscopic surgeon. A MicroScribe Digitizer (Immersion Corp., CA) with a reported accuracy of 70.2 mm was used to digitize in 3D the surfaces of the spherical markers and the peripheries of the insertion sites for the following soft tissue structures: anterior cruciate ligament (ACL) consisting of both anteromedial (AM) and posterolateral (PL) bundles, posterior cruciate ligament (PCL) consisting of both anterolateral (AL) and posteromedial (PM) bundles, medial collateral ligament (MCL), lateral collateral ligament (LCL), and popliteus tendon. The marker surfaces were digitized such that ‘point-clouds’ of the spherical surfaces were formed. The Mimics software (Materialise, Belgium) was used to segment the bones and spherical markers separately from the CT scans. As a result, 3D surface models of the femur, tibia, and spherical markers were generated. The centroids of spherical markers were obtained by fitting the marker surface data, digitization- or CT-acquired, by a least-squares geometric fitting algorithm. Given m surface points (m ranged about 600–2500 for the digitization data and 1200–15,000 for the CT data), the algorithm determined the four parameters of a sphere, the centroid coordinates (xc, yc, zc) and radius R of the sphere, by minimizing the following residual: vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u 2 ! m qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u1 X : ðxi xc Þ2 þ ðyi yc Þ2 þ ðzi zc Þ2 R Res ¼ t m i¼1
Spherical markers
PCL(AL,PM)
ð2Þ
where (xi, yi, zi) is a digitized insertion site point and (x*i, y*i, z*i) is the nearest point on the corresponding bone surface and n is the total number of the insertion points on each bone (n ranged 117–150). Note that the RMSD, mean distance (MD), and standard deviation of the distances (sD) have the following relation: RMSD2 ¼ MD2+sD2.
3. Results The proposed method was able to map the ligament insertion sites onto the bone surfaces (Fig. 1). The centroids of the spherical fiducial markers were successfully identified using the least-squares fitting algorithm. The average radii of the sphere markers in both modalities were within 72% of the nominal radius (Table 1). The residual errors were all less than 0.25 mm. The differences in inter-marker distances between the two modalities were less than 2%, confirming a small fiducial registration error (FRE)—the difference between homologous fiducial points after registration (Fitzpatrick et al., 1998). The optimization procedure reduced the TRE as measured by RMS distances from the insertion sites to the bone surface from 0.82 (70.21) mm to 0.59 (70.10) mm for the femur, from 0.78 (70.12) mm to 0.63 (70.10) for the tibia. The grand mean (7SD) distances from the insertion sites to the bone surface were reduced from 0.74(70.40) mm to 0.53 (70.29) mm for the femur,
ACL(PL,AM)
LCL Popliteus
MCL
LCL
ACL(PL,AM)
Spherical markers
ð1Þ
The centroids of three markers on each bone established a local coordinate system. Thus, two local coordinate systems were affixed to each bone, one based on digitization data and one on CT data. However, directly applying the transformation matrix between the two coordinate systems may not guarantee the proper positioning of the transformed insertion sites relative to the bone surface in the CT-based coordinate system (e.g., an insertion site could partially or completely lie beneath the bone surface). This is due to the inevitable target registration error (TRE), defined as the distance between the homologous points in both modalities (Fitzpatrick et al., 1998). An optimization correction procedure was formulated to alleviate the effect of TRE on the alignment of coordinate systems. The procedure varied the position of the most distorted marker (whose distance to the other two markers changed most from one modality to the other) within a range of 71 mm in all three dimensions in the digitization-based coordinate system, and through enumeration identified the position resulting in a global minimum root-mean-square distance (RMSD) from a digitized insertion site point to the bone surface. Mathematically, the following objective function was minimized: vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u n u1 X ½ðx xi Þ2 þ ðyi yi Þ2 þ ðzi zi Þ2 RMSD ¼ t n i¼1 i
2625
Fig. 1. The CT-based femur and tibia surface models with six spherical fiducial markers and insertion sites for five ligaments.
Table 1 Statistical summary of the radii of fitted spheres and residual fitting errors. CT
Digitization
Radii (mm) Femur Tibia
9.565 (70.035) 9.544 (70.065)
9.673 (70.117) 9.650 (70.145)
Fitting error (mm) Femur Tibia
0.163 (70.042) 0.139 (70.023)
0.168 (70.036) 0.164 (70.046)
The mean (7SD) values are averaged across all fiducial markers (3 markers per specimen 4 specimens ¼ 12 markers) on each bone within each modality.
Before correction
After correction LCL Popliteus
Popliteus LCL
Fig. 2. Insertion site mapping before and after the optimization-based correction. Note that before the correction, the lateral collateral ligament (LCL) insertion sites on the femur and the tibia were inside the bones.
from 0.69 (70.45) mm to 0.55 (70.37) mm for the tibia. Without the correction, the misalignment (‘‘tilting’’ and ‘‘translating’’) of the coordinate systems caused some insertion footprints to lie completely or partially underneath the bone surfaces and some others to be unrealistically farther away from the surfaces. With the correction, all digitized insertion footprints became visible (Fig. 2).
ARTICLE IN PRESS 2626
K. Li et al. / Journal of Biomechanics 42 (2009) 2624–2626
4. Discussion The technique established in this study will enable systematic investigation of the morphology of soft tissues in relation to 3D bone geometry in the knee, and lead to clinical tool development facilitating more precise, more anatomical, and better-navigated surgeries. A sufficiently accurate yet practical enabling technique has been lacking. The co-registration between an image modality and a physical space based on rigid-body points has been widely used in medical applications, but mostly for intra-operative or radiotherapy applications (cf. Maintz and Viergever, 1998), which required a sophisticated apparatus and a large number of fiducal markers. Practicality concerns impeded the use of these existing techniques for systematic studies of musculoskeletal anatomy. On the other hand, once such concerns are reduced or removed—for instance, as the feasibility to image a patient with fiducial markers on is improved—our technique can be combined with a surgical navigation system to acquire intraoperative data. The accuracy requirement presented another challenge. A few biomechanical studies employed ‘‘inter-modality registration’’ of CT and digitization data (Fischer et al., 2001; Gray et al., 2008). In fact, our exploration of a new method was preceded by testing the method by Fischer et al. (2001), which employed cubic registration blocks. We discovered that the reconstruction of a cubic block from CT, unlike a spherical fiducial marker, was sensitive to its orientation in relation to that of the CT slices; the accuracy or uncertainty, as indicated by the differences in the angles between adjacent planes of the registration block in both modalities, in determining the bone positions might be acceptable for estimating the gross knee joint kinematics, but not for mapping the ligament insertion sites. The screw-based co-registration used by Kaptein and van der Helm (2004) in mapping muscle attachment contours to the bone is not as accurate as the sphericalmarker-based co-registration; the same TRE measures, the attachment-to-bone distances, ranged mostly from 1 to 10 mm. The surfacebased method by Gray et al. (2008) relied on an iterative closest point (ICP) algorithm, and would require meticulous removal of all soft tissues to expose clean bone surfaces to be digitized. Further, the accuracy of ICP-based registration approach would be inferior (Maurer et al., 1998; Lee et al., 2008), while the point-based approach using spherical fiducial markers is considered a ‘‘gold standard’’ (West et al., 1997). Several sources of the mapping error have been effectively reduced by the two optimization procedures. The fiducial location error (FLE) (Fitzpatrick et al., 1998), defined in this study as the distance between the calculated centroid to the actual centroid (unknown), was contained by the sphere fitting procedure as evidenced by the small error in the radius of the fitted sphere. The discrepancy of the intermarker distances indicated the presence of the FRE. Yet, the FRE did not seem to affect the overall rigidity of the fiducial point configuration, which allowed another optimization procedure to alleviate the TRE by perturbing one of the fiducial points. Although in theory using more fiducial markers in the rigid-body point-based registration can decrease the TRE, in practice it may not be feasible to accommodate more markers with radius greater than a certain size.
Conflict of interest statement None.
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