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Journal of Biomechanics 38 (2005) 241–253
In vivo determination of normal and anterior cruciate ligament-deficient knee kinematics Douglas A. Dennisa,b, Mohamed R. Mahfouza,b,*, Richard D. Komisteka,b, William Hoffc a
University of Tennessee, 313 Perkins Hall, Knoxville, TN 37996, USA b Oak Ridge National Laboratory, Oak Ridge, TN 37831, USA c Colorado School of Mines, Division of Engineering, Golden, CO 80401, USA
Abstract The objective of the current study was to use fluoroscopy to accurately determine the three-dimensional (3D), in vivo, weightbearing kinematics of 10 normal and five anterior cruciate ligament deficient (ACLD) knees. Patient-specific bone models were derived from computed tomography (CT) data. 3D computer bone models of each subject’s femur, tibia, and fibula were recreated from the CT 3D bone density data. Using a model-based 3D-to-2D imaging technique registered CT images were precisely fit onto fluoroscopic images, the full six degrees of freedom motion of the bones was measured from the images. The computer-generated 3D models of each subject’s femur and tibia were precisely registered to the 2D digital fluoroscopic images using an optimization algorithm that automatically adjusts the pose of the model at various flexion/extension angles. Each subject performed a weightbearing deep knee bend while under dynamic fluoroscopic surveillance. All 10 normal knees experienced posterior femoral translation of the lateral condyle and minimal change in position of the medial condyle with progressive knee flexion. The average amount of posterior femoral translation of the lateral condyle was 21.07 mm, whereas the average medial condyle translation was 1.94 mm, in the posterior direction. In contrast, all five ACLD knees experienced considerable change in the position of the medial condyle. The average amount of posterior femoral translation of the lateral condyle was 17.00 mm, while the medial condyle translation was 4.65 mm, in the posterior direction. In addition, the helical axis of motion was determined between maximum flexion and extension. A considerable difference was found between the center of rotation locations of the normal and ACLD subjects, with ACLD subjects exhibiting substantially higher variance in kinematic patterns. r 2004 Elsevier Ltd. All rights reserved. Keywords: Kinematics; Helical axis; Registration; Anterior cruciate ligament deficient; Fluoroscopy; Normal knee
1. Introduction The ‘‘in vivo’’ measurement of dynamic knee kinematics is important for understanding the effects of joint injuries, diseases, and evaluating the outcome of surgical procedures. Researchers have used ‘‘in vitro’’ (cadavers), (Fukubayashi et al., 1982; Goldberg and Henderson, 1980; Hsieh and Walker, 1976; Mahoney et al., 1994; Markolf et al., 1976, 1979; Muller, 1983; O’Connor et al., 1990; Singerman et al., 1994) noninvasive (gait laboratories), (Andriacchi et al., 1986, 1994; Andriacchi, 1993; Draganich et al., 1987; Lafortune et al., 1992; Murphy et al., 1995; Wilson et al., 1996) and in vivo (roentgen stereophotogrammetry and fluoroscopy) *Corresponding author. Department of Biomedical Engineering, University of Tennessee, 313 Perkins Hall, Knoxville, TN 37996, USA. Tel.: +1-865-974-4159; fax: +1-865-974-7663. E-mail address:
[email protected] (M.R. Mahfouz). 0021-9290/$ - see front matter r 2004 Elsevier Ltd. All rights reserved. doi:10.1016/j.jbiomech.2004.02.042
(Chao, 1980; Dennis et al., 1998a,b. Kharrholm et al., 1994; Nilsson et al., 1991; Sarojak, 1998; Stiehl et al., 1995, 1997) approaches to assess human knee motion. Cadaveric and static X-ray measurement methods often do not accurately reflect loads encountered during typical movements, and often fail to reliably predict outcome. Therefore, treatments aimed at improving knee function should be evaluated using data obtained from dynamic measurement methods. This requires the determination of six degrees of freedom (DOF) pose (position and orientation) of objects to be measured during dynamic activities. The most commonly used methods for assessing dynamic movement rely upon skin-mounted markers or bone-implemented markers. However, external skinmarkers are often unable to accurately represent motion of the underlying bone due to movement of soft tissue relative to bone. To overcome the inherent inaccuracy of the skin-mounted markers, markers have been mounted
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on skeletal pins inserted into the underlying bones (Murphy, 1990), or inserted directly into the bones (Lafortune et al., 1992) to measure skeletal kinematics. Though these studies provide some of the best available quantitative data during movement, the requirement of skeletal pins or radio-opaque markers has limited application for human studies. The objective of the current study was to accurately determine the threedimensional (3D) kinematic patterns of normal and anterior cruciate ligament (ACL)-deficient (ACLD) knees, under in vivo weight-bearing activities using a novel intensity-based 3D-to-2D image registration method, similar to that previously utilized to analyze kinematics of total knee arthroplasty (TKA) (Banks, 1992; Banks and Hodge, 1996; Dennis et al., 1996,1998a,b; Hoff et al., 1998; Mahfouz, 2002; Mahfouz et al., 2003; Sarojak 1998; Stiehl et al., 1995; Zuffi et al., 1999.).
Fig. 1. Volume rendered images (top) showing both knees of a normal subject. The bottom images show the segmented bone models superimposed on the volume data.
2. Materials and methods
tibia and the proximal fibula (combined approximately 18 000 polygons) (Fig. 1).
2.1. Model creation
2.2. X-ray fluoroscopy
The computer software packages used in the current study were developed by the current authors with the aid of software development languages (Open Inventor C++ library, and C++ Qt GUI development library; TGS, San Diego, CA). Ten healthy normal volunteers with an average age of 37 years (range, 22–44 years), and average body mass of 76 kg participated in the study. The volunteers were scanned with MRI (Fast Spin Echo T2 FSE) and exhibited no lower extremity pathology or had any measurable ligamentous instability on clinical examination (pivot shift and Lachman exams). In addition, five patients with recently isolated ACL tear (4–6 weeks) with an average age 39 years (range, 25–47), and average body mass of 65 kg were also included in the study to compare their kinematics to the normal subjects. The five ACLD patients performed a KT-1000 test for laxity measurement and the manual max score was limited to 3 mm or less. Spiral computed tomography (CT) scans of the subjects’ (normal and ACLD) knees were made at levels ranging from 120 mm proximal to the joint to 120 mm distal to the joint. These scans were made at 1–2 mm intervals and the volumetric data of the knee joint was constructed at 0.5 mm interpolation in the transverse plane. Segmentation of the CT-scanned bone was automatically performed by applying a thresholding filter to the slices which isolated the bone from soft tissues. Manual intervention was conducted only when the thresholding filter failed. On completion of the segmentation process, the resulting data were used to create full 3D polygonal surface models for the distal femur (approximately 12 000 polygons), the proximal
The subjects were analyzed using a high-frequency pulsated video fluoroscopy unit (Radiographic and Data Solutions, Minneapolis, MN). All subjects gave informed consent to participate in this study. The study has been approved by an institutional research review board (IRRB #0607). Each subject subsequently performed weight-bearing deep knee bend activity while under fluoroscopic surveillance. During the deep knee bend activity, subjects were asked to begin in full extension and flex the knee of interest to maximum flexion (Fig. 2). The fluoroscope maximum frame rate is 30 frames/s. This frequency puts a limitation on the maximum frequency content of the kinematics data to 15 Hz to avoid aliasing (Nyquist criteria (Jain, 1989). In practice, all the activities that were analyzed with video fluoroscopy in the authors’ research have much lower bandwidth. The fluoroscopic images of the deep knee bending activity were downloaded to a workstation for processing. The fluoroscope is modeled as a perspective projection image formation model. The perspective projection model treats the fluoroscope sensor as consisting of an X-ray point source and a planar phosphor screen upon which the image is formed (Fig. 3). Although image distortion and non-uniform scaling can occur, these can be compensated for by careful calibration (Mahfouz et al., 2003). 2.3. 3D-to-2D registration Registering 3D models using 2D fluoroscopy images has been the subject of much research. Generally,
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Fig. 2. Single plane fluoroscopy allows the patient-free motion during deep knee bend activity in the plane between the X-ray source and the image intensifier.
Fig. 3. The imaging model for the fluoroscope is perspective projection. X-rays are emitted from a point source, pass through the object, and strike the image plane.
3D-to-2D registration methods fall into two different categories: feature-based methods and intensity-based methods (Penney et al., 1998). Feature-based methods rely on matching features of the model to observed features in the image. In X-ray images, this is usually done by matching points on the surface of the model to the observed silhouette in the image (Banks and Hodge, 1996; Hoff et al., 1998; Lavallee and Szeliski, 1995; Gueziec et al., 1998). Accuracy can be improved by using bi-plane (i.e., stereo) fluoroscopy (You et al., 2001) instead of single-plane (mono) imaging, but it would unacceptably constrain the motion of the patient. Intensity-based methods match the image values directly to a predicted image of the object. A predicted image is generated of the object in a hypothesized pose, and the pixel values are compared directly to the values in the actual input image, without trying to pre-segment the object from the image. A variety of image difference measures can be used, such as pattern intensity (Weese et al., 1997), gradient difference (Penney et al., 1998) and cross-correlation (Lemieux et al., 1994). Since the
measures are global in nature, they are robust to small amounts of clutter and occlusions. Whatever similarity measure is used, either in featureor intensity-based methods, it is still necessary to search over the space of possible poses for the pose with the best value of the similarity measure. This can be viewed as finding the minimum of a 6D function (three rotational, three translational DOF). A potential difficulty is that the function may contain local minima, or false solutions, which represent ambiguities in the pose of the model. For example, Fig. 4 shows a plot of the objective function for the registration of a femoral implant model to an image. The similarity measure was a type of cross-correlation (Mahfouz, 2002, Mahfouz et al., 2003). One of the two deep minima corresponds to the correct solution; the other corresponds to a false solution that is a rotated version of the correct solution. The silhouettes are very similar. There are a number of strategies for dealing with the problem of local minima. Pre-processing of the image to segment the object from the background can reduce the number of local minima (Banks and Hodge, 1996;
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Fig. 4. Objective function values as the femoral implant model is rotated about two axes. The two poses shown correspond to the two large minima. Note the many smaller local minima.
Lavallee and Szeliski, 1995; Gueziec et al., 1998; Zuffi et al., 1999). However, it may be difficult to know in advance which image points belong to the object instead of the background (Mahfouz et al., 2004). Convergence to the correct solution is more likely if one has a good initial guess (Mahfouz et al., 2003; Lavallee et al., 1996). The investigators have developed a 3D-to-2D registration intensity-based method that has been applied to knees, hips, shoulders, and other joints (Hoff et al., 1998; Mahfouz et al., 2001, 2003). The method is applicable to implant components as well as normal bones. It registers a surface model of the object to a single-plane fluoroscopy image, using a direct image-toimage similarity measure. Prior segmentation of the fluoroscopic image is not performed due to the inherent error created (Mahfouz et al., 2004). The disadvantage of this method is that it can result in numerous local minima that make it difficult to find the correct solution. This problem is avoided through utilization of a robust optimization algorithm (simulated annealing) that can escape local minima and find the global minimum (true solution) (Mahfouz et al., 2003). Accuracy tests were performed on cadaver images that were very similar to in vivo clinical X-ray fluoroscopy images, to allow a fair assessment of the algorithm. A completely independent method using an optical sensor was used for determining the ground truth, unlike other work that uses ground truth derived from X-ray data (Banks and Hodge, 1996; Lavallee and Szeliski, 1995; Penney et al., 1998; Weese et al., 1997). The registration method is highly accurate for measuring relative pose. The average errors in X, Y, and Z translations were 0.023, 0.086, and 1.054 mm respectively (standard deviations are 0.473, 0.449, and 3.031 mm). Likewise, the average errors in x, y, and z rotations were 0.068, 0.001, and 0.253 degrees (standard deviations are 0.942, 0.771, and 0.841 degrees). These numbers represent the errors in our process plus the errors in the independent measurement
system as well (i.e. the upper bound) (Mahfouz et al., 2003). If we view the knee joint in the sagittal (XY) plane, then the relative translational motion in the Z direction should be small and is not of interest in our application.
2.4. Analysis of knee kinematics On the workstation screen, perspective projection images of the model of the femur and the tibia (with the fibula) were processed according to their positions relative to the X-ray source and to the image intensifier. When both models of the femur and tibia are properly registered (overlaid), the relative pose between the two models should be the same as it was between the tibiofemoral joint at the time the X-ray image was created. Registering the models in the selected frames will result in calculating the relative pose between the two bones over the entire range of activity. The pose of each bone model is represented by a 4 4 homogeneous transformation matrix H that is comprised of rotation matrix and translation vector (see the Appendix). The rotation matrix Model fluoro Rxyz ðg; b; aÞ is given by (Craig, 1990) Model fluoro Rxyz ðg; b; aÞ
¼ Rz ðaÞRy ðbÞRx ðgÞ 2 3 CosðaÞ SinðaÞ 0 6 7 ¼ 4 SinðaÞ CosðaÞ 0 5 0 0 1 2 3 CosðbÞ 0 SinðbÞ 6 7 4 0 1 0 5 SinðbÞ 0 CosðbÞ 2 3 1 0 0 6 7 4 0 CosðgÞ SinðgÞ 5; 0
SinðgÞ
CosðgÞ
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2 Model fluoro Rxyz ðg; b; aÞ
r11
6 ¼ 4 r21 r31
r12 r22 r32
r13
3
7 r23 5; r33
g ¼ A tan 2ðr32 =CosðbÞ; r33 =CosðbÞÞ; qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi b ¼ A tan 2ðr31 ; r211 þ r221 Þ; a ¼ A tan 2ðr21 =CosðbÞ; r11 =CosðbÞÞ; where g, b, and a are the model’s angles of rotations about x, y, and z axes, respectively, and the translation vector is Model pfluoro ¼ Model xfluoro ; Model yfluoro ; Model zfluoro ÞT : Therefore, the relative pose of the femur with respect to the tibia is then calculated Fluoro Tib using the equation Tib Fem H ¼Fluoro HFem H: Two methods were utilized for analyzing the relative motion of the femur and tibia. The first was the method of screw axis decomposition, or the helical axis of motion (HAM). In this method, the axis in space about which the moving body rotates is determined. The
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location and orientation of the axis is defined with respect to the coordinate system of the tibia. If the knee was a simple hinge joint, with pure rotation about the medial axis, then the screw axis would be a stationary line perpendicular to the sagittal plane. This is not the case due to the complexity of knee motion which includes translation as well as rotation about other axes, resulting in the screw axis being not exactly perpendicular to the sagittal plane, and not fixed in space (Fig. 5). The screw axis was calculated between subsequent recordings of motion frames (i.e. between frames 1–2, 2–3, 3–4 etc.). The second method utilized to analyze the relative motion of the knee was to track the contact paths of the femur on the tibia. The minimum point on the surface of the medial and lateral condyles for each respective flexion angle was calculated automatically and projected down (vertical) onto the tibial plateau (Fig. 6). The intersection of these projected lines from the femoral condyles with the tibial plateau surface (in every flexion angle) are considered the contact paths of the femur on
Fig. 5. Helical axes of motion corresponding to nine increments of knee flexion from 0 to 120 , shown in the sagittal (A) and frontal (B) planes. The geometrical center of the femur rotating around the screw axis is demonstrated in the sagittal (C) and frontal (D) planes.
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Fig. 6. (A) A frontal view of the knee joint showing the minimum points on the femoral condyles projected on the tibial plateau. (B) A sagittal view of the knee joint showing three flexion angles and their corresponding lowest points projected on the tibial plateau (C) Anteroposterior view of the knee joint showing the three lowest points on the femur projected on the frontal plane. (D) Axial view of the minimum points projected on the tibial plateau.
the tibia because the articular surfaces and meniscus in CT are difficult to segment or model. These contact points are also located in the coordinate space of the tibia. This method is more standardized, and hence more consistent than other methods that use the centers of the posterior femoral condyles (Asano et al., 2001; Iwaki et al., 2000) because the process for obtaining these minimum points in our approach is automated and reproducible, thus eliminating human errors. On the other hand, the other methods require averaging of repeated measurements conducted by human operators with a period of time between each measurement which can introduce errors to the process. All anterior–posterior measurements are made with respect to a plane (frontal) that is located at the geometric center of the tibia the geometric center is calculated automatically. If the AP contact position of the condyle is anterior than this plane, the AP position is designated as positive. If the contact position is posterior than this plane, the AP position is designated as a negative value. For each selected frame of the X-ray video sequence, femorotibial contact paths were determined for the medial and lateral condyles and plotted with respect to knee flexion angle. The pivot position was determined by analysis of medial and lateral condylar contact positions at full
extension and 90 flexion. Initially, the contact positions of the medial and lateral condyles were determined. A line was constructed between these two points at full extension and at 90 knee flexion. An angle (y) then was determined between these two lines. If the full extension and 90 flexion lines converged on the medial half of the tibial insert in the coronal plane (apex of angle y, it was denoted that a medial pivot location occurred. If the lines converged on the lateral half of the tibial insert, it was denoted that a lateral pivot location had occurred. If these lines remained parallel to each other and the angle appeared to approach infinity, no convergence was found and it was denoted that there was no pivot location (Dennis et al., 2003).
3. Results 3.1. Normal knee While performing a deep knee bend, all 10 normal knees experienced posterior femoral translation of their lateral condyle and minimal change in the position of the medial condyle (Fig. 7, Table 1). The average amount of posterior femoral translation of the lateral condyle from 0 to 120 flexion was 21.07 mm (standard
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deviation 9.30), whereas the average medial condyle translation was 1.94 mm (standard deviation 1.86), in the posterior direction. The majority of the posterior femoral translation of the lateral condyle seemed to occur in the first 75 knee flexion (Fig. 8). Posterior femoral translation was not always continuous with
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Fig. 7. The average normal subjects medial and lateral condyle contact positions are plotted during a deep knee bend activity.
increasing knee flexion because small amounts of paradoxical anterior translation of femorotibial contact were observed in mid-to-deep flexion which was more common medially (90%) than laterally (30%). All 10 knees experienced a normal axial rotation pattern during a deep knee bend (tibia internally rotating with increased knee flexion, i.e., screw home mechanism), because the posterior translation of femorotibial contact laterally was greater than that observed medially (Fig. 7, Table 2). The average amount of axial rotation for the 10 subjects from 0 to 120 flexion was 23.67 (standard deviation, 9.56) in the normal direction. The majority of this rotation occurred in the first 30 knee flexion (average, 9.35 , Fig. 9). Rotational movements are best represented by describing HAM for the motion. This axis is an imaginary line in space, around which the femur rotates. Because of the out of plane motion of the knee (six DOF), the axis is almost never perpendicular to the sagittal plane The helical axis routinely translated posteriorly during progressive knee flexion (Fig. 10). Therefore, posterior femoral translation observed was due to a combination of pure AP linear motion and rotation of the femur relative to the tibia.
Table 1 Average anteroposterior femorotibial translation during deep knee bend activity
Table 2 Average axial femorotibial rotation during deep knee bend activity
Lateral condyle (mm)
Axial rotation (deg)
Average
STD
Average
STD
Average
STD
1.94 4.65
1.86 3.99
21.07 17.00
9.30 7.31
23.67 9.56
6.09 3.23
Normal ACLD
Lateral and Medial Anterior Posterior Translation For Normal Patients 15.00
Average LAP Average MAP
10.00
5.00 Translation in mm
Normal ACLD
Medial condyle (mm)
0.00
-5.00
-10.00
-15.00
-20.00 0
20
40
60 80 100 Flexion Angles in Degrees
120
140
Fig. 8. Anteroposterior translation of the contact points between the femur and the tibia during knee flexion.
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3.2. ACLD knee
knees, the helical axis translated posteriorly during increasing knee flexion (Fig. 14).
While performing a deep knee bend, all five subjects with an ACLD knee experienced posterior femoral translation of their lateral condyle with increased translation of the medial condyle when compared to normal knees (Fig. 11, Table 1). The average amount of posterior femoral translation of the lateral condyle from 0 to 120 of knee flexion was 17.00 mm (standard deviation, 7.31), while the medial condyle translation was 4.65 mm (standard deviation, 3.99), in the posterior direction. The majority of the posterior femoral translation of the lateral condyle occurred in the first 30 of knee flexion (Fig. 12). Similar to the normal knee, posterior femoral translation was not always continuous with increasing knee flexion as small amounts of paradoxical anterior translation of femorotibial contact were observed in mid-to-deep flexion which was more common laterally (80%) than medially (20%); opposite to the normal knee observation. All subjects experienced an axial rotation pattern similar to that of the normal knee subjects only during the first 30 (Fig. 13). After 30 the axial rotation pattern of the ACLD knees exhibited different behavior than normal knees (changing slope from positive to negative in Fig. 13). The average amount of axial rotation for the five subjects from 0 to 120 of flexion was 9.87 (standard deviation=3.23). Similar to normal
4. Discussion The current study determined that accurate 3D motion of normal and ACLD knees, under in vivo, weight-bearing conditions, can be determined using video fluoroscopy and a 3D-to-2D image registration process. The results differed from previous kinematic analyses of total knee arthroplasty (Dennis et al., 1996, 1998a,b) which typically demonstrated reduced magnitudes of both posterior femoral translation and axial rotation and an increased incidence and magnitude of paradoxical anterior femoral translation when compared to normal knees. Normal and ACLD knee subjects demonstrated similar pattern of posterior femoral translation during progressive knee flexion but they exhibited different axial rotation pattern after 30 of knee flexion (Fig. 15). Reduced mean magnitudes of posterior femoral translation laterally (normal, 21.07 mm; ACLD, 17.0 mm) and axial rotation (normal, 23.67.4 ; and ACLD 9.9 ) were
Axial Rotation Angle (Degrees)
Average Tibio-Femoral Axial Rotation 15.00 10.00 5.00 0.00 -5.00 -10.00 0
15
30
45
60
75
90
105
120
Flexion Angle (Degrees)
Fig. 9. Axial rotation of the femur during knee flexion for normal subjects.
Fig. 11. Axial (top) view of the minimum points projected on the tibial plateau of the five ACLD knees. Clear medial pivoting motion of the femur relative to the tibia is seen in all five knees.
Fig. 10. Helical axis of the normal knees showing the axis does not stay fixed in space during an entire movement through the available range of motion. Each column represents different view, each row is a different subject.
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Lateral and Medial Anterior Posterior Translation For ACLD Patients 15 .00
Average LAP Average MAP
10 .00
Flexion in Degrees
5. 00
0.00
-5. 00
-10.00
-15.00
-20.00 0
20
40
60
80
100
120
140
Translation in mm
Fig. 12. The average ACLD medial and lateral condyle contact positions are plotted during a deep knee bend activity.
Average Tibio-Femoral Axial Rotation For ACLD 10.00
Axial Rotation Angle (Degrees)
5.00
0.00
-5.00
-10.00
-15.00 0
15
30
45
60
75
90
105
120
Flexion Angle (Degrees)
Fig. 13. Axial rotation of the femur during knee flexion.
Fig. 14. Helical axis of the ACLD knees showing that axis does not stay fixed in space during progressive knee flexion.
observed in ACLD knees. Additionally, increased variability in knee kinematic patterns were observed in ACLD knees as compared with normal knees. Fig. 16 shows the point of intersection of the helical axis with the sagittal plane passing through the center of the tibia,
for each of the patients in both normal and ACLD groups. As can be seen, the dispersion of the intersection points is greater for the ACLD group than the normal group. Reduced magnitudes of both anteroposterior femorotibial translation (Brandsson et al., 2001) and
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Axial Rotation Normal Vs. ACLD 15.00
10.00
Translation in mm
5.00
0.00 Axial Rotation Normal Axial Rotation ACLD -5.00
-10.00
-15.00 0
20
40
60
80
100
120
140
Roration in Degrees
Fig. 15. Normal vs. ACLD axial rotation of the femur relative to the tibia during knee flexion.
Fig. 16. The intersection of the helical axes of motion with the sagittal plane, passing through the center of the tibia. The ellipsoids represent the covariance of each group of points, at a one-sigma level of significance.
axial rotation (Brandsson et al., 2001; Jonsson et al.,1989; Karrholm et al., 1988) after ACL disruption have similarly been observed in other reports of knee kinematics of ACLD knees. These studies, however, have typically reported lesser amounts of both anteroposterior translation and axial rotation than in
the present study. This may be related to the fact that most of these studies evaluated the knee in only a limited flexion range (i.e. 0–60 ) and/or tested under nonweight-bearing conditions. Posterior femoral translation was substantially greater laterally than medially in both normal and ACLD subjects, creating a medial pivot type of axial rotation pattern in which the tibia internally rotates relative to the femur as flexion progresses and externally rotates as the knee extends. Typically, as the knee approaches deep flexion, the medial femoral condyle translates a small amount anteriorly (o5 mm) while the lateral femoral condyle continues to translate posteriorly. However, in previous study of the authors’ (Komistek et al., 2003) one individual with a ‘‘normal’’ knee demonstrated substantial medial translation and exhibited a lateral pivot kinematic pattern (i.e., reverse screw home axial rotation pattern) in chair sit activity. Anteroposterior translation was not always unidirectional as flexion or extension progressed, particularly at the medial femorotibial articulation. For example, some individual subjects had initial posterior femoral rollback, then demonstrated paradoxical anterior translation of femorotibial contact, followed by a return to posterior femoral rollback in deep flexion. Therefore, a forward and back translational pattern occasionally was seen as flexion progressed. Both the incidence and magnitude of paradoxical anterior femoral translation observed progressive knee flexion was greater in ACLD knees (especially on the lateral articulation >5 mm, 80%).
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The paradoxical anterior femoral translation observed in the present investigation, particularly in ACLD knee subjects, may have undesirable consequences. This phenomenon results in more anterior axis of flexion and can reduce maximum knee flexion (Dennis et al, 1998a,b). Second, the quadriceps moment arm is decreased, resulting in reduced quadriceps efficiency. Lastly, articular cartilage shear forces will likely be increased, enhancing the risk of premature degenerative change, which is commonly observed in those with chronic ACL injuries (Beynnon et al., 2002). With knee flexion, the normal tibia typically internally rotates relative to the femur and conversely, externally rotates with knee extension (i.e., screw home mechanism) (Nordin and Frankel, 1980; Rosenberg et al., 1994; Shaw and Murray, 1974; Smillie, 1962; Van Dommelen and Fowler, 1989). Hypotheses advanced to explain the screw home mechanism include the length and tension of the cruciate and collateral ligaments (Brantigan and Voshell, 1941; Haines, 1941; Lewin, 1952) as well as differences in dimensions of both the medial and lateral femoral and tibial condyles (Nordin and Frankel, 1980; Roesnberg et al., 1994). The reduced amount of axial rotation and increased incidence of a reverse screw home axial rotation pattern observed in ACLD knee subjects may be detrimental, enhancing the risk of patellofemoral instability. Individual variances in the pattern and magnitudes of AP translation and axial rotation were observed among subjects in both groups. This likely is attributable, at least in part, to variances in ligamentous tension, muscle contraction patterns, anatomic condylar geometry variations and effort intensity among differing subjects. An advantage of the present experimental model is it allows analysis under in vivo, weight-bearing conditions throughout the entire range of knee flexion. The 3D kinematic process reduces potential error caused by motion between external skin markers and the underlying bone (Andriacchi et al., 1986). The importance of weight-bearing in kinematic evaluation of the knee is supported by the work of Hsieh and Walker (1976), who determined that in an unloaded knee joint, laxity is primarily determined by soft tissue constraints, whereas in the loaded knee joint, the geometric conformity of the joint surfaces plays the major role in controlling joint laxity. Additionally, abnormal anteroposterior femorotibial translation patterns following ACL disruption have been shown to be accentuated when tested under weight-bearing conditions (Beynnon et al., 2002; Brandsson et al., 2001; Kannus and Jarvinen, 1989; Roos et al., 1997). Methods which allow assessment during dynamic muscle contraction provide a superior estimation of true knee kinematics. Markolf et al. (1978), in an in vivo study of knee stability of 49 healthy subjects, found laxity measurements were
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reduced by as much as 50% with the addition of muscle contraction. In summary, the present analysis has demonstrated progressive posterior femoral translation and internal tibial rotation occurs in both normal and ACLD subjects during deep knee flexion. Reduced magnitudes of anteroposterior translation and axial rotation as well as increased variability in kinematic patterns was observed in ACLD subjects. Future studies will include evaluation of a larger number of subjects and reanalysis of subjects at sequential intervals to assess changes in kinematic patterns over time, particularly in those subjects with ACL injuries. In the present study, all ACLD subjects were evaluated relatively soon (o6 months) following their ACL injury. We suspect differences in kinematic patterns between normal and ACLD subjects may increase in chronic ACLD subjects due to attenuation of secondary soft tissue stabilizing structures over time.
Acknowledgements This work was supported by National Science Foundation, Grant #9729255, Radiographic and Data Solutions, Minneapolis, MN, USA.
Appendix. Background on pose estimation The pose of a rigid body {A} with respect to another coordinate system {B} can by a six be represented
T B element vector BA x ¼ B xAorg ; B yAorg ; z ; a; b; g ; Aorg
T where B pAorg ¼ B xAorg ; B yAorg ; B zAorg is the origin of frame {A} in frame {B}, and (a,b,g) are the angles of rotation of {A} about the (z, y, x) axes of {B}. An alternative representation of orientation is to use three elements of a quaternion; the conversion between Euler angles and quaternions is straightforward (Markolf et al., 1979; Craig, 1990). Equivalently, pose can be represented by a 4 4 homogeneous transformation matrix B B pAorg AR B H ¼ ; ðA:1Þ A 0 1 where BA R is the 3 3 rotation matrix corresponding to the angles (a,b,g). In this paper, we shall use the letter x to designate a six-element pose vector and the letter H to designate the equivalent 4 4 homogeneous transformation matrix. Homogeneous transformations are a convenient and elegant representation. TGiven a homogeneous point A p ¼ A xP ; A yP ; A zP ; 1 ; represented in coordinate system {A}, it may be transformed to coordinate system {B} with a simple matrix multiplication B p ¼BA HA p: The homogeneous matrix representing the pose of frame {B}
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with respect to frame {A} is just the inverse of the pose 1 B of {A} with respect to {B}, i.e., A B H ¼A H : Finally, if we know the pose of {A} with respect to {B}, and the pose of {B} with respect to {C}, then the pose of {A} with respect to {C} is easily given by the matrix B C multiplication C A H ¼B HA H:
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