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Patella tracking calculation from patellofemoral positions at finite angles of knee flexion
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Patella tracking calculation from patellofemoral positions at finite angles of knee flexion Jie Yao a,1, Bin Yang b,1, Yuxing Wang a, Yubo Fan a,c,∗ a
Key Laboratory for Biomechanics and Mechanobiology of Ministry of Education, School of Biological Science and Medical Engineering, Beijing Advanced Innovation Centre for Biomedical Engineering, Beihang University, 37 Xueyuan Road, Haidian District, Beijing, China b Peking University International Hospital, Bejing, China c National Research Center for Rehabilitation Technical Aids, Beijing, China
a r t i c l e
i n f o
Article history: Received 24 February 2018 Revised 13 July 2018 Accepted 24 July 2018 Available online xxx Keywords: Biomechanics Patellofemoral joint Patella tracking Motion capture In vitro experiment
a b s t r a c t Patellofemoral (PF) pain is a common knee disease. Patella tracking has a significant correlation with PF pain, therefore it could be used as an index for diagnosis and treatment evaluation. Previous research has proposed a method for measuring in vivo patella tracking by means of an interpolation algorithm. The present study aimed to quantify the effect of the interpolation parameters on the accuracy of the patella tracking with a motion capture experiment. The precise patella tracking of 5 knee specimens was collected and compared with the interpolated tracking. The results showed that the total interpolation error decreased to 2 mm with the number of interpolation angles increasing to 6. The number of interpolation reference points had a slight influence on the accuracy. The findings consolidated the feasibility of using interpolation to measure the in vivo patella tracking, and can help to optimize the accuracy and efficiency of the methodology. © 2018 IPEM. Published by Elsevier Ltd. All rights reserved.
1. Introduction Patellofemoral (PF) pain syndrome is a frequently diagnosed knee disorder. The incidence of this symptom is approximately 25% in the general population [1]. Anatomical features, such as patella alta, trochlear dysplasia, lateralized tibia tuberosity, abnormal Q angle, and imbalance of PF soft tissues, are predisposing factors of the PF pain and instability [2,3]. Previous studies reported that patella maltracking is highly correlated with the PF disorder, thereby could be crucial to diagnosis and pathological classification [4–6]. Correcting the PF tracking is a critical principle for the surgery. In total knee arthroplasty, the prosthesis design could affect the patella tracking, and the unphysiological patellar kinematics may be associated with the postoperative complication [7,8]. In PF ligament reconstruction, restoring the normal patella tracking plays an important role in the surgical outcome [9]. Therefore, patella tracking assessment has the potential to boost the understanding of pathology and optimize the treatment.
∗ Corresponding author at: Key Laboratory for Biomechanics and Mechanobiology of Ministry of Education, School of Biological Science and Medical Engineering, Beijing Advanced Innovation Centre for Biomedical Engineering, Beihang University, 37 Xueyuan Road, Haidian District, Beijing, China. E-mail address: [email protected] (Y. Fan). 1 Both the authors contributed equally to this work.
Patella tracking, however, has not been widely used in clinical diagnosis, or in rehabilitation assessment. Static imaging indexes with X-ray, computer tomography (CT), and magnetic resonance (MR) scanning are currently the prime basis for the diagnosis of PF disorders. However, static indexes may fail to accurately characterize the pathology of the joint. For example, recent study reported that the tibial tubercle to trochlear groove distance, which is a common static index for a variety of PF disorders, cannot accurately predict patella maltracking in a large percentage of patients [10]. Furthermore, Since the symptoms of individual joint abnormalities are often manifested at different knee flexion angles, it can be difficult to make accurate diagnosis with images at several specific knee flexion angles. For example, patella tracking in patients of patellar dislocation significantly differed from that in healthy people at knee flexion angles of 5–30° [11]. Symptoms of early osteoarthritis are often significant in radiographs with a knee flexion of approximately 45° [12]. The patellar motion in relation to the trochlear groove of the distal femur between patients of PF pain and healthy subjects are significantly different at knee flexion angle of 90–120° [13]. Patella tracking is an indispensable parameter for accurate diagnosis of PF disorder, especially in patients with unknown knee abnormalities. Previous studies have reported several methods to evaluate patellar tracking, including dynamic CT [5], dual-orthogonal fluoroscopic imaging system [14], dynamic MR imaging [15],
https://doi.org/10.1016/j.medengphy.2018.07.018 1350-4533/© 2018 IPEM. Published by Elsevier Ltd. All rights reserved.
Please cite this article as: J. Yao et al., Patella tracking calculation from patellofemoral positions at finite angles of knee flexion, Medical Engineering and Physics (2018), https://doi.org/10.1016/j.medengphy.2018.07.018
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Fig. 1. Diagram of the knee inserted with the markers.
optoelectronic motion capture [16], etc., but these methods require special equipment, or are limited by the radiation risk or unsatisfying precision. Therefore, a measurement method of patella tracking using routine clinical equipment is still necessary. Our previous study has proposed that approximate patella tracking could be derived from routine MRI or CT images of the PF joint at several angles of knee flexion [17]. The methodology could facilitate examination of PF kinematics in clinics. However, the influence of the interpolation parameters on tracking accuracy in full range of knee motion remains unquantified. The difference between the calculated and the realistic patella tracking needs to be characterized. The objective of this study was to calculate continuous patella tracking with the PF positions at finite angles of knee flexion (PF interpolation positions). The calculated tracking was validated with realistic patella tracking measured by motion capture equipment. The effect of interpolation parameters on the accuracy of the tracking was quantified. 2. Materials and methods 2.1. Specimens preparation Five fresh frozen knee specimens (female, 3 right-side and 2 left-side, average age 70, range 55–75 years old) were studied. The specimens were without any knee pathology as examined by physical assessment and MR images. Each specimen included 300 mm of femur above knee joint line and tibia below joint line, intact patella, all associated ligaments, muscles, tendons, fasciae, fat, and skin. The specimens were thawed out at room temperature for 24 h. The proximal femur was fixed onto a rigid support by clamping. Surgical suture was wrapped and looped over the proximal end of the quadriceps tendon. A 175 N load parallel to the femur shaft was applied on the quadriceps tendon, to simulate the quadriceps force in full range of knee motion. A force was applied on the distal tibia to drive the knee flexion-extension according to the inherent track. The force was not constant, it counteracted the quadriceps force and joint resistance, and kept the knee moving quasi-statically. The patella and femur motions were recorded with the motion capture system (Northern Digital Inc., Ontario, Canada). Totally 12 markers were fixed on the patella and proximal femur by steel pins inserted into the bone (Fig. 1). The skin and fat adjacent to the steel pin was carefully incised, in order to eliminate the influence of the soft tissue on the marker position during knee motion.
Fig. 2. 3D model of the PF joint with the markers. The model was developed with the CT images.
2.2. Data acquisition The knee was moved within a range of 135° and full extension with an angle velocity of approximately 20° per second to simulate a quasi-static knee motion. Each knee was flexed-extended 6 times. The marker positions were dynamically recorded with a sampling frequency of 30 Hz. The motions of patella and femur were calculated from the spatial coordinates of the markers according to the Lagrangian multiplier theorem [18]. In brief, the motion of the bone from an initial position to a certain position could be characterized with a 3∗ 3 rotation matrix (R) and a translation vector (v). R and v could be calculated with the coordinates of the markers fixed on the bone. Let ai (x,y,z) denotes the initial positions of the i-th marker, qi (x,y,z) denotes the final positions of the marker. They satisfy the following equations:
qi = Rai + v
(1)
Where R and v were unknown, ai and qi were obtained from the motion capture system. Introducing a functional f:
f (R, v ) =
m 1 (Rai + v − qi )T (Rai + v − qi ) m i=1
×(marker amount: m = 6 )
(2)
R and v can be determined by minimizing the f under the constraint condition that R is an orthogonal matrix. The movement of the patella relative to the femur was further determined with the following equation:
Rr = R−1 R p; f
vr = R−1 v p − R−1 vf f f
(3)
Where Rr is the relative rotation matrix, Rf is the femoral rotation matrix, Rp is the patellar rotation matrix, vr is the relative translation vector, vf is the femoral translation vector, vp is the patellar translation vector. To visualize the measured tracking of the patella and femur, the knee joint with the associated markers were scanned with CT (Philips, Brilliance 16, Holland). The slice thickness of scanning was 1 mm, the pixel size was 0.78 ∗ 0.78 mm2 . The threedimension (3D) models of the patella and femur were developed with the medical image processing software, Mimics (Materialise, Inc.,Belgium). The 3D models were shown in Fig. 2, and motion of the patella relative to the femur was shown in Supplementary material 1.
Please cite this article as: J. Yao et al., Patella tracking calculation from patellofemoral positions at finite angles of knee flexion, Medical Engineering and Physics (2018), https://doi.org/10.1016/j.medengphy.2018.07.018
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Fig. 3. Reference points and their trajectories. (a) Reference points at the top, bottom, medial, and lateral sites of the patella. (b) Realistic (red) and approximate (blue) trajectories (n = 3) of the reference points.
Fig. 4. Maximum deviations between calculated and realistic patellae throughout knee extension-flexion. The solid curve indicates the average of the patella maximum deviation in 5 specimens × 6 measurements. The dash curves indicate the standard deviations.
2.3. Calculation and validation of approximate patella tracking The patellar positions at n angles of knee flexion were used to interpolate the approximate patella tracking (n ≥ 2). At each angle of knee flexion, more than three non-collinear reference points on patella could determine the patella position. In the in vivo measurement, these reference points are determined on the joint model from the MR images. To simulate the in vivo measurement
and validate its applicability, the coordinates of reference points were calculated from the motion capture markers in this study, the points at the top, bottom, medial, and lateral site of the patella were used (Fig. 3-a). The trajectory of each reference point during knee flexion-extension was conducted with order-three spline algorithm. Briefly, let function a(θ ) = (x(θ ), y(θ ), z(θ )) denotes the radius vector of the point a during knee flexion-extension, θ is the knee flexion angle. a(θ ) could be represented with pieces of
Please cite this article as: J. Yao et al., Patella tracking calculation from patellofemoral positions at finite angles of knee flexion, Medical Engineering and Physics (2018), https://doi.org/10.1016/j.medengphy.2018.07.018
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Fig. 5. Influence of the n on the average maximum deviation between approximate and realistic patellae. (a) Deviation descended to 2.0 mm with the n increasing to 6, and further descended to 0.65 mm with the n increasing to 12, during knee flexed within 0–135°. (b) Deviation was lower than 2.0 mm as the n was greater than 5, and was lower than 1.0 mm as the n was greater than 6, during knee flexed within 21–135°. (c) Deviation was lower than 1.0 mm as the n was greater than 5, during knee flexed within 27–112°.
order-three polynomial functions a(i) (θ ):
⎧ a(1 ) (θ ), θ ∈ [θ1 , θ2 ] ⎪ ⎪ ⎨ ... a(i ) (θ ), θ ∈ [θi , θi+1 ] (i = 1, 2, . . . , n) a (θ ) = ⎪ ⎪ ... ⎩ a(n−1) (θ ), θ ∈ [θn−1 , θn ]
(4)
where n is the number of knee flexion angles used for interpolating the patella tracking. a(θ i ) were already obtained from the motion capture experiments. a(θ ) could be determined by satisfying the following conditions:
a˙ (i ) (θi+1 ) = a˙ (i+1) (θi+1 ); a¨ (i ) (θi+1 ) = a¨ i+1 (θi+1 ); a¨ (1 ) (θ1 ) = a¨ (n−1) (θn ) = 0;
(5)
where a˙ i (θ ) and a¨ i (θ ) are the first-order and second-order derivatives. The realistic and approximate trajectories of the reference points were shown in Fig. 3-b.
The patella tracking was then derived from the trajectories of the four reference points according to the Lagrangian multiplier theorem [18]. At any knee flexion angle θ , the approximate position of each point was calculated with the trajectory a(θ ) mentioned above. With the coordinates of the four reference points, the rotation matrix (R) and the translation vector (v) characterizing the patella motion could be determined with the Eqs. (1 and 2). With increasing the number of interpolation angles (n), the approximate tracking will converge to the realistic tracking. To validate the accuracy of the approximate patella tracking, and investigate the relationship between the number of interpolation angles (n) and the accuracy, the approximate tracking calculated was compared with the realistic tracking from motion capture experiment. At each point of the patella, the deviation between the approximate and realistic positions during knee motion was calculated. The maximum deviation was used as an accuracy index of the approximate patella tracking.
Please cite this article as: J. Yao et al., Patella tracking calculation from patellofemoral positions at finite angles of knee flexion, Medical Engineering and Physics (2018), https://doi.org/10.1016/j.medengphy.2018.07.018
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Fig. 6. Maximum deviations between approximate and realistic patellae when using difference reference points. (a) The positions of the reference points. (b) The averages of the patella maximum deviation in 5 specimens × 6 measurements in knee extension-flexion (n = 6). Legend: The reference point number influences slightly the accuracy of the calculated patella tracking.
3. Results The calculated (blue) and realistic (red) motions of the patellae were shown in the video (Supplementary material 2). In the case of n = 2, the calculated patella moved straightly from anterior side of the femur to distal side, the calculated tracking was far away from the realistic tracking. With increasing the number of interpolation angles (n), the calculated tracking got close to the realistic one. When n is greater than 5, the calculated patella almost overlapped with the realistic one. To quantify the accuracy of the calculated tracking, the maximum deviation between calculated and realistic patellae throughout the full range of knee motion were shown in Fig. 4. At each interval between two interpolation angles, the deviations approached zero at the boundary of the interval. The deviations approached peak value at near middle of the interval. When n was greater than 2, the maximum value of the average deviations throughout knee extension-flexion always occurred at the first interval. The deviation between the calculated and realistic patellae descended with increasing the n. Theoretically, the deviation would converge to 0 with the n increasing to infinity. In the present study, the maximum deviation descended rapidly to 2 mm with the n increasing to 6, and further descended gradually to 0.65 mm with the n increasing to 12 (Fig. 5-a). The descending rates varied at different ranges of knee flexion angle. When the knee flexion angle was within 21–135°, the maximum deviation was lower than 2.0 mm as the n was greater than 5, and was lower than 1.0 mm as the n was greater than 6 (Fig. 5-b). When the knee flexion angle was within 27–112°, the deviation was lower than 1.0 mm as the n was greater than 5 (Fig. 5-c). The reference point number influences slightly the accuracy of the calculated patella tracking. The reference point numbers of 4, 5, 6, and 150 were used to determine the tracking of the patella. The positions of the reference points are shown in Fig. 6-a. With increasing point number, the deviation in position varied by up to 0.3%. 4. Discussion In our previous study, the in vivo patella tracking was calculated. The geometry of PF joint at several angles of knee flexion
was developed from MR images. The approximate patella tracking was derived from these discrete joint models with the spline interpolation algorithm [17]. Although the calculated tracking was within the range of experimental measurements in literature [4,19], calculation error in full range of knee motion remains unquantified. The errors come from two parts: (1) The reconstruction and registration of bone models can induce errors. The errors could be minimized by improving the reconstruction and registration algorithm. (2) The interpolation of patella tracking from finite PF positions can also cause error. The present study quantified the influence of the interpolation angles (n) on the tracking accuracy in full range of knee motion. Thus the number of interpolation angles (n) could be customized according to the needs of accuracy and knee motion ranges. Given the allowable deviation of 2 mm, interpolation angles of 6 would be sufficient to conduct the patella tracking; Given the interested knee flexion range of 27–112°, interpolation angles of 5 would suffice with the deviation less than 1 mm. The deviation between the calculated and the realistic patella tracking decreased with increasing number of interpolation angles. However, the maximum deviation remained greater than 0.65 mm with the number of interpolation angles increasing to 12. The small deviation could not be eliminated because the patella was not fully constrained during knee motion. The deviation mainly occurred near knee extension, because the constraint on patella was weakest at this position. With the patella sliding into the femoral trochlea, the deviation reduced and remained slightly changed. The number of reference points has little influence on accuracy. Theoretically, three non-collinear points on the patella could determine the patella position. However, because of the errors in measurement and numeric calculation, one point’s error may affect the overall accuracy. Therefore, Lagrangian multiplier theorem was applied to conduct the patella position with non-collinear points more than 3, which could reduce the error caused by any individual point. The present study used 4, 5, 6, and 150 reference points to calculate the patella tracking. The results were almost the same among them, which also indicated the reliability of the method. In the present study, the interpolation angles were evenly set in full range of knee motion. For example, interpolation angles of knee flexion were 0°, 27°, 54°, 81°, 108°, 135° when the number of interpolation angles was 6. At each interval between two
Please cite this article as: J. Yao et al., Patella tracking calculation from patellofemoral positions at finite angles of knee flexion, Medical Engineering and Physics (2018), https://doi.org/10.1016/j.medengphy.2018.07.018
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interpolation angles, maximum deviation occurred near the middle of the interval, and the deviation approached zero at the boundary of the interval. The overall maximum deviation occurred in the first interval near knee extension as mentioned above. If more interpolation angles were placed near knee extension, rather than arranged evenly, the maximum deviation throughout the knee extension-flexion may be further reduced. In the future studies, the arrangement of the interpolation angles would be optimized to increase accuracy of the method. The present study has some limitations. The loading condition in this in vitro experiment was different from the in vivo conditions, which will change patella tracking. However, change of the tracking was relatively small compared with the patella motion in full range of knee flexion, the trend of the patella motion calculated with the presented method is still available to provide information of joint pathology. Furthermore, the specimens were from old people with healthy knee joint. The patella tracking may be different from that of the patient with knee disease to a certain extent. Another limitation is that when the present method was applied in vivo measurement, interpolation, geometry reconstruction, and model registration are three most important factors that contribute to the total error of tracking calculation. The errors could be decreased respectively by algorithm optimizations. This study aimed to investigate the error of tracking caused by interpolation. We are also improving the algorithms of geometry reconstruction and model registration, to improve the accuracy and efficiency of the methodology. Furthermore, this method was mainly available for analyzing quasi-static and relaxed joint motion yet cannot investigate the effect of speed and loading on the motion. A device that can apply loading on knee during MR scanning will be designed in our future study, it can be used to investigate joint tracking at different in vivo loading conditions. 5. Conclusion The present study quantified the effect of interpolation parameters on the accuracy of the Patella tracking with a motion capture experiment. The results showed that the total interpolation error decreased to 2 mm with the number of interpolation angles increasing to 6. The number of interpolation reference points had slight influence on the accuracy. The findings consolidated the feasibility of using interpolation to measure in vivo patella tracking, and can help to optimize the accuracy and efficiency of the methodology. Funding This study was supported by grants from the National Natural Science Foundation of China (NSFC. 11502014 and 11421202), Young Elite Scientist Sponsorship Program by CAST(YESS 2015QNRC001), The open foundation of State Key Laboratory (SMFA16K01), Peking University International Hospital Research Fund (YN2016QN08), National Key Research and Development Program (2016YFC1101904 and 2016YFB1101101), the National Key Lab of Virtual Reality Technology, Defense Industrial technology Development Program (JCKY2016601B009), and 111 Project (B13003). Ethical approval
Acknowledgment The authors would like to convey their appreciation to Dr. Yang Chen of Peking Union Medical College Hospital for her very kind suggestion and encouragement. The authors thanked all the participants in this study. The sponsor had no role in any aspect of the study, including data collection and analysis, manuscript preparation, or authorization for publication. Supplementary materials Supplementary material associated with this article can be found, in the online version, at doi:10.1016/j.medengphy.2018.07. 018. References [1] Wilson T. The measurement of patellar alignment in patellofemoral pain syndrome: are we confusing assumptions with evidence. J Orthop Sports Phys Ther 2007;37(6):330–41. [2] Steensen RN, Bentley JC, Trinh TQ, Backes JR, Wiltfong RE. The prevalence and combined prevalences of anatomic factors associated with recurrent patellar dislocation: a magnetic resonance imaging study. Am J Sports Med 2015;43(4):921–7. [3] Amis AA. Current concepts on anatomy and biomechanics of patellar stability. Sports Med Arthrosc 2007;15(2):48–56. [4] Sheehan FT, Derasari A, Brindle TJ, Alter KE. Understanding patellofemoral pain with maltracking in the presence of joint laxity: complete 3D in vivo patellofemoral and tibiofemoral kinematics. J Orthop Res 2009;27(5):561–70. [5] Elias JJ, Soehnlen NT, Guseila LM, Cosgarea AJ. Dynamic tracking influenced by anatomy in patellar instability. Knee 2016;23(3):450–5. [6] Song CY, Lin JJ, Jan MH, Lin YF. The role of patellar alignment and tracking in vivo: the potential mechanism of patellofemoral pain syndrome. Phys Ther Sport 2011;12(3):140–7. [7] Tanikawa H, Tada M, Harato K, Okuma K, Nagura T. Influence of total knee arthroplasty on patellar kinematics and patellofemoral pressure. J Arthroplast. 2017;32(1):280–5 https://www.sciencedirect.com/science/article/ pii/S0883540316303382. [8] Bertin KC, Lloyd WW. Effect of total knee prosthesis design on patellar tracking and need for lateral retinacular release. J Arthroplast. 2013;28(5):772–7. [9] Stephen JM, Kittl C, Williams A, Zaffagnini S, Marcheggiani Muccioli GM, Fink C, et al. Effect of medial patellofemoral ligament reconstruction method on patellofemoral contact pressures and kinematics. Am J Sports Med 2016;44(5):1186–94. [10] Carlson VR, Sheehan FT, Shen A, Yao L, Jackson JN, Boden BP. The relationship of static tibial tubercle-trochlear groove measurement and dynamic patellar tracking. Am J Sports Med 2017;45(8):1856–63. [11] Fujita Y, Tsuda E, Yamamoto Y, Naraoka T, Kimura Y, Sasaki S, et al. Quantitative analysis of dynamic patellar tracking in patients with lateral patellar instability using a simple video system. Knee 2016;23(4):604–9. [12] Boegard TL, Rudling O, Petersson IF, Jonsson K. Distribution of MR-detected cartilage defects of the patellofemoral joint in chronic knee pain. Osteoarthritis Cartil 2003;11(7):494–8. [13] Narkbunnam R, Chareancholvanich K, Hanroongroj T. Sagittal plane evaluation of patellofemoral movement in patellofemoral pain patients with no evidence of maltracking. Knee Surg Sports Traumatol Arthrosc 2015;23(4):986–90. [14] Nha KW, Papannagari R, Gill TJ, Van de Velde SK, Freiberg AA, Rubash HE, et al. In vivo patellar tracking: clinical motions and patellofemoral indices. J Orthop Res 2008;26(8):1067–74. [15] Sheehan FT, Zajac FE, Drace JE. Using cine phase contrast magnetic resonance imaging to non-invasively study in vivo knee dynamics. J Biomech 1998;31(1):21–6. [16] Wilson NA, Press JM, Koh JL, Hendrix RW, Zhang LQ. In vivo noninvasive evaluation of abnormal patellar tracking during squatting in patients with patellofemoral pain. J Bone Joint Surg Am Vol 2009;91(3):558–66. [17] Yao J, Yang B, Niu WX, Zhou JW, Wang YX, Gong H, et al. In vivo measurements of patellar tracking and finite helical axis using a static magnetic resonance based methodology. Med Eng Phys 2014;36(12):1611–17. [18] Spoor CW, Veldpaus FE. Rigid body motion calculated from spatial co-ordinates of markers. J Biomech 1980;13(4):391–3. [19] Amis AA, Senavongse W, Bull AM. Patellofemoral kinematics during knee flexion-extension: an in vitro study. J Orthop Res: Off Publ Orthop Res Soc 2006;24(12):2201–11.
An ethical approval was given by the Biomedical Ethics Committee of Peking University International Hospital (No. 2016-043).
Please cite this article as: J. Yao et al., Patella tracking calculation from patellofemoral positions at finite angles of knee flexion, Medical Engineering and Physics (2018), https://doi.org/10.1016/j.medengphy.2018.07.018
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Medical Engineering and Physics 0 0 0 (2018) 1–6
Contents lists available at ScienceDirect
Medical Engineering and Physics journal homepage: www.elsevier.com/locate/medengphy
Patella tracking calculation from patellofemoral positions at finite angles of knee flexion Jie Yao a,1, Bin Yang b,1, Yuxing Wang a, Yubo Fan a,c,∗ a
Key Laboratory for Biomechanics and Mechanobiology of Ministry of Education, School of Biological Science and Medical Engineering, Beijing Advanced Innovation Centre for Biomedical Engineering, Beihang University, 37 Xueyuan Road, Haidian District, Beijing, China b Peking University International Hospital, Bejing, China c National Research Center for Rehabilitation Technical Aids, Beijing, China
a r t i c l e
i n f o
Article history: Received 24 February 2018 Revised 13 July 2018 Accepted 24 July 2018 Available online xxx Keywords: Biomechanics Patellofemoral joint Patella tracking Motion capture In vitro experiment
a b s t r a c t Patellofemoral (PF) pain is a common knee disease. Patella tracking has a significant correlation with PF pain, therefore it could be used as an index for diagnosis and treatment evaluation. Previous research has proposed a method for measuring in vivo patella tracking by means of an interpolation algorithm. The present study aimed to quantify the effect of the interpolation parameters on the accuracy of the patella tracking with a motion capture experiment. The precise patella tracking of 5 knee specimens was collected and compared with the interpolated tracking. The results showed that the total interpolation error decreased to 2 mm with the number of interpolation angles increasing to 6. The number of interpolation reference points had a slight influence on the accuracy. The findings consolidated the feasibility of using interpolation to measure the in vivo patella tracking, and can help to optimize the accuracy and efficiency of the methodology. © 2018 IPEM. Published by Elsevier Ltd. All rights reserved.
1. Introduction Patellofemoral (PF) pain syndrome is a frequently diagnosed knee disorder. The incidence of this symptom is approximately 25% in the general population [1]. Anatomical features, such as patella alta, trochlear dysplasia, lateralized tibia tuberosity, abnormal Q angle, and imbalance of PF soft tissues, are predisposing factors of the PF pain and instability [2,3]. Previous studies reported that patella maltracking is highly correlated with the PF disorder, thereby could be crucial to diagnosis and pathological classification [4–6]. Correcting the PF tracking is a critical principle for the surgery. In total knee arthroplasty, the prosthesis design could affect the patella tracking, and the unphysiological patellar kinematics may be associated with the postoperative complication [7,8]. In PF ligament reconstruction, restoring the normal patella tracking plays an important role in the surgical outcome [9]. Therefore, patella tracking assessment has the potential to boost the understanding of pathology and optimize the treatment.
∗ Corresponding author at: Key Laboratory for Biomechanics and Mechanobiology of Ministry of Education, School of Biological Science and Medical Engineering, Beijing Advanced Innovation Centre for Biomedical Engineering, Beihang University, 37 Xueyuan Road, Haidian District, Beijing, China. E-mail address: [email protected] (Y. Fan). 1 Both the authors contributed equally to this work.
Patella tracking, however, has not been widely used in clinical diagnosis, or in rehabilitation assessment. Static imaging indexes with X-ray, computer tomography (CT), and magnetic resonance (MR) scanning are currently the prime basis for the diagnosis of PF disorders. However, static indexes may fail to accurately characterize the pathology of the joint. For example, recent study reported that the tibial tubercle to trochlear groove distance, which is a common static index for a variety of PF disorders, cannot accurately predict patella maltracking in a large percentage of patients [10]. Furthermore, Since the symptoms of individual joint abnormalities are often manifested at different knee flexion angles, it can be difficult to make accurate diagnosis with images at several specific knee flexion angles. For example, patella tracking in patients of patellar dislocation significantly differed from that in healthy people at knee flexion angles of 5–30° [11]. Symptoms of early osteoarthritis are often significant in radiographs with a knee flexion of approximately 45° [12]. The patellar motion in relation to the trochlear groove of the distal femur between patients of PF pain and healthy subjects are significantly different at knee flexion angle of 90–120° [13]. Patella tracking is an indispensable parameter for accurate diagnosis of PF disorder, especially in patients with unknown knee abnormalities. Previous studies have reported several methods to evaluate patellar tracking, including dynamic CT [5], dual-orthogonal fluoroscopic imaging system [14], dynamic MR imaging [15],
https://doi.org/10.1016/j.medengphy.2018.07.018 1350-4533/© 2018 IPEM. Published by Elsevier Ltd. All rights reserved.
Please cite this article as: J. Yao et al., Patella tracking calculation from patellofemoral positions at finite angles of knee flexion, Medical Engineering and Physics (2018), https://doi.org/10.1016/j.medengphy.2018.07.018
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Fig. 1. Diagram of the knee inserted with the markers.
optoelectronic motion capture [16], etc., but these methods require special equipment, or are limited by the radiation risk or unsatisfying precision. Therefore, a measurement method of patella tracking using routine clinical equipment is still necessary. Our previous study has proposed that approximate patella tracking could be derived from routine MRI or CT images of the PF joint at several angles of knee flexion [17]. The methodology could facilitate examination of PF kinematics in clinics. However, the influence of the interpolation parameters on tracking accuracy in full range of knee motion remains unquantified. The difference between the calculated and the realistic patella tracking needs to be characterized. The objective of this study was to calculate continuous patella tracking with the PF positions at finite angles of knee flexion (PF interpolation positions). The calculated tracking was validated with realistic patella tracking measured by motion capture equipment. The effect of interpolation parameters on the accuracy of the tracking was quantified. 2. Materials and methods 2.1. Specimens preparation Five fresh frozen knee specimens (female, 3 right-side and 2 left-side, average age 70, range 55–75 years old) were studied. The specimens were without any knee pathology as examined by physical assessment and MR images. Each specimen included 300 mm of femur above knee joint line and tibia below joint line, intact patella, all associated ligaments, muscles, tendons, fasciae, fat, and skin. The specimens were thawed out at room temperature for 24 h. The proximal femur was fixed onto a rigid support by clamping. Surgical suture was wrapped and looped over the proximal end of the quadriceps tendon. A 175 N load parallel to the femur shaft was applied on the quadriceps tendon, to simulate the quadriceps force in full range of knee motion. A force was applied on the distal tibia to drive the knee flexion-extension according to the inherent track. The force was not constant, it counteracted the quadriceps force and joint resistance, and kept the knee moving quasi-statically. The patella and femur motions were recorded with the motion capture system (Northern Digital Inc., Ontario, Canada). Totally 12 markers were fixed on the patella and proximal femur by steel pins inserted into the bone (Fig. 1). The skin and fat adjacent to the steel pin was carefully incised, in order to eliminate the influence of the soft tissue on the marker position during knee motion.
Fig. 2. 3D model of the PF joint with the markers. The model was developed with the CT images.
2.2. Data acquisition The knee was moved within a range of 135° and full extension with an angle velocity of approximately 20° per second to simulate a quasi-static knee motion. Each knee was flexed-extended 6 times. The marker positions were dynamically recorded with a sampling frequency of 30 Hz. The motions of patella and femur were calculated from the spatial coordinates of the markers according to the Lagrangian multiplier theorem [18]. In brief, the motion of the bone from an initial position to a certain position could be characterized with a 3∗ 3 rotation matrix (R) and a translation vector (v). R and v could be calculated with the coordinates of the markers fixed on the bone. Let ai (x,y,z) denotes the initial positions of the i-th marker, qi (x,y,z) denotes the final positions of the marker. They satisfy the following equations:
qi = Rai + v
(1)
Where R and v were unknown, ai and qi were obtained from the motion capture system. Introducing a functional f:
f (R, v ) =
m 1 (Rai + v − qi )T (Rai + v − qi ) m i=1
×(marker amount: m = 6 )
(2)
R and v can be determined by minimizing the f under the constraint condition that R is an orthogonal matrix. The movement of the patella relative to the femur was further determined with the following equation:
Rr = R−1 R p; f
vr = R−1 v p − R−1 vf f f
(3)
Where Rr is the relative rotation matrix, Rf is the femoral rotation matrix, Rp is the patellar rotation matrix, vr is the relative translation vector, vf is the femoral translation vector, vp is the patellar translation vector. To visualize the measured tracking of the patella and femur, the knee joint with the associated markers were scanned with CT (Philips, Brilliance 16, Holland). The slice thickness of scanning was 1 mm, the pixel size was 0.78 ∗ 0.78 mm2 . The threedimension (3D) models of the patella and femur were developed with the medical image processing software, Mimics (Materialise, Inc.,Belgium). The 3D models were shown in Fig. 2, and motion of the patella relative to the femur was shown in Supplementary material 1.
Please cite this article as: J. Yao et al., Patella tracking calculation from patellofemoral positions at finite angles of knee flexion, Medical Engineering and Physics (2018), https://doi.org/10.1016/j.medengphy.2018.07.018
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Fig. 3. Reference points and their trajectories. (a) Reference points at the top, bottom, medial, and lateral sites of the patella. (b) Realistic (red) and approximate (blue) trajectories (n = 3) of the reference points.
Fig. 4. Maximum deviations between calculated and realistic patellae throughout knee extension-flexion. The solid curve indicates the average of the patella maximum deviation in 5 specimens × 6 measurements. The dash curves indicate the standard deviations.
2.3. Calculation and validation of approximate patella tracking The patellar positions at n angles of knee flexion were used to interpolate the approximate patella tracking (n ≥ 2). At each angle of knee flexion, more than three non-collinear reference points on patella could determine the patella position. In the in vivo measurement, these reference points are determined on the joint model from the MR images. To simulate the in vivo measurement
and validate its applicability, the coordinates of reference points were calculated from the motion capture markers in this study, the points at the top, bottom, medial, and lateral site of the patella were used (Fig. 3-a). The trajectory of each reference point during knee flexion-extension was conducted with order-three spline algorithm. Briefly, let function a(θ ) = (x(θ ), y(θ ), z(θ )) denotes the radius vector of the point a during knee flexion-extension, θ is the knee flexion angle. a(θ ) could be represented with pieces of
Please cite this article as: J. Yao et al., Patella tracking calculation from patellofemoral positions at finite angles of knee flexion, Medical Engineering and Physics (2018), https://doi.org/10.1016/j.medengphy.2018.07.018
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Fig. 5. Influence of the n on the average maximum deviation between approximate and realistic patellae. (a) Deviation descended to 2.0 mm with the n increasing to 6, and further descended to 0.65 mm with the n increasing to 12, during knee flexed within 0–135°. (b) Deviation was lower than 2.0 mm as the n was greater than 5, and was lower than 1.0 mm as the n was greater than 6, during knee flexed within 21–135°. (c) Deviation was lower than 1.0 mm as the n was greater than 5, during knee flexed within 27–112°.
order-three polynomial functions a(i) (θ ):
⎧ a(1 ) (θ ), θ ∈ [θ1 , θ2 ] ⎪ ⎪ ⎨ ... a(i ) (θ ), θ ∈ [θi , θi+1 ] (i = 1, 2, . . . , n) a (θ ) = ⎪ ⎪ ... ⎩ a(n−1) (θ ), θ ∈ [θn−1 , θn ]
(4)
where n is the number of knee flexion angles used for interpolating the patella tracking. a(θ i ) were already obtained from the motion capture experiments. a(θ ) could be determined by satisfying the following conditions:
a˙ (i ) (θi+1 ) = a˙ (i+1) (θi+1 ); a¨ (i ) (θi+1 ) = a¨ i+1 (θi+1 ); a¨ (1 ) (θ1 ) = a¨ (n−1) (θn ) = 0;
(5)
where a˙ i (θ ) and a¨ i (θ ) are the first-order and second-order derivatives. The realistic and approximate trajectories of the reference points were shown in Fig. 3-b.
The patella tracking was then derived from the trajectories of the four reference points according to the Lagrangian multiplier theorem [18]. At any knee flexion angle θ , the approximate position of each point was calculated with the trajectory a(θ ) mentioned above. With the coordinates of the four reference points, the rotation matrix (R) and the translation vector (v) characterizing the patella motion could be determined with the Eqs. (1 and 2). With increasing the number of interpolation angles (n), the approximate tracking will converge to the realistic tracking. To validate the accuracy of the approximate patella tracking, and investigate the relationship between the number of interpolation angles (n) and the accuracy, the approximate tracking calculated was compared with the realistic tracking from motion capture experiment. At each point of the patella, the deviation between the approximate and realistic positions during knee motion was calculated. The maximum deviation was used as an accuracy index of the approximate patella tracking.
Please cite this article as: J. Yao et al., Patella tracking calculation from patellofemoral positions at finite angles of knee flexion, Medical Engineering and Physics (2018), https://doi.org/10.1016/j.medengphy.2018.07.018
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Fig. 6. Maximum deviations between approximate and realistic patellae when using difference reference points. (a) The positions of the reference points. (b) The averages of the patella maximum deviation in 5 specimens × 6 measurements in knee extension-flexion (n = 6). Legend: The reference point number influences slightly the accuracy of the calculated patella tracking.
3. Results The calculated (blue) and realistic (red) motions of the patellae were shown in the video (Supplementary material 2). In the case of n = 2, the calculated patella moved straightly from anterior side of the femur to distal side, the calculated tracking was far away from the realistic tracking. With increasing the number of interpolation angles (n), the calculated tracking got close to the realistic one. When n is greater than 5, the calculated patella almost overlapped with the realistic one. To quantify the accuracy of the calculated tracking, the maximum deviation between calculated and realistic patellae throughout the full range of knee motion were shown in Fig. 4. At each interval between two interpolation angles, the deviations approached zero at the boundary of the interval. The deviations approached peak value at near middle of the interval. When n was greater than 2, the maximum value of the average deviations throughout knee extension-flexion always occurred at the first interval. The deviation between the calculated and realistic patellae descended with increasing the n. Theoretically, the deviation would converge to 0 with the n increasing to infinity. In the present study, the maximum deviation descended rapidly to 2 mm with the n increasing to 6, and further descended gradually to 0.65 mm with the n increasing to 12 (Fig. 5-a). The descending rates varied at different ranges of knee flexion angle. When the knee flexion angle was within 21–135°, the maximum deviation was lower than 2.0 mm as the n was greater than 5, and was lower than 1.0 mm as the n was greater than 6 (Fig. 5-b). When the knee flexion angle was within 27–112°, the deviation was lower than 1.0 mm as the n was greater than 5 (Fig. 5-c). The reference point number influences slightly the accuracy of the calculated patella tracking. The reference point numbers of 4, 5, 6, and 150 were used to determine the tracking of the patella. The positions of the reference points are shown in Fig. 6-a. With increasing point number, the deviation in position varied by up to 0.3%. 4. Discussion In our previous study, the in vivo patella tracking was calculated. The geometry of PF joint at several angles of knee flexion
was developed from MR images. The approximate patella tracking was derived from these discrete joint models with the spline interpolation algorithm [17]. Although the calculated tracking was within the range of experimental measurements in literature [4,19], calculation error in full range of knee motion remains unquantified. The errors come from two parts: (1) The reconstruction and registration of bone models can induce errors. The errors could be minimized by improving the reconstruction and registration algorithm. (2) The interpolation of patella tracking from finite PF positions can also cause error. The present study quantified the influence of the interpolation angles (n) on the tracking accuracy in full range of knee motion. Thus the number of interpolation angles (n) could be customized according to the needs of accuracy and knee motion ranges. Given the allowable deviation of 2 mm, interpolation angles of 6 would be sufficient to conduct the patella tracking; Given the interested knee flexion range of 27–112°, interpolation angles of 5 would suffice with the deviation less than 1 mm. The deviation between the calculated and the realistic patella tracking decreased with increasing number of interpolation angles. However, the maximum deviation remained greater than 0.65 mm with the number of interpolation angles increasing to 12. The small deviation could not be eliminated because the patella was not fully constrained during knee motion. The deviation mainly occurred near knee extension, because the constraint on patella was weakest at this position. With the patella sliding into the femoral trochlea, the deviation reduced and remained slightly changed. The number of reference points has little influence on accuracy. Theoretically, three non-collinear points on the patella could determine the patella position. However, because of the errors in measurement and numeric calculation, one point’s error may affect the overall accuracy. Therefore, Lagrangian multiplier theorem was applied to conduct the patella position with non-collinear points more than 3, which could reduce the error caused by any individual point. The present study used 4, 5, 6, and 150 reference points to calculate the patella tracking. The results were almost the same among them, which also indicated the reliability of the method. In the present study, the interpolation angles were evenly set in full range of knee motion. For example, interpolation angles of knee flexion were 0°, 27°, 54°, 81°, 108°, 135° when the number of interpolation angles was 6. At each interval between two
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interpolation angles, maximum deviation occurred near the middle of the interval, and the deviation approached zero at the boundary of the interval. The overall maximum deviation occurred in the first interval near knee extension as mentioned above. If more interpolation angles were placed near knee extension, rather than arranged evenly, the maximum deviation throughout the knee extension-flexion may be further reduced. In the future studies, the arrangement of the interpolation angles would be optimized to increase accuracy of the method. The present study has some limitations. The loading condition in this in vitro experiment was different from the in vivo conditions, which will change patella tracking. However, change of the tracking was relatively small compared with the patella motion in full range of knee flexion, the trend of the patella motion calculated with the presented method is still available to provide information of joint pathology. Furthermore, the specimens were from old people with healthy knee joint. The patella tracking may be different from that of the patient with knee disease to a certain extent. Another limitation is that when the present method was applied in vivo measurement, interpolation, geometry reconstruction, and model registration are three most important factors that contribute to the total error of tracking calculation. The errors could be decreased respectively by algorithm optimizations. This study aimed to investigate the error of tracking caused by interpolation. We are also improving the algorithms of geometry reconstruction and model registration, to improve the accuracy and efficiency of the methodology. Furthermore, this method was mainly available for analyzing quasi-static and relaxed joint motion yet cannot investigate the effect of speed and loading on the motion. A device that can apply loading on knee during MR scanning will be designed in our future study, it can be used to investigate joint tracking at different in vivo loading conditions. 5. Conclusion The present study quantified the effect of interpolation parameters on the accuracy of the Patella tracking with a motion capture experiment. The results showed that the total interpolation error decreased to 2 mm with the number of interpolation angles increasing to 6. The number of interpolation reference points had slight influence on the accuracy. The findings consolidated the feasibility of using interpolation to measure in vivo patella tracking, and can help to optimize the accuracy and efficiency of the methodology. Funding This study was supported by grants from the National Natural Science Foundation of China (NSFC. 11502014 and 11421202), Young Elite Scientist Sponsorship Program by CAST(YESS 2015QNRC001), The open foundation of State Key Laboratory (SMFA16K01), Peking University International Hospital Research Fund (YN2016QN08), National Key Research and Development Program (2016YFC1101904 and 2016YFB1101101), the National Key Lab of Virtual Reality Technology, Defense Industrial technology Development Program (JCKY2016601B009), and 111 Project (B13003). Ethical approval
Acknowledgment The authors would like to convey their appreciation to Dr. Yang Chen of Peking Union Medical College Hospital for her very kind suggestion and encouragement. The authors thanked all the participants in this study. The sponsor had no role in any aspect of the study, including data collection and analysis, manuscript preparation, or authorization for publication. Supplementary materials Supplementary material associated with this article can be found, in the online version, at doi:10.1016/j.medengphy.2018.07. 018. References [1] Wilson T. The measurement of patellar alignment in patellofemoral pain syndrome: are we confusing assumptions with evidence. J Orthop Sports Phys Ther 2007;37(6):330–41. [2] Steensen RN, Bentley JC, Trinh TQ, Backes JR, Wiltfong RE. The prevalence and combined prevalences of anatomic factors associated with recurrent patellar dislocation: a magnetic resonance imaging study. Am J Sports Med 2015;43(4):921–7. [3] Amis AA. Current concepts on anatomy and biomechanics of patellar stability. Sports Med Arthrosc 2007;15(2):48–56. [4] Sheehan FT, Derasari A, Brindle TJ, Alter KE. Understanding patellofemoral pain with maltracking in the presence of joint laxity: complete 3D in vivo patellofemoral and tibiofemoral kinematics. J Orthop Res 2009;27(5):561–70. [5] Elias JJ, Soehnlen NT, Guseila LM, Cosgarea AJ. Dynamic tracking influenced by anatomy in patellar instability. Knee 2016;23(3):450–5. [6] Song CY, Lin JJ, Jan MH, Lin YF. The role of patellar alignment and tracking in vivo: the potential mechanism of patellofemoral pain syndrome. Phys Ther Sport 2011;12(3):140–7. [7] Tanikawa H, Tada M, Harato K, Okuma K, Nagura T. Influence of total knee arthroplasty on patellar kinematics and patellofemoral pressure. J Arthroplast. 2017;32(1):280–5 https://www.sciencedirect.com/science/article/ pii/S0883540316303382. [8] Bertin KC, Lloyd WW. Effect of total knee prosthesis design on patellar tracking and need for lateral retinacular release. J Arthroplast. 2013;28(5):772–7. [9] Stephen JM, Kittl C, Williams A, Zaffagnini S, Marcheggiani Muccioli GM, Fink C, et al. Effect of medial patellofemoral ligament reconstruction method on patellofemoral contact pressures and kinematics. Am J Sports Med 2016;44(5):1186–94. [10] Carlson VR, Sheehan FT, Shen A, Yao L, Jackson JN, Boden BP. The relationship of static tibial tubercle-trochlear groove measurement and dynamic patellar tracking. Am J Sports Med 2017;45(8):1856–63. [11] Fujita Y, Tsuda E, Yamamoto Y, Naraoka T, Kimura Y, Sasaki S, et al. Quantitative analysis of dynamic patellar tracking in patients with lateral patellar instability using a simple video system. Knee 2016;23(4):604–9. [12] Boegard TL, Rudling O, Petersson IF, Jonsson K. Distribution of MR-detected cartilage defects of the patellofemoral joint in chronic knee pain. Osteoarthritis Cartil 2003;11(7):494–8. [13] Narkbunnam R, Chareancholvanich K, Hanroongroj T. Sagittal plane evaluation of patellofemoral movement in patellofemoral pain patients with no evidence of maltracking. Knee Surg Sports Traumatol Arthrosc 2015;23(4):986–90. [14] Nha KW, Papannagari R, Gill TJ, Van de Velde SK, Freiberg AA, Rubash HE, et al. In vivo patellar tracking: clinical motions and patellofemoral indices. J Orthop Res 2008;26(8):1067–74. [15] Sheehan FT, Zajac FE, Drace JE. Using cine phase contrast magnetic resonance imaging to non-invasively study in vivo knee dynamics. J Biomech 1998;31(1):21–6. [16] Wilson NA, Press JM, Koh JL, Hendrix RW, Zhang LQ. In vivo noninvasive evaluation of abnormal patellar tracking during squatting in patients with patellofemoral pain. J Bone Joint Surg Am Vol 2009;91(3):558–66. [17] Yao J, Yang B, Niu WX, Zhou JW, Wang YX, Gong H, et al. In vivo measurements of patellar tracking and finite helical axis using a static magnetic resonance based methodology. Med Eng Phys 2014;36(12):1611–17. [18] Spoor CW, Veldpaus FE. Rigid body motion calculated from spatial co-ordinates of markers. J Biomech 1980;13(4):391–3. [19] Amis AA, Senavongse W, Bull AM. Patellofemoral kinematics during knee flexion-extension: an in vitro study. J Orthop Res: Off Publ Orthop Res Soc 2006;24(12):2201–11.
An ethical approval was given by the Biomedical Ethics Committee of Peking University International Hospital (No. 2016-043).
Please cite this article as: J. Yao et al., Patella tracking calculation from patellofemoral positions at finite angles of knee flexion, Medical Engineering and Physics (2018), https://doi.org/10.1016/j.medengphy.2018.07.018