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The effect of volume sub-sampling on motion estimation of joints via MR imaging
Computerized Medical Imaging and Graphics 33 (2009) 242–246
Contents lists available at ScienceDirect
Computerized Medical Imaging and Graphics journal homepage: www.elsevier.com/locate/compmedimag
The effect of volume sub-sampling on motion estimation of joints via MR imaging D. Sekaran ∗ Medical Image Processing Group, University of Pennsylvania, 313 Rollingwood Rd. Roanoke Rapids, NC 27870, United States
a r t i c l e
i n f o
Article history: Received 29 July 2008 Received in revised form 21 December 2008 Accepted 6 January 2009 Keywords: MRI Joint imaging Joint kinematics Image sub-sampling Registration
a b s t r a c t The technique of three-dimensional imaging is being explored as a means of better understanding the morphology and kinematics of the foot and ankle. To capture information about the dynamic nature of the joint, MRI pulse sequences are used to rapidly acquire single-slice kinematic images, which are then used to track the motion of the bone or other tissues. This approach cannot capture true 3D motion information. On the other hand, full 3D acquisitions are time consuming. A more time-efficient alternative to this method that may give accurate 3D motion information may be to use select MRI slices, instead of full resolution 3D models, that may be just enough to capture the vital information needed to track motion. This was tested by removing slices from already acquired full kinematic MRI datasets and progressively removing slices to determine up to what level data can be eliminated and still achieve accurate motion tracking. We evaluated the ability of the reduced data set in tracking motion in terms of both volume overlap and actual motion estimated, and compared these with the results from the full resolution data. We based our analysis of accuracy on the ability to transform the reduced images from one position of the foot to another. In tracking the motion of the bones of the tarsal joints, we were able to reduce the number of slices to about 25% of the full data set while maintaining an accurate representation of motion within about 0.5 mm of translation and 0.5 degrees of rotation of the motion estimated from full data. © 2009 Elsevier Ltd. All rights reserved.
1. Introduction The foot and ankle have been the subject of many investigations, as they are one of the most common sources of personal injury today. Efforts to better understand and treat these ailments have led to studies which strive to characterize the morphology, architecture, and kinematics of the foot. A comprehensive understanding in this field would help ensure that each patient would receive the appropriate diagnosis and treatment for his specific ailments. The traditional techniques for studying morphology and kinematics include the dissection of cadavers and taking radiographic projections in two dimensions. However, both of these methods have their drawbacks—the invasive dissections yield unrealistic motion and cannot be performed on live patients, and a 2D perspective cannot provide a good representation of the 3D morphology, architecture, and movement of joints. For this reason, considerable work has been carried out in order to create 3D representations of the foot and ankle via 3D imaging techniques. The main idea behind 3D imaging techniques is to acquire full 3D internal anatomic information via an imaging modality such as
∗ Tel.: +1 252 532 5343. E-mail address: [email protected]. 0895-6111/$ – see front matter © 2009 Elsevier Ltd. All rights reserved. doi:10.1016/j.compmedimag.2009.01.002
computed tomography or magnetic resonance imaging. Images are acquired in each of a number of positions of the foot [1–6]. Usually, a positioning device is used to hold the joint in each position. Often stress is applied to study the flexibility characteristics of the joint or to test how the bones move under normal loading conditions [11,12]. How different types of injuries affect joint flexibility and motion characteristics and how different surgical reconstructions differ in their ability to restore flexibility and motion have been studied via multi position stress MRI [11,12]. Such images have also been used in building subject specific biomechanical models of the ankle complex and in verifying how realistic such models are in predicting joint mechanical behavior [17]. However studies are also carried out to examine bone motion without stress or in the condition of an open kinematic chain to understand differences among foot architectures, both normal and abnormal [7–10]. One of the main issues with imaging is to capture the true dynamic nature of the information about the joint in images. It would be ideal if enough 3D data sets can be acquired on the fly while the joint is subjected to a particular motion in the scanner. Unfortunately this is not feasible at present, and perhaps will not be feasible for quite some time to come. Although CT imaging has become extremely rapid in recent times, enabling the acquisition of a full volume image within a second, in this application, its use in patients may be hazardous due to excessive X-ray radiation,
D. Sekaran / Computerized Medical Imaging and Graphics 33 (2009) 242–246
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Fig. 1. A slice of the calcaneus bone, taken from two different positions.
especially because multiple 3D image sets will have to be acquired. Although MRI is the safer option by far, it is also much slower. However, MRI pulse sequences have been developed for rapid single-slice kinematic imaging of joints [13–15]. The main drawback of such studies is that they permit the examination of 2D planar motions only. These studies and the possibility of a few-sliceonly rapid MRI led us naturally to the investigation undertaken in this paper. Suppose a full resolution 3D image In is acquired in one, say neutral, position of the joint, and suppose the goal is tracking the rigid motion of bones. If it is feasible, from a combination of image segmentation and registration techniques to accurately track the motion of a bone defined in In in different positions of the joint from rapidly acquired “a few-slice-only” MR images, then there is no need to acquire full resolution image sets in all other positions. This can reduce the acquisition time in kinematic studies considerably. All of the MRI scenarios described above – stress MRI, open kinematic chain MRI, and dynamic MRI – can benefit from the reduction ensuing from a “few-slice-only” study if it would become feasible. For example, in the first two situations, the total acquisition time will be reduced improving patient comfort especially in pain invoking positions. In the third situation, it may become feasible to carry out a 3D dynamic study instead of a single-slice kinematic study. Thus the investigation carried out in this paper can be beneficial to all three scenarios mentioned above. A fourth scenario, not discussed above, is imaging in natural weight-bearing conditions, such as in upright MRI scanners. Although this still does not correspond to the most natural condition of joint loading that occurs during various activities (such as walking, running, climbing, jumping), it comes closest to being a naturally active condition among the four scenarios, and here too the results of our study will be applicable. We close this discussion by stating that studies in all four scenarios have produced a useful understanding of the joint flexibility characteristics, architecture, kinematics and mechanics. In the present study, our focus is on the first two scenarios. Motivated by the above consideration, in this short paper, based on existing full resolution image data for different positions of the foot, an investigation is carried out as to how the accuracy of motion analysis may suffer as fewer and fewer slices in the image set for subsequent positions are taken in the analysis. 2. Materials and methods For this study, we used data from 10 different feet. MR images were taken of the foot and ankle in two different positions with the primary focus on the calcaneus, talus, navicular and cuboid bones. Each image set consisted of approximately 60 sagittal slice images.
A 1.5T commercial GE Signa MRI scanner was used. Two 3-in. single-loop RF coils and one 5-in. single- loop coil were placed on the two sides of the ankle and underneath the heel. They were configured as a multi coil receiver. The scanning protocol consisted of a 3D Fast Gradient Echo pulse sequence with a TR/TE/flip angle of 11.5 ms/2.4 ms/60◦ , a 512 × 256 in-plane acquisition matrix, a ±31.2 receiver bandwidth, and a 180 mm field of view. Sixty 2.1 mm-thick contiguous sagittal slices were collected to cover the foot from the medial to the lateral aspect. The first imaged position corresponded to the neutral joint position [7,11] and the second represented an inversion with a 3.4 Nm moment applied via the positioning device. An Ankle Loading Device was used as described in [11] for holding and stressing the foot and for acquiring the images. Let I1 and I2 denote the original acquired images corresponding to the two different positions of the same foot. Multiple slices were removed from the set I2 , using the software system 3DVIEWNIX. The removal of slices was evenly distributed, and resulted in reduced sets consisting of anywhere between 20 and 50 percent of the original number of slices. The MR slices which captured the polar ends of the bones contained superficial data, so they were omitted from the study to reduce unnecessary error. Before forming any three-dimensional models, the binary slices were always interpolated via shape-based interpolation, in order to smoothen the edges and reduce errors in analysis. The goal of this analysis then was to determine, how close the motion of a given bone (say, talus) determined by matching its full image I1 to the sub-sampled image J2 is, to the motion estimated by matching the full resolution data I1 to I2 . Let the respective binary images representing the particular bone be I1b , I2b , and J2b . In these images, the entire bone region – boundary and interior – is represented by 1-valued voxels. Since the segmentations I1b , I2b of the original images, carried out carefully in a slice-by-slice manner, were available for this study, the creation of J2b was straightforward. In all registration experiments, the gray image pertaining to just the bone of interest was obtained by masking with the binary image corresponding to the bone (Fig. 1). That is, the gray image I1 = I1 ∩ I1b was registered with J2 = J2i ∩ J2bi where J2i denotes an image resulting from linear interpolation of J2 , and J2bi denotes a binary image resulting from shape-based interpolation of J2b . The similarity metric that was used for registration was mutual information [16]. J2 was created at five different subsampling levels: 100%, 50%, 33%, 25% and 20%. Per our notation, at 100% sub-sampling, J2 = I2 which means no slices were removed, and thus the percent indicates the fraction of the slices retained. The registration experiments were performed separately for the four bones of the peritalar complex of the foot: talus, calcaneus, navicular, and cuboid. In all cases, the registration of I1 to J2 yielded the rigid transformation matrix.
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Fig. 2. Transformed models of the calcaneus, talus, cuboid, and navicular bones (clockwise from top left).
Two methods were used to assess the influence of sub-sampling on motion analysis. 2.1. Method 1 In this method, the accuracy of registration was expressed by a measure of overlap between the original full resolution segmented region I1b that was transformed by the rigid registration transform and the sub-sampled binary image J2b . The transformed I1b will be b . To make the overlap measure fair, I b was interpodenoted by I1t 1t b so that the slices in I b matched with those in lated to yield I1ti 1ti b = I b since J b = I b , and no J2b . When sub-sampling level is 100% I1ti 1t 2 2 interpolation is needed. The overlap measure OM was normalized, so that, in the ideal case of 100% sub-sampling level, OM equaled 1.
OM =
b ∩ J b |/|I b ∪ J b | |I1ti 2 2 1ti b ∩ I b |/|I b ∪ I b | |I1t 2 1t 2
Here |x| denotes the number of 1-valued voxels in the binary image x. 2.2. Method 2 This method focused on the accuracy of estimating the actual movement of bones rather than on the delineation of bones. Accuracy was separately quantified for rotation and translation which resulted from helical axes motion description from one position to another. The translation and rotation coming from matching I1 to I2 ∩ I2b was considered perfect, and deviations, from these values, of the translation and rotation derived by matching I1 to J2 were computed. 3. Results The sub-sampled datasets (J2b ) were interpolated and converted into three-dimensional models using 3DVIEWNIX. The models were then superimposed upon the 3D representation from the transb ), to provide a qualitative perspective of formed original data (I1t the accuracy of the transformation (Fig. 2). The model from the original data is represented in pink. The blue shows the models formed when the original data are reduced by 50%, and the green models were formed from 20% of the data. The slices on each end of the bones contained inconsequential data which varied widely between the two positions. In this study, these slices were
removed for the sake of reducing unnecessary error in the transformation process; this led to the models being flattened out on the ends. The normalized overlap measure OM was computed for the various sub-sampling schemes and bones. We calculated the mean values over the 10 datasets (Table 1). Variations in the accuracy of estimating translation and rotation are illustrated. A sample comparison of variation across subjects (Fig. 3) and the general trends of variation across the four bones are demonstrated (Fig. 4). Due to their similarity, the deviations across all three helical axes were averaged for the purpose of simplifying the graphs. 4. Discussion Using Method 1, we were able to qualitatively observe the effects of reducing the number of slices. It appears that halving the number of slices does not seem to have a significant effect on the quality of the 3D model (Fig. 1). Many of the fine details of the bones are still present, and the transformation appears to be accurate. However, the dataset which contains only 20% of the original slices is noticeably cruder, with only the basic shape of the bone being evident. However, the OM values are useful as a reference to be used alongside the 3D models. These values are sufficiently high, being over 0.9, even when two thirds of the slices are discarded. We note that, in evaluating image segmentation methods using an overlap measure (true positive), even an 85% overlap (OM = 0.85) is considered to be very good. While 90% overlap can be considered to be very good, over 95% signals excellent agreement with truth. Interestingly, discarding about one half of the data does not seem to make a significant dent on the overlap measure. It should also be noted that the calcaneus and talus – the larger bones – yield noticeably higher OM values than the navicular and cuboid bones. This seems reasonable, as the sets for the larger bones contain more overall data, and would thus be less affected by the removal of slices. The Table 1 Mean OM values over 10 datasets for the four bones for the different sub-sampling schemes.
20% 25% 33% 50% 100%
Calcaneus
Talus
Navicular
Cuboid
0.8858 0.9121 0.9412 0.9749 1.0000
0.9032 0.9231 0.9427 0.9672 1.0000
0.8273 0.8597 0.9048 0.9571 1.0000
0.8548 0.8960 0.9522 0.9653 1.0000
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trend becomes more evident for any datasets which are more than 50% reduced. While the first method of evaluation is helpful in understanding how registration is influenced by sub-sampling I2 , the second method yields directly the discrepancy in motion estimation as a function of the degree of sub-sampling. It was ineffective to analyze motion information in terms of percent error, so the average deviations of rotation and translation were computed and compared. The data for the individual subjects displayed the same general trends for all cases, with few major outliers. We note that the degree of inversion rotation varied among the feet studied within our database and not particularly controlled to make them uniform. When the results are averaged across subjects and categorized by bone, the quality of the reduced datasets and its influence on motion estimation are much clearer. An especially strong trend is observed in the variance of rotation. For all four bones, reducing the data by 50% results in a deviation of approximately 0.5 degrees from the original coordinates. With respect to the qualitative observations from Method 1, this seems to be the range for acceptable quality. The values for the other reduced datasets yielded equally narrow ranges of results, all within 1 degree of each other. The trends for translation were present, but not as strong; they appeared to be more dependent on which bone was represented. However, the ranges were still within 1 mm, regardless of the amount of reduction. Once again, smaller bones such as the navicular and cuboid displayed noticeably lower quality in the reduced datasets. Similar to rotation, the 50% reduced data resulted in deviations lower than 0.5 mm. While the above results seem very promising, we cannot generalize and claim the same degree of data reduceability for other Fig. 4. Average variance of rotation (in degrees) and translation (in mms) for all subjects, categorized by bone.
types of joint motion. Because of the high degree of anisotropy of sampling involved, it is conceivable that the results may have a dependency on several factors such as the orientation of the slice plane, the shape of the bone, and the type of motion. A more comprehensive study is needed to fully understand the influence of these factors on the results.
5. Conclusion
Fig. 3. Comparison of variation of rotation (in degrees) and translation (in mms) for calcaneus bone—categorized by subjects.
Given the rapid growth of technology today, it may be possible to routinely use three-dimensional models to understand the morphology and kinematics of joints in the clinic in the near future. As of now, MR imaging is the safest and most effective way to acquire data for these studies. From the results of this experiment, we can conclude that it may be possible to drastically decrease the number of slices in a 3D model while maintaining a level of quality which would allow for effective modeling of joint movement. When the data are reduced by 50%, the resulting models nearly replicate the quality of the original, and the coordinates of transformation are accurate to within half a degree of rotation and 0.5 mm of translation. On the other hand, it is also clear that the size of the bone matters as well; reducing the slices of smaller bones could result in too much data loss to create an effective model. This observation should be noted when considering the target bones of a scan. However, in the larger picture, these results strongly suggest that it is possible to significantly decrease MR acquisition time, perhaps as much as two thirds, by lowering the number of necessary images for a functional 3D model.
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Acknowledgements The author is grateful to Dr. Udupa and the MIPG members for the resources made available and for guidance during his internship which led to this paper. References [1] Hirsch BE, Udupa JK, Goncalves R, Roberts D. Kinematics of joints of the foot via three-dimensional magnetic resonance images. In: Proceedings of the First Conference on Visualization in Biomedical Computing, VBC’90. 1990. p. 232–7. [2] Baker HH, Bruckner JS, Langdon LH. Estimating ankle rotational constraints from anatomic structure. In: Visualization in Biomedical Computing: SPIE Proceedings, 1808. 1992. p. 422–32. [3] Garg A, Walker PS. Prediction of total knee motion using a three-dimensional computer-graphics model. Journal of Biomechanics 1990;33:45–58. [4] Crisco JJ, McGovern, RD, Wolfe SW. Three-dimensional joint kinematics using bone surface registration: a computer assisted approach with an application to the wrist joint in vivo, MICCAI. 1998. p. 696–699. [5] von Eisenhart-Rothe R, Siebert M, Bringmann C, Vogl T, Englmeier K-H, Graichen H. A new in vivo technique for determination of 3D kinematics and contact areas of the patelo-femoral and tibio-femoral joint. Journal of Biomechanics 2004;37:927–34. [6] Goto A, Moritomo H, Murase T, Oka K, Sugamoto K, Arimura T, et al. In vivo threedimensional wrist motion analysis using magnetic resonance imaging and volume-based registration. Journal of Orthopaedic Research 2005;23:750–6. [7] Udupa JK, Hirsch BE, Samarasekera S, Hillstrom H, Bauer G, Kneeland B. Analysis of in vivo 3D internal kinematics of the joints of the foot. IEEE Transactions on Biomedical Engineering 1998;45(11):1387–96. [8] Stindel E, Udupa JK, Hirsch BE, Odhner D, Couture C. 3D MR image analysis of the morphology of the rear foot: application to classification of bones. Computerized Medical Imaging and Graphics 1999;23(2):75–83.
[9] Stindel E, Udupa JK, Hirsch BE, Odhner D. A characterization of the geometric architecture of the peritalar joint complex via MRI: an aid to classification of feet. IEEE Transactions on Medical Imaging 1999;18:753–63. [10] Stindel E, Udupa JK, Hirsch BE, Odhner D. An in vivo analysis of the peri-talar joint complex based on MR imaging. IEEE Transactions on Biomedical Engineering 2001;48(2):236–47. [11] Siegler S, Udupa JK, Ringleb SI, Imhauser CW, Hirsch BE, Odhner D, et al. Mechanics of the ankle and subtalar joints revealed through a three-dimensional stress MRI technique. Journal of Biomechanics 2005;38(3):567–78. [12] Ringleb SI, Udupa JK, Siegler S, Imhauser CW, Hirsch BE, Liu J, et al. The effect of ankle ligament damage an surgical reconstruction on the mechanics of the ankle and subtalar joints revealed by three-dimensional stress MRI. Journal of Orthopaedic Research 2005;23(4):743–9. [13] Wolf P, Luechinger R, Boesiger P. An MR imaging procedure to measure tarsal bone rotations. Journal of Biomechanical Engineering 2007;129:931–6. [14] Ward SR, Shellock FG, Terk MR, Salsich GB, Powers CM. Assessment of patellofemoral relationships using kinematic MRI: comparison between qualitative and quantitative methods. Journal of Magnetic Resonance Imaging 2002;16:74–9. [15] Witonski D, Goraj B. Patellar motion analyzed by kinematic and dynamic axial magnetic resonance imaging in patients with anterior knee pain syndrome. Archives of Orthopaedic and Trauma Surgery 1999;119:46–9. [16] Wells III WM, Viola P, Atsumi H, Nakajuma S, Kikinis R. Multi-modal volume registration by maximization of mutual information. Medical Image Analysis 1996;1(1):35–51. [17] Imhauser CW, Siegler S, Udupa JK, Toy JR. Subject-specific models of the hindfoot reveal a relationship between morphology and passive mechanical properties. Journal of Biomechanics 2008;41:1342–9. Deepak Sekaran is currently an undergraduate at Washington University in Saint Louis. This fall, he will complete his B.S. in Neuroscience. He is currently working for a neuroimaging lab at the Washington University School of Medicine. He plans to enter medical school and eventually begin practice as a physician.
Contents lists available at ScienceDirect
Computerized Medical Imaging and Graphics journal homepage: www.elsevier.com/locate/compmedimag
The effect of volume sub-sampling on motion estimation of joints via MR imaging D. Sekaran ∗ Medical Image Processing Group, University of Pennsylvania, 313 Rollingwood Rd. Roanoke Rapids, NC 27870, United States
a r t i c l e
i n f o
Article history: Received 29 July 2008 Received in revised form 21 December 2008 Accepted 6 January 2009 Keywords: MRI Joint imaging Joint kinematics Image sub-sampling Registration
a b s t r a c t The technique of three-dimensional imaging is being explored as a means of better understanding the morphology and kinematics of the foot and ankle. To capture information about the dynamic nature of the joint, MRI pulse sequences are used to rapidly acquire single-slice kinematic images, which are then used to track the motion of the bone or other tissues. This approach cannot capture true 3D motion information. On the other hand, full 3D acquisitions are time consuming. A more time-efficient alternative to this method that may give accurate 3D motion information may be to use select MRI slices, instead of full resolution 3D models, that may be just enough to capture the vital information needed to track motion. This was tested by removing slices from already acquired full kinematic MRI datasets and progressively removing slices to determine up to what level data can be eliminated and still achieve accurate motion tracking. We evaluated the ability of the reduced data set in tracking motion in terms of both volume overlap and actual motion estimated, and compared these with the results from the full resolution data. We based our analysis of accuracy on the ability to transform the reduced images from one position of the foot to another. In tracking the motion of the bones of the tarsal joints, we were able to reduce the number of slices to about 25% of the full data set while maintaining an accurate representation of motion within about 0.5 mm of translation and 0.5 degrees of rotation of the motion estimated from full data. © 2009 Elsevier Ltd. All rights reserved.
1. Introduction The foot and ankle have been the subject of many investigations, as they are one of the most common sources of personal injury today. Efforts to better understand and treat these ailments have led to studies which strive to characterize the morphology, architecture, and kinematics of the foot. A comprehensive understanding in this field would help ensure that each patient would receive the appropriate diagnosis and treatment for his specific ailments. The traditional techniques for studying morphology and kinematics include the dissection of cadavers and taking radiographic projections in two dimensions. However, both of these methods have their drawbacks—the invasive dissections yield unrealistic motion and cannot be performed on live patients, and a 2D perspective cannot provide a good representation of the 3D morphology, architecture, and movement of joints. For this reason, considerable work has been carried out in order to create 3D representations of the foot and ankle via 3D imaging techniques. The main idea behind 3D imaging techniques is to acquire full 3D internal anatomic information via an imaging modality such as
∗ Tel.: +1 252 532 5343. E-mail address: [email protected]. 0895-6111/$ – see front matter © 2009 Elsevier Ltd. All rights reserved. doi:10.1016/j.compmedimag.2009.01.002
computed tomography or magnetic resonance imaging. Images are acquired in each of a number of positions of the foot [1–6]. Usually, a positioning device is used to hold the joint in each position. Often stress is applied to study the flexibility characteristics of the joint or to test how the bones move under normal loading conditions [11,12]. How different types of injuries affect joint flexibility and motion characteristics and how different surgical reconstructions differ in their ability to restore flexibility and motion have been studied via multi position stress MRI [11,12]. Such images have also been used in building subject specific biomechanical models of the ankle complex and in verifying how realistic such models are in predicting joint mechanical behavior [17]. However studies are also carried out to examine bone motion without stress or in the condition of an open kinematic chain to understand differences among foot architectures, both normal and abnormal [7–10]. One of the main issues with imaging is to capture the true dynamic nature of the information about the joint in images. It would be ideal if enough 3D data sets can be acquired on the fly while the joint is subjected to a particular motion in the scanner. Unfortunately this is not feasible at present, and perhaps will not be feasible for quite some time to come. Although CT imaging has become extremely rapid in recent times, enabling the acquisition of a full volume image within a second, in this application, its use in patients may be hazardous due to excessive X-ray radiation,
D. Sekaran / Computerized Medical Imaging and Graphics 33 (2009) 242–246
243
Fig. 1. A slice of the calcaneus bone, taken from two different positions.
especially because multiple 3D image sets will have to be acquired. Although MRI is the safer option by far, it is also much slower. However, MRI pulse sequences have been developed for rapid single-slice kinematic imaging of joints [13–15]. The main drawback of such studies is that they permit the examination of 2D planar motions only. These studies and the possibility of a few-sliceonly rapid MRI led us naturally to the investigation undertaken in this paper. Suppose a full resolution 3D image In is acquired in one, say neutral, position of the joint, and suppose the goal is tracking the rigid motion of bones. If it is feasible, from a combination of image segmentation and registration techniques to accurately track the motion of a bone defined in In in different positions of the joint from rapidly acquired “a few-slice-only” MR images, then there is no need to acquire full resolution image sets in all other positions. This can reduce the acquisition time in kinematic studies considerably. All of the MRI scenarios described above – stress MRI, open kinematic chain MRI, and dynamic MRI – can benefit from the reduction ensuing from a “few-slice-only” study if it would become feasible. For example, in the first two situations, the total acquisition time will be reduced improving patient comfort especially in pain invoking positions. In the third situation, it may become feasible to carry out a 3D dynamic study instead of a single-slice kinematic study. Thus the investigation carried out in this paper can be beneficial to all three scenarios mentioned above. A fourth scenario, not discussed above, is imaging in natural weight-bearing conditions, such as in upright MRI scanners. Although this still does not correspond to the most natural condition of joint loading that occurs during various activities (such as walking, running, climbing, jumping), it comes closest to being a naturally active condition among the four scenarios, and here too the results of our study will be applicable. We close this discussion by stating that studies in all four scenarios have produced a useful understanding of the joint flexibility characteristics, architecture, kinematics and mechanics. In the present study, our focus is on the first two scenarios. Motivated by the above consideration, in this short paper, based on existing full resolution image data for different positions of the foot, an investigation is carried out as to how the accuracy of motion analysis may suffer as fewer and fewer slices in the image set for subsequent positions are taken in the analysis. 2. Materials and methods For this study, we used data from 10 different feet. MR images were taken of the foot and ankle in two different positions with the primary focus on the calcaneus, talus, navicular and cuboid bones. Each image set consisted of approximately 60 sagittal slice images.
A 1.5T commercial GE Signa MRI scanner was used. Two 3-in. single-loop RF coils and one 5-in. single- loop coil were placed on the two sides of the ankle and underneath the heel. They were configured as a multi coil receiver. The scanning protocol consisted of a 3D Fast Gradient Echo pulse sequence with a TR/TE/flip angle of 11.5 ms/2.4 ms/60◦ , a 512 × 256 in-plane acquisition matrix, a ±31.2 receiver bandwidth, and a 180 mm field of view. Sixty 2.1 mm-thick contiguous sagittal slices were collected to cover the foot from the medial to the lateral aspect. The first imaged position corresponded to the neutral joint position [7,11] and the second represented an inversion with a 3.4 Nm moment applied via the positioning device. An Ankle Loading Device was used as described in [11] for holding and stressing the foot and for acquiring the images. Let I1 and I2 denote the original acquired images corresponding to the two different positions of the same foot. Multiple slices were removed from the set I2 , using the software system 3DVIEWNIX. The removal of slices was evenly distributed, and resulted in reduced sets consisting of anywhere between 20 and 50 percent of the original number of slices. The MR slices which captured the polar ends of the bones contained superficial data, so they were omitted from the study to reduce unnecessary error. Before forming any three-dimensional models, the binary slices were always interpolated via shape-based interpolation, in order to smoothen the edges and reduce errors in analysis. The goal of this analysis then was to determine, how close the motion of a given bone (say, talus) determined by matching its full image I1 to the sub-sampled image J2 is, to the motion estimated by matching the full resolution data I1 to I2 . Let the respective binary images representing the particular bone be I1b , I2b , and J2b . In these images, the entire bone region – boundary and interior – is represented by 1-valued voxels. Since the segmentations I1b , I2b of the original images, carried out carefully in a slice-by-slice manner, were available for this study, the creation of J2b was straightforward. In all registration experiments, the gray image pertaining to just the bone of interest was obtained by masking with the binary image corresponding to the bone (Fig. 1). That is, the gray image I1 = I1 ∩ I1b was registered with J2 = J2i ∩ J2bi where J2i denotes an image resulting from linear interpolation of J2 , and J2bi denotes a binary image resulting from shape-based interpolation of J2b . The similarity metric that was used for registration was mutual information [16]. J2 was created at five different subsampling levels: 100%, 50%, 33%, 25% and 20%. Per our notation, at 100% sub-sampling, J2 = I2 which means no slices were removed, and thus the percent indicates the fraction of the slices retained. The registration experiments were performed separately for the four bones of the peritalar complex of the foot: talus, calcaneus, navicular, and cuboid. In all cases, the registration of I1 to J2 yielded the rigid transformation matrix.
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D. Sekaran / Computerized Medical Imaging and Graphics 33 (2009) 242–246
Fig. 2. Transformed models of the calcaneus, talus, cuboid, and navicular bones (clockwise from top left).
Two methods were used to assess the influence of sub-sampling on motion analysis. 2.1. Method 1 In this method, the accuracy of registration was expressed by a measure of overlap between the original full resolution segmented region I1b that was transformed by the rigid registration transform and the sub-sampled binary image J2b . The transformed I1b will be b . To make the overlap measure fair, I b was interpodenoted by I1t 1t b so that the slices in I b matched with those in lated to yield I1ti 1ti b = I b since J b = I b , and no J2b . When sub-sampling level is 100% I1ti 1t 2 2 interpolation is needed. The overlap measure OM was normalized, so that, in the ideal case of 100% sub-sampling level, OM equaled 1.
OM =
b ∩ J b |/|I b ∪ J b | |I1ti 2 2 1ti b ∩ I b |/|I b ∪ I b | |I1t 2 1t 2
Here |x| denotes the number of 1-valued voxels in the binary image x. 2.2. Method 2 This method focused on the accuracy of estimating the actual movement of bones rather than on the delineation of bones. Accuracy was separately quantified for rotation and translation which resulted from helical axes motion description from one position to another. The translation and rotation coming from matching I1 to I2 ∩ I2b was considered perfect, and deviations, from these values, of the translation and rotation derived by matching I1 to J2 were computed. 3. Results The sub-sampled datasets (J2b ) were interpolated and converted into three-dimensional models using 3DVIEWNIX. The models were then superimposed upon the 3D representation from the transb ), to provide a qualitative perspective of formed original data (I1t the accuracy of the transformation (Fig. 2). The model from the original data is represented in pink. The blue shows the models formed when the original data are reduced by 50%, and the green models were formed from 20% of the data. The slices on each end of the bones contained inconsequential data which varied widely between the two positions. In this study, these slices were
removed for the sake of reducing unnecessary error in the transformation process; this led to the models being flattened out on the ends. The normalized overlap measure OM was computed for the various sub-sampling schemes and bones. We calculated the mean values over the 10 datasets (Table 1). Variations in the accuracy of estimating translation and rotation are illustrated. A sample comparison of variation across subjects (Fig. 3) and the general trends of variation across the four bones are demonstrated (Fig. 4). Due to their similarity, the deviations across all three helical axes were averaged for the purpose of simplifying the graphs. 4. Discussion Using Method 1, we were able to qualitatively observe the effects of reducing the number of slices. It appears that halving the number of slices does not seem to have a significant effect on the quality of the 3D model (Fig. 1). Many of the fine details of the bones are still present, and the transformation appears to be accurate. However, the dataset which contains only 20% of the original slices is noticeably cruder, with only the basic shape of the bone being evident. However, the OM values are useful as a reference to be used alongside the 3D models. These values are sufficiently high, being over 0.9, even when two thirds of the slices are discarded. We note that, in evaluating image segmentation methods using an overlap measure (true positive), even an 85% overlap (OM = 0.85) is considered to be very good. While 90% overlap can be considered to be very good, over 95% signals excellent agreement with truth. Interestingly, discarding about one half of the data does not seem to make a significant dent on the overlap measure. It should also be noted that the calcaneus and talus – the larger bones – yield noticeably higher OM values than the navicular and cuboid bones. This seems reasonable, as the sets for the larger bones contain more overall data, and would thus be less affected by the removal of slices. The Table 1 Mean OM values over 10 datasets for the four bones for the different sub-sampling schemes.
20% 25% 33% 50% 100%
Calcaneus
Talus
Navicular
Cuboid
0.8858 0.9121 0.9412 0.9749 1.0000
0.9032 0.9231 0.9427 0.9672 1.0000
0.8273 0.8597 0.9048 0.9571 1.0000
0.8548 0.8960 0.9522 0.9653 1.0000
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trend becomes more evident for any datasets which are more than 50% reduced. While the first method of evaluation is helpful in understanding how registration is influenced by sub-sampling I2 , the second method yields directly the discrepancy in motion estimation as a function of the degree of sub-sampling. It was ineffective to analyze motion information in terms of percent error, so the average deviations of rotation and translation were computed and compared. The data for the individual subjects displayed the same general trends for all cases, with few major outliers. We note that the degree of inversion rotation varied among the feet studied within our database and not particularly controlled to make them uniform. When the results are averaged across subjects and categorized by bone, the quality of the reduced datasets and its influence on motion estimation are much clearer. An especially strong trend is observed in the variance of rotation. For all four bones, reducing the data by 50% results in a deviation of approximately 0.5 degrees from the original coordinates. With respect to the qualitative observations from Method 1, this seems to be the range for acceptable quality. The values for the other reduced datasets yielded equally narrow ranges of results, all within 1 degree of each other. The trends for translation were present, but not as strong; they appeared to be more dependent on which bone was represented. However, the ranges were still within 1 mm, regardless of the amount of reduction. Once again, smaller bones such as the navicular and cuboid displayed noticeably lower quality in the reduced datasets. Similar to rotation, the 50% reduced data resulted in deviations lower than 0.5 mm. While the above results seem very promising, we cannot generalize and claim the same degree of data reduceability for other Fig. 4. Average variance of rotation (in degrees) and translation (in mms) for all subjects, categorized by bone.
types of joint motion. Because of the high degree of anisotropy of sampling involved, it is conceivable that the results may have a dependency on several factors such as the orientation of the slice plane, the shape of the bone, and the type of motion. A more comprehensive study is needed to fully understand the influence of these factors on the results.
5. Conclusion
Fig. 3. Comparison of variation of rotation (in degrees) and translation (in mms) for calcaneus bone—categorized by subjects.
Given the rapid growth of technology today, it may be possible to routinely use three-dimensional models to understand the morphology and kinematics of joints in the clinic in the near future. As of now, MR imaging is the safest and most effective way to acquire data for these studies. From the results of this experiment, we can conclude that it may be possible to drastically decrease the number of slices in a 3D model while maintaining a level of quality which would allow for effective modeling of joint movement. When the data are reduced by 50%, the resulting models nearly replicate the quality of the original, and the coordinates of transformation are accurate to within half a degree of rotation and 0.5 mm of translation. On the other hand, it is also clear that the size of the bone matters as well; reducing the slices of smaller bones could result in too much data loss to create an effective model. This observation should be noted when considering the target bones of a scan. However, in the larger picture, these results strongly suggest that it is possible to significantly decrease MR acquisition time, perhaps as much as two thirds, by lowering the number of necessary images for a functional 3D model.
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