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Precision of cortical bone reconstruction based on 3D CT scans
Computerized Medical Imaging and Graphics 33 (2009) 235–241
Contents lists available at ScienceDirect
Computerized Medical Imaging and Graphics journal homepage: www.elsevier.com/locate/compmedimag
Precision of cortical bone reconstruction based on 3D CT scans Jianping Wang a , Ming Ye a , Zhongtang Liu b , Chengtao Wang a,∗ a b
School of Mechanical Engineering, Shanghai Jiao Tong University, 800 DongChuan Road, Shanghai 200240, China Department of Orthopedics, Shanghai Sixth People’s Hospital, Shanghai 200233, China
a r t i c l e
i n f o
Article history: Received 25 March 2008 Received in revised form 22 December 2008 Accepted 6 January 2009 Keywords: Reconstruction Human bone Geometric anatomy model Medical image processing CT scan
a b s t r a c t The precision and accuracy of human cortical bone reconstruction using 3D CT scans was evaluated using machined bone segments. Both linear and angular errors were measured. Cadaver adult femoral and tibial cortical bone segments were obtained and machined in six orthogonal planes with a precision milling machine. CT scans were then obtained and the bone segments were reconstructed as digital replicas. Dimensional and angular measurements errors were evaluated for the machined bone segments and the results were compared with known dimensions based on milling machine settings to calculate errors due to scanning and model reconstruction. The model dimensional error in the coronal, sagittal and axial directions had a mean of 0.21 mm, with standard a deviation of 0.12 mm and a maximum error of 0.47 mm. The mean percent error was 0.74% and the maximum percent error was 1.9%. The angular error of models in the coronal, sagittal and axial directions was calculated, yielding a mean of 0.47◦ with a standard deviation of 0.37◦ and a maximum of 1.33◦ . The error in the cross-sectional axial direction had a mean of 0.54 mm with a maximum error of 0.83 mm, depending on the slice interval. The main error source was of the image processing, which was about 70% of the total error. We found that machining cortical bone segments prior to CT scanning is an effective method for accuracy evaluation of CT-based bone reconstruction. This method can provide a reference for assessing the sensitivity, reliability and accuracy of CT-based applications in the study of movement, finite element modeling, and prosthesis construction. Crown Copyright © 2009 Published by Elsevier Ltd. All rights reserved.
1. Introduction 3D medical imaging has been used for diagnosis, treatment planning and so on for decades [1]. As one of the method of 3D medical imaging, Computed Tomography (CT) reconstruction was usually used to create three-dimensional models of organs for modeling [2]. Typical orthopaedics-related modeling applications are relative kinematics of the normal knee in vivo [3–8], finite element analysis (FEA) [9–12] and custom-made implants design [13]. To create such models, it is usually necessary to process CT data to extract the bone geometry and then to convert this geometry into the form required by the specific application [14]. To be used in biomechanical modeling, reconstruction (extracted bone geometry) must satisfy topological consistency (it completely bounds a closed volume) and topological correctness (it accurately describes the bone geometry) [15]. Then, the particular and digital accuracy of reconstruction based on CT must be well known, which can be the references for further sensitivity analysis to determine the reliability and accuracy of movement and FE analysis [11,12]. Some researches had been
∗ Corresponding author. Tel.: +86 1350 1748168. E-mail address: [email protected] (C. Wang).
done about the accuracy of reconstruction based on CT. Scheerlinck et al. validated and assessed inter- and intra-observer reliability of a CT-scan based measurement tool to evaluate morphological characteristics of the bone-cement-stem complex of hip implants in cadaver femurs [16]. Viceconti et al. reported the accuracy of Discretized Marching Cube (DMC) and Standard Marching Cube (SMC) methods of three dimensional segmentation algorithms that was used to reconstruct the geometry of a human femur [14]. Hildebolt et al. had done an accuracy analysis of CT-based reconstruction of skull by measuring special points of five skulls respectively using a 3D electromagnetic digitizer and 3D reconstructed images CT based [17]. The former studies were mainly done by measuring feature points of bone or focused on parts of the whole reconstruction procedure. However, the accuracy analysis by measuring special bone points has limitations in identifying the anatomical landmarks [17]. Additionally, from the application point of view, the errors of entire reconstruction and the error contribution of each step of entire reconstruction are of more interest. As the contour of human bone was of complex surfaces, there was no golden standard to compare the difference of complex surfaces between real human bone and replicas one. However, 3D complex surface can be approximated by facets as long as the sampling is sufficiently dense [18,19]. In this work, the accuracy of
0895-6111/$ – see front matter. Crown Copyright © 2009 Published by Elsevier Ltd. All rights reserved. doi:10.1016/j.compmedimag.2009.01.001
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Fig. 1. 3D clouds models of in vivo human knee. (A) 3D clouds models of extensional knee. (B) 3D clouds models of nominal 30◦ flexional knee. (C): 3D clouds models of nominal 60◦ flexional knee. (D) 3D clouds models of nominal 90◦ flexional knee. (E) 3D clouds models of nominal 120◦ flexional knee.
regular bone segments was acquired by measuring the distances and angles between planes of six machined orthogonal planes from the bone segments. By measuring distances between arbitrary point pairs on bone surface of reconstruction, the error of which can be directly denoted by accuracy test results of regular
bone segments, and the accuracy of complex surface bone could be predicted. In this research, for the target of indirect prediction of reconstruction accuracy of complex surfaces bone, the accuracy of the entire reconstruction of machined bone segments (six
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orthogonal planes were machined out of bone segments) was assessed by acquiring not only linear but also angular errors in different directions from machined bone segments. As the direct bone measurement in vivo was almost impossible, machined cadaver bone segments test was designed and implemented, in which the relative regular bones reconstruction error was measured to avoid the error based on the identification of anatomical landmarks. The linear distances and angles between planes of machined bone segments of the cadaver bone were manually measured before CT-scanning. Finally, the digital replicas were also measured after the machined bones were reconstructed using the same method employed for the volunteer’s knee (used as an application example). These two data sets were measured respectively in the first step and final step of the reconstructed process. The overall error due to bone model reconstruction was illustrated by comparing the reconstructed model to the original machined model. The entire linear and angular reconstruction errors in different directions were obtained. These results can be used as a reference for accuracy prediction of CT-based bone reconstruction and the further sensitivity, reliability and accuracy of CT-based applications of movement analysis, FEA and so on. 2. Materials and methods
Fig. 2. Flowchart of the procedure and methods in this research. The left part in this flowchart is the principle target of this precision assessment. The right part is the procedure and methods of precision assessment used in this research.
2.1. 3D models reconstruction of human bone acquisition Computed tomography was performed using a spiral-imaging installation (GE Light Speed VCT 64). Medical-imaging protocols were standard: the slice interval was set to 1 mm, the image matrix was 512 × 512, the scan diameter was between 100 mm (upper limb) and 100 mm (lower limb), and the number of slices was 226. The collected images were stored on CD-ROM in DICOM 3.0 format prior to further processing. The stack of CT image slices was reconstructed into a 3D bone model using Marching Cubes algorithm, in which the three-dimensional reconstruction calculation had a maximal error of 1/2 pixel (mostly at a maximum of 0.2 mm). As an applied example, three-dimensional models of five flexional knees were reconstructed and were input into Imageware12.1 (EDS, USA) software as shown in Fig. 1, as which the procedure of test bones reconstruction were the same. 2.2. Experimental validation of model reconstruction In this research, the linear and angular errors of regular bone (six machined planes from the bone segments) segments in three orthogonal directions were tested to evaluate the accuracy of CTbased regular bones reconstruction, which can be used to denote the accuracy of linear distance between arbitrary points on human bone surface. Then, cadaver bone test was conducted. A flowchart of the procedure and methods in this research was illustrated in Fig. 2. The mean value of original and repeated caliper measurements was used as physical dimension, and the 3D reconstruction precision was defined as the difference between the software measurement and the physical measurement. Two femur and two tibia bones of human cadavers (provided by the Changzheng Hospital, Second Military Medical University, Shanghai) were cut into ten segments of three linear intervals – 20 mm, 50 mm, and 100 mm – and were machined with the Almighty Rocker milling machine (Model: X5325, China Hangzhou Milling Machine Manufacturing Co. Ltd.), which had a working accuracy of 0.02 mm/300 mm2 denoted by Flatness, as shown in Fig. 3A. One of the ten segments was shown in Fig. 3B. Six planar surfaces were machined from each segment, as shown in Fig. 4, forming a hexahedron. In fact, it is almost impossible to machine bones of complex surfaces into perfect box-shaped models. Six surfaces were extended,
respectively, based on machined surfaces, and rounded chamfers brought no error to caliper measurements. The measurements of real model and digital replica in this research were implemented on the machined planes. The machined bones were scanned and reconstructed using the same method as described in Section 2.1. Segment scanning procedure was illustrated in Fig. 3C, and one of the reconstructed segments is shown in Fig. 3D. The distance between each pair of parallel machined surfaces, is denoted by a1 , b1 , and c1 . The angles of the two adjacent surfaces such as surfaces 3–4, 3–5, 6–4, and 6–5 are, respectively, denoted by ˛1 , ˇ1 , 1 , 1 . All values for the distances and angles are the average of three data-points, which were manually measured. The software measurements for distances and angles are represented by, a2 , b2 , and c2 , and, ˛2 , ˇ2 , 2 , and 2 . The values of these measurements were acquired with the same method described above using the software package, Imageware12.1 (EDS, USA). The percentage of variation between sets of measurements (real vs. phantom) was determined with the following formula. %error =
3 Xi1 − Xi2 /3 i=1
X
× 100
where Xi1 is the real measurement, Xi2 is the phantom measurement, and X is the mean value of the real and phantom measurements. The reliability of measurements was assessed by calculating the mean of differences in the repeated measurements and their standard deviation. Student’s t-tests (SAS Institute Inc., 2002) were used to test the null hypotheses that there were no differences between the means of original vs. repeat measurements. To compare techniques, averages of measurements (original vs. repeats) were used. 3. Result Two sets of data were compared and computed in the software package of MATLAB. The maximum error of each data point was defined as the difference of the real and phantom measurements. Each bone segment was positioned along the principal axis of the marrow cavity, which was also the cross-sectional axis of the CT scan. Errors in the cross-sectional axial direction depended on slice
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Fig. 3. Validation of reconstruction of knee. (A) One of segments was machined. (B) One of segments machined. (C) Segments scanned by CT. (D) One of reconstructive segments.
Fig. 4. Machined planar surfaces of one of machined bones, where 1, 2, 3, 4, 5 and 6 refer to six machined surfaces that were mutual orthogonal. These six surfaces were extended, respectively, based on machined surfaces. The measurement of real and digital replicas in this research was implemented on the machined surfaces.
interval, which was set at 1 mm with potential maximum linear error in this direction to be 1 mm with an average error of 0.5 mm. The cross-sectional axial direction was defined as z-axial direction, and the other two directions were defined as the x-axial direction and y-axial direction. The terms of a, b and c refer to the linear error data of x, y, z directions, respectively, and d was used to denote the linear error of the sample’s combined dimensions of a and b. The terms of ˛, ˇ, and refer to the angular error data between adjacent planes which were vertical to the cross-section, i.e., surfaces of 3–4, surfaces of 3–5, surfaces of 6–4, surfaces of 6–5, respectively, and ı was used to denote the angular error of sample combined ˛, ˇ, and . The value of maximum, average error and standard deviation were given in Table 1, and terms of ıa , ıb , ıc , ı˛ , ıˇ , ı , ı refer to the error of a, b, c, ˛, ˇ, and , respectively. As shown in Tables 1 and 2, test results of the point clouds in the coronal and sagittal directions were 0.47 mm of maximum linear error, 0.21 mm of mean linear error, 0.12 mm of linear standard deviation, 0.74% of mean percent error, 1.9% of maximum percent error, 0.47◦ of angular average error, 0.37◦ of angular standard deviation, and 1.33◦ of maximum angular error. Errors in cross-sectional axial direction depended on the slice interval, and the maximum
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Table 2 Length a, b, c of segments and the percent errors.
Table 1 Maximum, average error and Standard deviation of segments. Number
ıa (mm)
ıb (mm)
ıc (mm)
ı˛ (◦ )
ıˇ (◦ )
ı (◦ )
ı (◦ )
Number
a (mm)
b (mm)
c (mm)
%ıa
%ıb
%ıc
1 2 3 4 5 6 7 8 9 10
0.22 0.10 0.20 −0.23 0.47 0.30 −0.41 −0.24 0.17 0.025
0.05 0.14 −0.13 0.33 0.10 −0.03 −0.27 −0.25 0.35 0.19
−0.83 −0.62 −0.76 −0.27 −0.45 −0.56 −0.73 −0.25 −0.47 −0.50
−0.1 −0.08 −0.97 −0.03 −0.42 −0.2 −0.13 −0.23 −0.3 −0.12
−0.97 −1.3 −0.58 −0.52 −0.48 −0.42 −0.2 −0.2 −0.93 −0
−0.9 −1.27 −0.6 −0.75 −0.42 −0.48 −0.17 −0.12 −0.3 −0.03
−0.78 −0.42 −0.07 −0.78 −0.37 −1.33 −0.88 −0.28 −0.37 −0.15
1 2 3 4 5 6 7 8 9 10
24.13 21.79 24.01 31.15 24.8 34.75 29.5 23.3 27.45 43.2
31.08 25.49 22.58 28.76 32.18 29.03 29.3 35.5 40.1 31.3
103.97 101.2 91.62 56.23 54.3 87.3 23.7 50.3 27.23 27.25
0.93 0.46 0.81 −0.72 1.9 0.86 −1.4 −1 0.61 0.06
0.18 0.55 −0.58 1.2 0.3 0.11 0.92 −0.7 0.86 0.62
−0.8 −0.62 −0.82 −0.49 −0.83 −0.64 −3.08 −0.5 −1.7 −1.8
0.18 0.11
0.54 0.19
0.26 0.26
0.56 0.38
0.5 0.37
0.54 0.39
Mean S.D.
0.23 0.12
Maximum, average error and standard deviation of segments, where the terms of ıa , ıb , ıc , ı␣ , ı , ı␥ , ı refer to the error value of a, b, c, ˛, ˇ, and , respectively. The terms of a, b and c, respectively, refer to the linear error data of x, y, z, and d was used to denote the linear error of sample combined a and b. The terms of ˛, ˇ, and refer to the angular error data between adjacent planes which were vertical to the cross-sectional surface, respectively.
Length a, b, c of segments and the percent errors of dimension, where %ıa , %ıb , %ıc were the percent error of dimension of a, b and c, respectively.
error was 1 mm and the maximum percent error of dimension was 3.08%. The discrete cloud points of linear and angular errors were computed with software package MATLAB, as shown in Fig. 5. 4. Discussion
Fig. 5. Distribution of errors of length and angle, where ı, , and , respectively, refer to the error value, mean, and standard deviation of real vs. phantom. The ordinate refer to the value of errors. The abscissa refer to the number of measurements on segments of bones.
As the contour of human bone was of complex geometry, there was no golden standard to compare the difference of complex surfaces between real human bone and replica one. However, 3D complex surface can be approximated by facets as long as the sampling is sufficiently dense [18,19]. Therefore, the accuracy of human bone reconstruction can be predicted by measuring the distances between arbitrary point pairs on human bone surface. In this research, the linear and angular errors of regular bone (six machined orthogonal planes from the bone segments) segments in three directions were tested to evaluate the accuracy of CT-based regular bones reconstruction, which can be used to denote the distances error between arbitrary points on human bone surface. Then the accuracy of human bone reconstruction could be predicted. Figs. 6 and 7 show this basic consideration. For anatomical geometry of human bone, generally the comparison measurement of choice of special landmarks was used to assess the accuracy of bone reconstructed model. However, the accuracy analysis by measuring feature bone points has limitations in identifying the anatomical landmarks [17]. In our research, additional error due to feature point selection was avoided. During the entire geometrical reconstruction process, the CT imaging modality is known to be of very high precision, with essentially no geometric magnification error [17]. Principal error-prone factors include: (1) the error of the image processing, (2) the measurement error introduced by caliper or protractor, and (3) the error brought in during machining. The first factor was considered as the main error source. In this investigation, the stack of CT image slices was reconstructed with Marching Cubes algorithm, which was widely adopted in surgery simulation applications [20]. The three-dimensional reconstruction calculation has a maximal error of 1/2 pixel size (mostly at a maximum 0.2 mm). Viceconti et al. reported that the peak error of SMC methods of three dimensional segmentation algorithms was 0.9 mm and an average error comparable to the pixel size (0.3 mm) [9]. For the second error source, the division value, measurement range and maximum permissible indicating errors of calipers and protractors, respectively, were 0.02 mm and 0.03◦ , 0–300 mm and 0–360◦ , ±0.02–±0.07 and 0.03◦ . The uncertainty of measurement result was respectively 0.02 mm of calipers [21,22]. These errors were no more than 10% of mean error of entire reconstruction. For the third error source, there is no doubt that there existed errors in machining of regular bone. However, all the measurement and reconstruction of bone in this study were conducted after bone seg-
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Fig. 6. Approximation of complex surface. (A) Part of complex bone surface. (B) Approximated by facets. (C) Points cloud.
ments were machined. Moreover, the measurement comparison between real bone and phantom was focused on the same bone and the same corresponding locations. So, the third error source was mainly from the choice of the corresponding locations. The measurement error introduced by the choice of location depended on the working accuracy of Milling Machine denoted by Flatness, which was 0.02 mm/300 mm2 and the measurement error introduced by caliper or protractor. One test was applied to analyze the measurement error introduced by choice of location, the mean error was of 0.04 mm of linear error and 0.08◦ of angular error, and the maximum error was of 0.08 mm of dimension and 0.12◦ of angular error. The results of the Student’s t-tests indicate that none of the differences between the means of measurements were significant (P > 0.05). It suggested that the error during machining was less than 20% of mean error of entire reconstruction. The above researches focused on the different stages of entire reconstruction. It was suggested that among the three error sources, the big part of the error was due to the image processing step, and it contributed 70% of average error of entire reconstruction. However, from the application point of view, errors of entire reconstruction are of more interest. The accuracy of a three-dimensional model derived from routine preoperative CT scans when used to model a specific dry human calvarium had been assessed by Remmler et al. [23]. In their study it was declared that the transcranial linear error did not exceed 1%. The difference of error values may be introduced by different methods of image processing algorithms. Scheerlinck et al. found that the average accuracy for measuring contours of femur bone was 0.27 mm (2.57%) [16]. Hildebolt et al. has shown a case of accuracy of CT-based reconstruction of skull of 0.2–3.4 mm of average error, of 0.14–13.18% of percent error, of 0.1 mm of minimum error, of 5.5 mm of maximum error [17]. In our research, models in coronal and sagittal axial direction was 0.47 mm of maximum linear error, 0.21 mm of mean linear error, and 0.12 mm of linear Standard deviation. The average percent error was 0.74% and the maximum percent error was 3.08%. Compared to others’ research results cited above, errors of our method for the entire reconstruction procedure were smaller. The reason is that different reconstruction and measurement procedures adopted. Models in our research were based on surfaces of the machined bone rather than complex markers. Hildebolt et
al. also noted that their analysis effectiveness was limited by the measurements of bone anatomical landmarks of dry skulls – 24% of anatomical landmarks could not be performed because they could not be confidently identified. In addition, for measurement of in vivo human bone, variations in shape, position, and orientation of feature points themselves between the real bone and phantom images result in additional errors. Consequently, errors encountered in the reconstruction of 3D medical images can be magnified. By comparing these results, it is evident that errors caused by selection methods and magnitudes of feature points themselves were avoided and there was a relatively minor error in our research. The accuracy of bone reconstruction depends on not only location precision but also orientation precision. In this research, linear and angular errors in different directions were given, and the precision of location was defined as error of linear distance and the precision of direction was defined as angular error. Angular error, which was denoted as difference between angles of two adjacent surfaces on the bone, was measured to be 0.47◦ of average angular error, 0.37◦ of angular standard deviation, and 1.33◦ of maximum angular error. The present measurement method provides more comprehensive and accurate method to calculate human bone reconstruction accuracy. Error sources and the reliability of reconstruction algorithm and measurement method had been discussed. For the anatomical geometry analysis of human bone, feature landmarks generally were selected to assess the accuracy of bone reconstructed. These kinds of accuracy analysis by special bone points measurements have limitations. However, direct measurement in vivo was almost impossible and the contour complexity of human bone, there was no golden standard to compare the difference of complex surfaces between real human bone and replica one. In this research, the accuracy of CT-based reconstruction was assessed by measuring the linear and angular errors of regular bone (six machined planes from the bone segments) segments in three directions. Because 3D complex surface can be approximated by facets, the method proposed in this research could be adopted to indirectly predict the accuracy of complex bone surface reconstructed. The results of this work suggested that bone reconstruction accuracy analysis method with bone models composed of orthogonal planar surfaces was an effective and applicable method. It could also be a reference for the further sensitivity, reliability and accuracy assessment of CT-based applications such as movement analysis and FEA. The method used in this research could be extended to other reconstruction applications of laser scan, MRI and so on.
Acknowledgements
Fig. 7. The accuracy of human bone reconstruction indirectly predicted by measuring the distances and angles between planes.
This work was primarily supported by the “Biotribological Key Basic Research of Implanting Prothesis” project funded by the National Natural Science Foundation of China (50535050) and further support was from the “Mechanical Virtual Human of China” project funded by the National Natural Science Foundation of China (30530230). We would also like to thank Anupam Pathak from the University of Michigan for his contributions and insight.
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[15] Ning P, Bloomenthal J. An evaluation of implicit surface tilers. IEEE Computer Graphics and Applications 1993;13(6):33–41. [16] Scheerlinck T, de Mey J, Deklerck R. In vitro analysis of the cement mantle of femoral hip implants: development and validation of a CTscan based measurement tool. Journal of Orthopaedic Research 2005;23: 698–704. [17] Hildebolt CF, Vannier MW, Knapp RH. Validation study of skull three dimensional computerized tomography measurements. American Journal of Physical Anthropology 1990;82:283–94. [18] Brandt W. Theory and application of the skeleton representation of continuous shapes. Dissertation. University of California, Davis; December 1991. [19] Huang Y-C, Lai J-Y. A fast error comparison method for massive STL data. Advances in Engineering Software 2008;39(12):962–72. [20] Cheung JT, Zhang M, Leung AK, Fan YB. Three-dimensional finite element analysis of the foot during standing – a material sensitivity study. Journal of Biomechanics 2005;38:1045–54. [21] Man Y, Sun C. The uncertainty assessment of measurement result of calipers. Enterprise Standardization 2003;06:26–7. [22] Yang Y, Zhao W. The uncertainty assessment of measurement result of protractors. Enterprise Standardization 2006;09:26. [23] Remmler D, Olson L, Ekstrom R, Duke D, Matamoros A, Matthews D, Ullrich CG. Pre-surgical CT/FEA for craniofacial distraction: I. Methodology, development, and validation of the cranial finite element model. Medical Engineering and Physics 1998;20:607–19. Wang Jianping was born in Jiaozuo, China, in 1969. He is a Ph.D. scholar in School of Mechanical Engineering, Shanghai Jiao Tong University, Shanghai, China. He specialized Biomedical Manufacturing and Mechanical Engineering for a while previously. Ye Ming is an Assistant Professor in Institute of Biomedical Manufacturing and Life Quality Engineering, School of Mechanical Engineering, Shanghai Jiao Tong University, Shanghai, China. Liu Zhongtang is an Assistant Professor in Department of Orthopedics, Shanghai Sixth People’s Hospital, Shanghai, China. Wang Chengtao is a Professor in Institute of Biomedical Manufacturing and Life Quality Engineering, School of Mechanical Engineering, Shanghai Jiao Tong University, Shanghai, China.
Contents lists available at ScienceDirect
Computerized Medical Imaging and Graphics journal homepage: www.elsevier.com/locate/compmedimag
Precision of cortical bone reconstruction based on 3D CT scans Jianping Wang a , Ming Ye a , Zhongtang Liu b , Chengtao Wang a,∗ a b
School of Mechanical Engineering, Shanghai Jiao Tong University, 800 DongChuan Road, Shanghai 200240, China Department of Orthopedics, Shanghai Sixth People’s Hospital, Shanghai 200233, China
a r t i c l e
i n f o
Article history: Received 25 March 2008 Received in revised form 22 December 2008 Accepted 6 January 2009 Keywords: Reconstruction Human bone Geometric anatomy model Medical image processing CT scan
a b s t r a c t The precision and accuracy of human cortical bone reconstruction using 3D CT scans was evaluated using machined bone segments. Both linear and angular errors were measured. Cadaver adult femoral and tibial cortical bone segments were obtained and machined in six orthogonal planes with a precision milling machine. CT scans were then obtained and the bone segments were reconstructed as digital replicas. Dimensional and angular measurements errors were evaluated for the machined bone segments and the results were compared with known dimensions based on milling machine settings to calculate errors due to scanning and model reconstruction. The model dimensional error in the coronal, sagittal and axial directions had a mean of 0.21 mm, with standard a deviation of 0.12 mm and a maximum error of 0.47 mm. The mean percent error was 0.74% and the maximum percent error was 1.9%. The angular error of models in the coronal, sagittal and axial directions was calculated, yielding a mean of 0.47◦ with a standard deviation of 0.37◦ and a maximum of 1.33◦ . The error in the cross-sectional axial direction had a mean of 0.54 mm with a maximum error of 0.83 mm, depending on the slice interval. The main error source was of the image processing, which was about 70% of the total error. We found that machining cortical bone segments prior to CT scanning is an effective method for accuracy evaluation of CT-based bone reconstruction. This method can provide a reference for assessing the sensitivity, reliability and accuracy of CT-based applications in the study of movement, finite element modeling, and prosthesis construction. Crown Copyright © 2009 Published by Elsevier Ltd. All rights reserved.
1. Introduction 3D medical imaging has been used for diagnosis, treatment planning and so on for decades [1]. As one of the method of 3D medical imaging, Computed Tomography (CT) reconstruction was usually used to create three-dimensional models of organs for modeling [2]. Typical orthopaedics-related modeling applications are relative kinematics of the normal knee in vivo [3–8], finite element analysis (FEA) [9–12] and custom-made implants design [13]. To create such models, it is usually necessary to process CT data to extract the bone geometry and then to convert this geometry into the form required by the specific application [14]. To be used in biomechanical modeling, reconstruction (extracted bone geometry) must satisfy topological consistency (it completely bounds a closed volume) and topological correctness (it accurately describes the bone geometry) [15]. Then, the particular and digital accuracy of reconstruction based on CT must be well known, which can be the references for further sensitivity analysis to determine the reliability and accuracy of movement and FE analysis [11,12]. Some researches had been
∗ Corresponding author. Tel.: +86 1350 1748168. E-mail address: [email protected] (C. Wang).
done about the accuracy of reconstruction based on CT. Scheerlinck et al. validated and assessed inter- and intra-observer reliability of a CT-scan based measurement tool to evaluate morphological characteristics of the bone-cement-stem complex of hip implants in cadaver femurs [16]. Viceconti et al. reported the accuracy of Discretized Marching Cube (DMC) and Standard Marching Cube (SMC) methods of three dimensional segmentation algorithms that was used to reconstruct the geometry of a human femur [14]. Hildebolt et al. had done an accuracy analysis of CT-based reconstruction of skull by measuring special points of five skulls respectively using a 3D electromagnetic digitizer and 3D reconstructed images CT based [17]. The former studies were mainly done by measuring feature points of bone or focused on parts of the whole reconstruction procedure. However, the accuracy analysis by measuring special bone points has limitations in identifying the anatomical landmarks [17]. Additionally, from the application point of view, the errors of entire reconstruction and the error contribution of each step of entire reconstruction are of more interest. As the contour of human bone was of complex surfaces, there was no golden standard to compare the difference of complex surfaces between real human bone and replicas one. However, 3D complex surface can be approximated by facets as long as the sampling is sufficiently dense [18,19]. In this work, the accuracy of
0895-6111/$ – see front matter. Crown Copyright © 2009 Published by Elsevier Ltd. All rights reserved. doi:10.1016/j.compmedimag.2009.01.001
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Fig. 1. 3D clouds models of in vivo human knee. (A) 3D clouds models of extensional knee. (B) 3D clouds models of nominal 30◦ flexional knee. (C): 3D clouds models of nominal 60◦ flexional knee. (D) 3D clouds models of nominal 90◦ flexional knee. (E) 3D clouds models of nominal 120◦ flexional knee.
regular bone segments was acquired by measuring the distances and angles between planes of six machined orthogonal planes from the bone segments. By measuring distances between arbitrary point pairs on bone surface of reconstruction, the error of which can be directly denoted by accuracy test results of regular
bone segments, and the accuracy of complex surface bone could be predicted. In this research, for the target of indirect prediction of reconstruction accuracy of complex surfaces bone, the accuracy of the entire reconstruction of machined bone segments (six
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orthogonal planes were machined out of bone segments) was assessed by acquiring not only linear but also angular errors in different directions from machined bone segments. As the direct bone measurement in vivo was almost impossible, machined cadaver bone segments test was designed and implemented, in which the relative regular bones reconstruction error was measured to avoid the error based on the identification of anatomical landmarks. The linear distances and angles between planes of machined bone segments of the cadaver bone were manually measured before CT-scanning. Finally, the digital replicas were also measured after the machined bones were reconstructed using the same method employed for the volunteer’s knee (used as an application example). These two data sets were measured respectively in the first step and final step of the reconstructed process. The overall error due to bone model reconstruction was illustrated by comparing the reconstructed model to the original machined model. The entire linear and angular reconstruction errors in different directions were obtained. These results can be used as a reference for accuracy prediction of CT-based bone reconstruction and the further sensitivity, reliability and accuracy of CT-based applications of movement analysis, FEA and so on. 2. Materials and methods
Fig. 2. Flowchart of the procedure and methods in this research. The left part in this flowchart is the principle target of this precision assessment. The right part is the procedure and methods of precision assessment used in this research.
2.1. 3D models reconstruction of human bone acquisition Computed tomography was performed using a spiral-imaging installation (GE Light Speed VCT 64). Medical-imaging protocols were standard: the slice interval was set to 1 mm, the image matrix was 512 × 512, the scan diameter was between 100 mm (upper limb) and 100 mm (lower limb), and the number of slices was 226. The collected images were stored on CD-ROM in DICOM 3.0 format prior to further processing. The stack of CT image slices was reconstructed into a 3D bone model using Marching Cubes algorithm, in which the three-dimensional reconstruction calculation had a maximal error of 1/2 pixel (mostly at a maximum of 0.2 mm). As an applied example, three-dimensional models of five flexional knees were reconstructed and were input into Imageware12.1 (EDS, USA) software as shown in Fig. 1, as which the procedure of test bones reconstruction were the same. 2.2. Experimental validation of model reconstruction In this research, the linear and angular errors of regular bone (six machined planes from the bone segments) segments in three orthogonal directions were tested to evaluate the accuracy of CTbased regular bones reconstruction, which can be used to denote the accuracy of linear distance between arbitrary points on human bone surface. Then, cadaver bone test was conducted. A flowchart of the procedure and methods in this research was illustrated in Fig. 2. The mean value of original and repeated caliper measurements was used as physical dimension, and the 3D reconstruction precision was defined as the difference between the software measurement and the physical measurement. Two femur and two tibia bones of human cadavers (provided by the Changzheng Hospital, Second Military Medical University, Shanghai) were cut into ten segments of three linear intervals – 20 mm, 50 mm, and 100 mm – and were machined with the Almighty Rocker milling machine (Model: X5325, China Hangzhou Milling Machine Manufacturing Co. Ltd.), which had a working accuracy of 0.02 mm/300 mm2 denoted by Flatness, as shown in Fig. 3A. One of the ten segments was shown in Fig. 3B. Six planar surfaces were machined from each segment, as shown in Fig. 4, forming a hexahedron. In fact, it is almost impossible to machine bones of complex surfaces into perfect box-shaped models. Six surfaces were extended,
respectively, based on machined surfaces, and rounded chamfers brought no error to caliper measurements. The measurements of real model and digital replica in this research were implemented on the machined planes. The machined bones were scanned and reconstructed using the same method as described in Section 2.1. Segment scanning procedure was illustrated in Fig. 3C, and one of the reconstructed segments is shown in Fig. 3D. The distance between each pair of parallel machined surfaces, is denoted by a1 , b1 , and c1 . The angles of the two adjacent surfaces such as surfaces 3–4, 3–5, 6–4, and 6–5 are, respectively, denoted by ˛1 , ˇ1 , 1 , 1 . All values for the distances and angles are the average of three data-points, which were manually measured. The software measurements for distances and angles are represented by, a2 , b2 , and c2 , and, ˛2 , ˇ2 , 2 , and 2 . The values of these measurements were acquired with the same method described above using the software package, Imageware12.1 (EDS, USA). The percentage of variation between sets of measurements (real vs. phantom) was determined with the following formula. %error =
3 Xi1 − Xi2 /3 i=1
X
× 100
where Xi1 is the real measurement, Xi2 is the phantom measurement, and X is the mean value of the real and phantom measurements. The reliability of measurements was assessed by calculating the mean of differences in the repeated measurements and their standard deviation. Student’s t-tests (SAS Institute Inc., 2002) were used to test the null hypotheses that there were no differences between the means of original vs. repeat measurements. To compare techniques, averages of measurements (original vs. repeats) were used. 3. Result Two sets of data were compared and computed in the software package of MATLAB. The maximum error of each data point was defined as the difference of the real and phantom measurements. Each bone segment was positioned along the principal axis of the marrow cavity, which was also the cross-sectional axis of the CT scan. Errors in the cross-sectional axial direction depended on slice
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Fig. 3. Validation of reconstruction of knee. (A) One of segments was machined. (B) One of segments machined. (C) Segments scanned by CT. (D) One of reconstructive segments.
Fig. 4. Machined planar surfaces of one of machined bones, where 1, 2, 3, 4, 5 and 6 refer to six machined surfaces that were mutual orthogonal. These six surfaces were extended, respectively, based on machined surfaces. The measurement of real and digital replicas in this research was implemented on the machined surfaces.
interval, which was set at 1 mm with potential maximum linear error in this direction to be 1 mm with an average error of 0.5 mm. The cross-sectional axial direction was defined as z-axial direction, and the other two directions were defined as the x-axial direction and y-axial direction. The terms of a, b and c refer to the linear error data of x, y, z directions, respectively, and d was used to denote the linear error of the sample’s combined dimensions of a and b. The terms of ˛, ˇ, and refer to the angular error data between adjacent planes which were vertical to the cross-section, i.e., surfaces of 3–4, surfaces of 3–5, surfaces of 6–4, surfaces of 6–5, respectively, and ı was used to denote the angular error of sample combined ˛, ˇ, and . The value of maximum, average error and standard deviation were given in Table 1, and terms of ıa , ıb , ıc , ı˛ , ıˇ , ı , ı refer to the error of a, b, c, ˛, ˇ, and , respectively. As shown in Tables 1 and 2, test results of the point clouds in the coronal and sagittal directions were 0.47 mm of maximum linear error, 0.21 mm of mean linear error, 0.12 mm of linear standard deviation, 0.74% of mean percent error, 1.9% of maximum percent error, 0.47◦ of angular average error, 0.37◦ of angular standard deviation, and 1.33◦ of maximum angular error. Errors in cross-sectional axial direction depended on the slice interval, and the maximum
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Table 2 Length a, b, c of segments and the percent errors.
Table 1 Maximum, average error and Standard deviation of segments. Number
ıa (mm)
ıb (mm)
ıc (mm)
ı˛ (◦ )
ıˇ (◦ )
ı (◦ )
ı (◦ )
Number
a (mm)
b (mm)
c (mm)
%ıa
%ıb
%ıc
1 2 3 4 5 6 7 8 9 10
0.22 0.10 0.20 −0.23 0.47 0.30 −0.41 −0.24 0.17 0.025
0.05 0.14 −0.13 0.33 0.10 −0.03 −0.27 −0.25 0.35 0.19
−0.83 −0.62 −0.76 −0.27 −0.45 −0.56 −0.73 −0.25 −0.47 −0.50
−0.1 −0.08 −0.97 −0.03 −0.42 −0.2 −0.13 −0.23 −0.3 −0.12
−0.97 −1.3 −0.58 −0.52 −0.48 −0.42 −0.2 −0.2 −0.93 −0
−0.9 −1.27 −0.6 −0.75 −0.42 −0.48 −0.17 −0.12 −0.3 −0.03
−0.78 −0.42 −0.07 −0.78 −0.37 −1.33 −0.88 −0.28 −0.37 −0.15
1 2 3 4 5 6 7 8 9 10
24.13 21.79 24.01 31.15 24.8 34.75 29.5 23.3 27.45 43.2
31.08 25.49 22.58 28.76 32.18 29.03 29.3 35.5 40.1 31.3
103.97 101.2 91.62 56.23 54.3 87.3 23.7 50.3 27.23 27.25
0.93 0.46 0.81 −0.72 1.9 0.86 −1.4 −1 0.61 0.06
0.18 0.55 −0.58 1.2 0.3 0.11 0.92 −0.7 0.86 0.62
−0.8 −0.62 −0.82 −0.49 −0.83 −0.64 −3.08 −0.5 −1.7 −1.8
0.18 0.11
0.54 0.19
0.26 0.26
0.56 0.38
0.5 0.37
0.54 0.39
Mean S.D.
0.23 0.12
Maximum, average error and standard deviation of segments, where the terms of ıa , ıb , ıc , ı␣ , ı , ı␥ , ı refer to the error value of a, b, c, ˛, ˇ, and , respectively. The terms of a, b and c, respectively, refer to the linear error data of x, y, z, and d was used to denote the linear error of sample combined a and b. The terms of ˛, ˇ, and refer to the angular error data between adjacent planes which were vertical to the cross-sectional surface, respectively.
Length a, b, c of segments and the percent errors of dimension, where %ıa , %ıb , %ıc were the percent error of dimension of a, b and c, respectively.
error was 1 mm and the maximum percent error of dimension was 3.08%. The discrete cloud points of linear and angular errors were computed with software package MATLAB, as shown in Fig. 5. 4. Discussion
Fig. 5. Distribution of errors of length and angle, where ı, , and , respectively, refer to the error value, mean, and standard deviation of real vs. phantom. The ordinate refer to the value of errors. The abscissa refer to the number of measurements on segments of bones.
As the contour of human bone was of complex geometry, there was no golden standard to compare the difference of complex surfaces between real human bone and replica one. However, 3D complex surface can be approximated by facets as long as the sampling is sufficiently dense [18,19]. Therefore, the accuracy of human bone reconstruction can be predicted by measuring the distances between arbitrary point pairs on human bone surface. In this research, the linear and angular errors of regular bone (six machined orthogonal planes from the bone segments) segments in three directions were tested to evaluate the accuracy of CT-based regular bones reconstruction, which can be used to denote the distances error between arbitrary points on human bone surface. Then the accuracy of human bone reconstruction could be predicted. Figs. 6 and 7 show this basic consideration. For anatomical geometry of human bone, generally the comparison measurement of choice of special landmarks was used to assess the accuracy of bone reconstructed model. However, the accuracy analysis by measuring feature bone points has limitations in identifying the anatomical landmarks [17]. In our research, additional error due to feature point selection was avoided. During the entire geometrical reconstruction process, the CT imaging modality is known to be of very high precision, with essentially no geometric magnification error [17]. Principal error-prone factors include: (1) the error of the image processing, (2) the measurement error introduced by caliper or protractor, and (3) the error brought in during machining. The first factor was considered as the main error source. In this investigation, the stack of CT image slices was reconstructed with Marching Cubes algorithm, which was widely adopted in surgery simulation applications [20]. The three-dimensional reconstruction calculation has a maximal error of 1/2 pixel size (mostly at a maximum 0.2 mm). Viceconti et al. reported that the peak error of SMC methods of three dimensional segmentation algorithms was 0.9 mm and an average error comparable to the pixel size (0.3 mm) [9]. For the second error source, the division value, measurement range and maximum permissible indicating errors of calipers and protractors, respectively, were 0.02 mm and 0.03◦ , 0–300 mm and 0–360◦ , ±0.02–±0.07 and 0.03◦ . The uncertainty of measurement result was respectively 0.02 mm of calipers [21,22]. These errors were no more than 10% of mean error of entire reconstruction. For the third error source, there is no doubt that there existed errors in machining of regular bone. However, all the measurement and reconstruction of bone in this study were conducted after bone seg-
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Fig. 6. Approximation of complex surface. (A) Part of complex bone surface. (B) Approximated by facets. (C) Points cloud.
ments were machined. Moreover, the measurement comparison between real bone and phantom was focused on the same bone and the same corresponding locations. So, the third error source was mainly from the choice of the corresponding locations. The measurement error introduced by the choice of location depended on the working accuracy of Milling Machine denoted by Flatness, which was 0.02 mm/300 mm2 and the measurement error introduced by caliper or protractor. One test was applied to analyze the measurement error introduced by choice of location, the mean error was of 0.04 mm of linear error and 0.08◦ of angular error, and the maximum error was of 0.08 mm of dimension and 0.12◦ of angular error. The results of the Student’s t-tests indicate that none of the differences between the means of measurements were significant (P > 0.05). It suggested that the error during machining was less than 20% of mean error of entire reconstruction. The above researches focused on the different stages of entire reconstruction. It was suggested that among the three error sources, the big part of the error was due to the image processing step, and it contributed 70% of average error of entire reconstruction. However, from the application point of view, errors of entire reconstruction are of more interest. The accuracy of a three-dimensional model derived from routine preoperative CT scans when used to model a specific dry human calvarium had been assessed by Remmler et al. [23]. In their study it was declared that the transcranial linear error did not exceed 1%. The difference of error values may be introduced by different methods of image processing algorithms. Scheerlinck et al. found that the average accuracy for measuring contours of femur bone was 0.27 mm (2.57%) [16]. Hildebolt et al. has shown a case of accuracy of CT-based reconstruction of skull of 0.2–3.4 mm of average error, of 0.14–13.18% of percent error, of 0.1 mm of minimum error, of 5.5 mm of maximum error [17]. In our research, models in coronal and sagittal axial direction was 0.47 mm of maximum linear error, 0.21 mm of mean linear error, and 0.12 mm of linear Standard deviation. The average percent error was 0.74% and the maximum percent error was 3.08%. Compared to others’ research results cited above, errors of our method for the entire reconstruction procedure were smaller. The reason is that different reconstruction and measurement procedures adopted. Models in our research were based on surfaces of the machined bone rather than complex markers. Hildebolt et
al. also noted that their analysis effectiveness was limited by the measurements of bone anatomical landmarks of dry skulls – 24% of anatomical landmarks could not be performed because they could not be confidently identified. In addition, for measurement of in vivo human bone, variations in shape, position, and orientation of feature points themselves between the real bone and phantom images result in additional errors. Consequently, errors encountered in the reconstruction of 3D medical images can be magnified. By comparing these results, it is evident that errors caused by selection methods and magnitudes of feature points themselves were avoided and there was a relatively minor error in our research. The accuracy of bone reconstruction depends on not only location precision but also orientation precision. In this research, linear and angular errors in different directions were given, and the precision of location was defined as error of linear distance and the precision of direction was defined as angular error. Angular error, which was denoted as difference between angles of two adjacent surfaces on the bone, was measured to be 0.47◦ of average angular error, 0.37◦ of angular standard deviation, and 1.33◦ of maximum angular error. The present measurement method provides more comprehensive and accurate method to calculate human bone reconstruction accuracy. Error sources and the reliability of reconstruction algorithm and measurement method had been discussed. For the anatomical geometry analysis of human bone, feature landmarks generally were selected to assess the accuracy of bone reconstructed. These kinds of accuracy analysis by special bone points measurements have limitations. However, direct measurement in vivo was almost impossible and the contour complexity of human bone, there was no golden standard to compare the difference of complex surfaces between real human bone and replica one. In this research, the accuracy of CT-based reconstruction was assessed by measuring the linear and angular errors of regular bone (six machined planes from the bone segments) segments in three directions. Because 3D complex surface can be approximated by facets, the method proposed in this research could be adopted to indirectly predict the accuracy of complex bone surface reconstructed. The results of this work suggested that bone reconstruction accuracy analysis method with bone models composed of orthogonal planar surfaces was an effective and applicable method. It could also be a reference for the further sensitivity, reliability and accuracy assessment of CT-based applications such as movement analysis and FEA. The method used in this research could be extended to other reconstruction applications of laser scan, MRI and so on.
Acknowledgements
Fig. 7. The accuracy of human bone reconstruction indirectly predicted by measuring the distances and angles between planes.
This work was primarily supported by the “Biotribological Key Basic Research of Implanting Prothesis” project funded by the National Natural Science Foundation of China (50535050) and further support was from the “Mechanical Virtual Human of China” project funded by the National Natural Science Foundation of China (30530230). We would also like to thank Anupam Pathak from the University of Michigan for his contributions and insight.
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