Projective geometric model for automatic determination of X-ray-emitting source of a standard radiographic system

Depósito de Investigación de la Universidad de Sevilla https://idus.us.es/ This is an Accepted Manuscript of an article published by Elsevier in Computers in Biology and Medicine, Vol. 99 on August 2018, available at: https://doi.org/10.1016/j.compbiomed.2018.06.016 © 2018 Elsevier. En idUS Licencia Creative Commons CC BY-NC-ND Projective geometric model for automatic determination of X-rayemitting source of a standard radiographic system Laura García-Ruesgas* University of Seville, Department of Engineering Graphics. Isla de la Cartuja, Camino de los Descubrimientos, s/n, Sevilla 41092, Spain. Rafael Álvarez-Cuervo University of Oviedo, Department of Construction and Manufacturing Engineering. Campus de Gijón, Gijón 33203, Spain. Francisco Valderrama-Gual University of Seville, Department of Engineering Graphics. Isla de la Cartuja, Camino de los Descubrimientos, s/n, Sevilla 41092, Spain. José Ignacio Rojas-Sola University of Jaen, Department of Engineering Graphics, Design and Projects. Campus de las Lagunillas, s/n. Jaén 23071, Spain. * Corresponding author University of Seville. Isla de la Cartuja, Camino de los Descubrimientos, s/n, Sevilla 41092, Spain. Telephone: 34 (954) 486161 E-mail: [email protected] This is the accepted version for publication of the manuscript published in Computers in Biology and Medicine. Please, cite the published version. García-Ruesgas, L; Álvarez-Cuervo, R; Valderrama-Gual, F; Rojas-Sola, JI. Projective geometric model for automatic determination of X-ray-emitting source of a standard radiographic system. Comput. Biol. Med. 2018; 99: 209-220. DOI: https://doi.org/10.1016/j.compbiomed.2018.06.016 This accepted version of the manuscript is deposited under a CC-BY-NC-ND license. 2 Projective geometric model for automatic determination of X-ray emitting source of a standard radiographic system ABSTRACT Background and objective: Currently, many orthopedic operations are planned by analyzing X-rays. The exact position of the focus is needed to calculate the real size of an object that is represented in conical projection, although in practice, this position is difficult to determine using current X-ray commercial systems. In this paper, a new geometric model is proposed in order to determine accurately, practically, and economically the location of the emitting source of commercial imaging systems using a single standard X-ray image. Method: The method requires a specific reference locator object to be positioned in the visual field of radiographic image. Because this object cannot implement ideal geometric points, but instead works with small spheres, it was necessary to experimentally validate the proposed methodology. The implemented software that was developed to validate the model was used in four series of tests. In these tests, we studied the influence on the final result of: 1. the selection of a specific set of markers in radiography, 2. the focus position variation in relation to radiograph and 3. the possible rotated angle of locator object about Z axis. Results: The results for 164 tests that were performed with this software showed that the expected error for 99.5% of values ranges with maximum error of [-0.35%, +0.39%], which shows that the model is independent of the design of locator object and its position and orientation in the radiographic field. The software used to validate the proposed model has been found useful to verify its reliability, effectiveness, ease of implementation, and accuracy. 3 Conclusions: This model is effective to calculate the precise position of the X-ray focus of any standard radiographic system accurately. Keywords: projective geometric model, X-ray focus emitter, standard radiograph, specific locator reference object, MATLAB. 4 1. Introduction Currently, total knee arthroplasty is an efficient and reliable approach in orthopedic surgery [1,2]. Most surgery patients report satisfactory functional results and significant pain reduction, thus notably improving their quality of life [3]. Therefore, the number of primary and revision prostheses in recent years has increased markedly, especially in younger patients [4], and implant survival is estimated at 92.3% at 15 years after surgery [5]. A common procedure used in preoperative planning [6] of knee replacements consists of analyzing X-rays previously made for the patient. In view of the daily nature of hip and knee arthroplasties, it is becoming increasingly important to make accurate measurements on standard radiographic images in order to perform postoperative surveillance [7], to improve the design of prosthetic systems [8], to perform clinical studies concerning implants reliability which assess their possible wear or deterioration over time [9,10] or to determine the variation of the relative position of the prosthetic elements over long time periods [11]. In daily work, many analyses of radiographs are performed by measuring on them directly or by using two or three-dimensional overlapping templates, in order to determine by visual comparison, the optimal size of the implant to be inserted [12]. Despite being imprecise, the measurements with templates help surgeons adapt the knee implants and align the joint properly [13]. An inherent problem in these methods is that X-rays do not show the real size of radiographed objects, because the image is generated by conical projections. This involves the loss of information needed to perform certain studies [14], leading to the use of more costly techniques such as Computed Tomography (CT) [15] or Roentgen 5 Stereophotogrammetry Analysis techniques (RSA) [16], which also expose the patient to higher doses of radiation. The pinpointing of the X-ray focus position will enable these studies to be undertaken economically and with the necessary precision. In this paper, we propose a new projective geometric model to determine accurately the position in space of the emitting source of X-rays (X-RFE, X-Ray Focus Emitter) of a standard commercial radiographic system, i.e. to specify the ideal geometric point from which radiation of X-ray commercial tube is emitted. The commercial radiographic systems used in industrial and medical applications do not permit to determine the position of the focus with the accuracy required to perform high precision measurements on radiographed objects. The problem is the X-ray emitting tubes emit radiation from an inaccessible point inside a glass flask. In addition, tubes are placed inside lead capsules in order to avoid radiation leaks and their position, in relation to Xray machines, prevent any attempt to measure the position of focus directly. The proposed model works with radiographic images which were made with standard systems, (i.e. with a single focus), unlike RSA analysis systems, which require special radiographic machines with two focuses that act simultaneously. In practice, most hospitals do not have RSA systems because of their high price. All development of the proposed model is based on geometry that is associated with conical projection and classic concepts of homography have been applied. As it is known, it is essential to place a specific reference locator object in the visual field of radiographic image to restore the focus position in relation to projection plane. Because of the fact that, in practice, it is impossible to implement ideal geometric points, it was necessary to experimentally validate the proposed model and we used 6 small tantalum spheres for that. Tantalum was used because it is a radio opaque and physiologically inert material which is often used in medicine field. So therefore, the objectives of this research are: - First of all, to develop a geometric model that can be implemented as a practical computer program, for the accurate determination of Cartesian coordinates of the X-ray emitting source of any standard radiographic system. - To ensure that the aforementioned model can be applied in systems with a single Xray source and in experimental situations using a single radiographic image. - To design a specific Locator Object of Reference Markers composed of small tantalum spheres suitable for calculating the position of the focus of a radiation system from a single standard radiographic image. - The geometric model should be independent of Locator Object of Reference Markers used [17], as well as its position and orientation, considering the limitations of geometric arrangement of its markers and its number. 2. Materials and Methods 2.1 Geometric foundation It is proposed as a methodology to structure the development of the geometric model the ‘projective unity’ [18]. This methodology consists of the necessary minimum elements to define the conical projection of a point of the three-dimensional space: the focus or projection center, X-RFE, which is the reference point from which the conical projection is made, a marker –in theory, the O point of the space of the geometric center of said markerand its ф(O) projection on the projection plane (the radiographic image). The proposed problem satisfies the way homography is set out partially, so some of its equations are applied in the development of the projective model. 7 The coordinates (x,y) of the projected points are known in relation to a specific point of the plane, in the proposed case, the upper left corner of the radiograph. A single projective unit cannot determine the three coordinates of the projection center and hence more than one will be needed. Specifically, according to the principles of the homography, in order to restore an object, at least four projective units must be considered and the relative distances between four points of the object and their corresponding projections have to be known. The mathematical development requires a series of points distributed in space for which the coordinates are known in relation to a given reference system to be determined. Thus, the aim is to find the focus of a conical projection from the relative distances between points in space and the relative distances between the projections of these points. Initially, the position of points in space is arbitrary. To simplify, the origin of coordinates has been placed in the projection center. Since only relative distances between points will be considered, any point that is incorporated must be defined according to the first point O. Therefore, the distance h is to be determined in relation to the radiograph plane, and the coordinates of the point O (O1, O2, O3) in relation to focus, which is the origin of the coordinates. This information will allow to position the radiograph in relation to focus since coordinates of its upper left corner (X_rd, Y_rd , h) will be known. Once the first point O has been selected, the three needed rest points P1, P2 and P3, whose coordinates may be any one at first must be selected (figure 1). 8 Figure 1. General sketch of the proposed problem The relative distances between the last three selected points in relation to the first one O are known: (1) Distances between points projections of the object are also known: (2) These distances will be our only starting data and they will be measured in the radiograph. The parametric equation of the line through two points will be used as a mathematical model of each projective unity. This equation is very simplified because the origin of coordinates was placed in the focus: (3) As the plane is located at a height h in relation to focus, z = h and therefore λ = h/x3: (4) 15 Figure 7. Flow chart of software application 16 Figure 8. LEFERX interface displaying focus coordinates A special feature to consider when calculation is done is that two different reference systems are used. The first one, absolute reference system, has its origin of coordinates in the focus, while the second, relative reference system, has the origin of coordinates in the upper left corner of the radiograph. Projections of selected markers are chosen in this last system. It is taken into account that directions of locator object contour, match up with directions of the absolute reference system. Axes directions of this system are known on radiograph because they are aligned with the arrays of markers located at the top and underside of locator object. Axes directions of relative reference system are parallel to radiograph´s contour. Both reference systems must be moved to the first chosen marker to solve the model. This allows to calculate the existing rotated angle between both systems which must be taken into account to calculate focus coordinates. To this end, it is necessary to transform the relative distances calculated in relation to relative reference system, into distances in relation to absolute reference system (algorithm). Radiograph resolution is 17 also needed. Once focus position is calculated in space, its orthogonal projection is displayed on radiograph (figure 9). If we denoted the rotated angle as α, then the coordinates on the axes of the absolute reference system (x and y) would be calculated as follows in relation to the coordinates of the axes of the radiograph (x' and y'): (7) Figure 9. Representation of focus position on radiograph Finally, it´s enough to add the two known vectors (figure 10) to know the position of the origin of relative system, with regard to the X-ray emitting focus. Figure 10. Calculation method of coordinates of the upper left corner of image in relation to the focus 18 3. Results The inability to use geometric points as markers in practice [22], forced us to validate model accuracy experimentally under real use conditions with markers that are made up of small spheres, in order to determine the order of magnitude of the mistakes that this could cause. For that purpose, four series of tests were designed, the first three with synthetic radiographs, and the last series with real radiographs. This analysis methodology, using images both from virtual models and radiographs has previously been used for other experiments of identifying medical images with other techniques [23]. Obviously, in virtual tests, the exact position of the X-ray focus is known, so its position error can be calculated easily. On the other hand, it is necessary to mention that the set of obtained samples follows a normal distribution, so [µ-kσ, µ+kσ] interval corresponds to 99.5% of the values for k=2.578. 3.1. First series of tests The influence in final result of the selection of a set of four markers or other in radiograph was studied [24,25], and for this purpose, four synthetic radiographs were generated. Three mistake-estimate parameters were considered like in the rest of tests: the coordinates of focus determined, the coordinates of the first chosen marker that allowed to verify that mistakes show no characteristic pattern, and the resolution of radiographic image. Furthermore, fourteen different combinations of the four markers needed were selected in each synthetic radiograph to calculate the spatial position of focus, these combinations being equals in the four radiographs. A total of 56 tests were performed. One of the four analyzed synthetic radiographs was generated with the following parameters: h = 764,8mm O = (-46, -61, 469´5) Resolution = 4 pixels/mm 19 (X_rd, Y_rd) = ( -177´25, -214´875) The results in the other three radiographs, generated with completely different parameters to the first one, were similar. Figure 11. Error in the calculation of focus coordinates depending on selected markers Results of the first eleven combinations of the set of four markers show 99, 5% of the values are within a range of maximum error of [-0,35%, +0,39%] which is considered acceptable. Error variation in the height of the focus from the radiograph in relation to the exact position of focus does not exceed 0.3% in the confidence interval, leaving the maximum measured error in just over 2 mm in this case. The exact position of the first chosen marker is known in virtual tests, as it was said, and we can study its position error in order to look for possible patterns. In all tests that were performed, it was verified that there is no correlation between the error variation curves. As expected, it can be observed that there is a correlation between the position errors of this marker and those of the focus. There is no an appreciable error pattern in X and Y coordinates of the focus position either. In the last three combinations of set of markers, the third and fourth selected markers were placed in the same vertical line and this case has no solution as it was already said. 20 The implemented program prevents this situation beforehand by warning with an error message. 3.2. Second series of tests The influence on the final result of the focus position variation in relation to the radiograph in the directions of the X, Y and Z axes was studied. In specific, fifteen variations of position were considered for each axis of the coordinates, which were taken in increments of five millimeters over a range of ± 30mm, together with two other cases at distances of ± 50mm in order to check the behavior in the most remote areas. In Z axis, two more cases were added at distances of ±100 millimeters in relation to initial position. A total of 47 tests were performed in this series. In all cases the same markers were chosen in order to not to introduce any additional factor that could influence in result. As an example, the results when the focus of X-ray machine is moved in Y axis direction are shown. Results in the other two directions were similar. The radiograph was generated with the following parameters: h = 1200 mm O = (-50, -60, 900) Resolution = 5 pixels/mm (X_rd, Y_rd) = ( -166, -220) Figure 12. Error in the calculation of focus coordinates when the focus is moved in the Y axis direction 21 The results in the fifteen positions clearly show that expected error for 99.5% of the values considered is in a range of maximum error of [ -0.03%, +0.02%], so that, the influence on the result of the variation of the focus position in Y axis is practically null. The result of the error for h was, in all cases, lower than in the first series, as it in no case exceeded 0.01%, which was expected because there are no restrictions or dependences in the model regarding the focus position. As in the first series, no correlation was found between the percentage of position errors of the reference marker and the position errors in X and Y are also less than 0.01% in all test cases. 3.3. Third series of tests The influence on the final result of the possible rotated angle (α) about Z axis (the only axis is possible for a turn to take place in the normal real use of the locator object, which is set on a table of the x-ray machine) between the axes of the reference systems used, was studied. Angular variations of 15° in a range between 0° and 90° were considered first. To ensure the reliability of the model, regardless of the angle at which the locator object is rotated, a second analysis was made by turning the locator object an angle of 360° in intervals of 30°. Finally, since in practice it is not feasible to turn the locator object 360°, a third study was performed at angular intervals of 5° within ± 15°. It was necessary to consider the possibility of turning the locator object 180°. As in the previous case, the same markers were always chosen. In this section, the results by rotating the locator object 360º at 30º intervals are shown. The results in the rest of the tests were similar. Initial radiograph was generated with the following parameters: h = 1200 mm O = (-50, -60, 900) Resolution = 5 pixels/mm (X_rd, Y_rd) = ( -166, -220) 22 Figure 13. Error in the calculation of α and focus coordinates when the locator object is rotated The results in the twelve radiographs show that the expected error for 99.5% of the values is in a range of maximum error of ± 0.12%, which is considered acceptable. As expected, because of the trigonometric functions that implement programming languages, results of the error for h, show a relatively high error. This is because we must add this error to the signaling error, which increases the mean measure to a maximum of 0.06%, greater than that measured in the second series. Like the position errors of the reference marker, there is nothing different to note in this series of tests, nor errors of X and Y. Results of α error were illustrated in figure 13, as α is a value which influences on calculus of radiograph position in relation to focus. 3.4. Fourth series of tests Previous series of tests confirmed the effectiveness and accuracy of the proposed model using spherical markers [26]. However, a last series of tests was performed with real radiographs using a standard equipment for clinical use. Real radiographs were made only with the locator object, without the presence of the patient, because any movement of the latter could falsify the results. As it is not possible to use the real focus position in relation to the radiograph, in order to estimate the error, a first position was accepted as actual focus position, which was calculated by LEFERX software. Afterwards, the 23 focus of the X-ray machine was moved precise known distances in the three space coordinates and it was proven that differences between calculated solutions matched up with displacements that were done. Series of ten radiographs were made in each space coordinate at intervals of 10 millimeters each, so the range of variation between the first and the last one was 90 millimeters. Figure 14. Error in the calculation of focus coordinates when the focus is moved in the Z axis direction Results show 99.5% of the values are within a range of maximum error of [ -0.23%, +0.17%] which is considered acceptable. Results in all tests that were performed in the first experimental series, suggest the validity, in practice, of the proposed model and allow to affirm that small variations in the signaling of the centers of real markers generates minimal alterations in the result, and also that, the selection of the four markers is independent of the method accuracy. Based on results of second series of tests, it can be stated that the influence on the result of the variation of the focus position in the three coordinate axes is practically null and the independence of the geometric model is guaranteed in relation to the focus position. Results of the third series of tests proved the independence of the geometrical model in relation to the focus orientation on radiographs. On the other hand, the results of tests on 24 real radiographs hardly differed from those made on virtual radiographs, in relation to the error range of the coordinates. Therefore, it can be concluded that if tantalum markers which diameter is 0.5 mm are used, then it will be possible to ensure that the maximum error in each coordinate of the focus position on the radiograph will not exceed 0.8%. Accuracy is high enough to state that the geometric model that was proposed is reliable and accurate. Also, it can be said that the locator object prototype that was manufactured performs the objectives that were set initially, providing the necessary information to calculate the position of the focus in space. 4. Discussion Taking measurements on standard radiographs is a common method used in preoperative planning and postoperative surveillance in biomechanics field. Biomechanical studies made from radiological images are safe, versatile and economical but today, they suffer from poor precision [14,27], although an automated measurement method is used [28], which limits their field of application. In this paper, we present a novel geometric model based on projections in order to calculate accurately, practically and economically, the exact position of the focus required to perform accurate measurements over radiographs. The model was implemented as a practical computer program by a multidisciplinary group of engineers, providing a useful and friendly tool which can be used with any standard radiographic system, that is, a system available in any hospital. It is important to highlight that this model can be applied in systems with a single focus and a single radiographic image. A novel reference locator object was also designed and manufactured, which avoids introducing markers inside the human body. Model effectiveness was demonstrated by performing virtual simulations with synthetic radiographs prior to performing tests with 31 Design, analysis and verification of a knee joint oncological prosthesis finite element model Computers in Biology and Medicine, 54 (1) (2014), pp. 53-60 doi.org/10.1016/j.compbiomed.2014.08.021 [9] S. Lasurt-Bachs, P. Torner, F. Maculé, E. Prats, F. Menéndez-García, J. RíosGuillermo, A. Torrents Cross-linked polyethylene does not reduce wear in total knee arthroplasty Revista española de cirugía ortopédica y traumatologia, 62 (3) (2018), pp. 197-203. doi.org/10.1016/j.recote.2018.03.004 [10] H. Haider Reference Module in Materials Science and Materials Engineering Comprehensive Biomaterials II, 7 (2017), pp. 152-174 doi.org/10.1016/B978-0-12-803581-8.09359-0 [11] A.L.L. Oliveira, E.G. Cueva, R.T. Carvalho Failure analysis of the tibial component baseplate after total knee arthroplasty Engineering Failure Analysis , 36 (2014), pp. 147-154 doi.org/10.1016/j.engfailanal.2013.10.012 [12] A.C. Peek, B. Bloch, J. Auld How useful is templating for total knee replacement component sizing? The Knee, 19 (4) (2012), pp. 266-269 doi.org/10.1016/j.knee.2011.03.010 [13] M. Ettinger, L. Claassen, P. Paes, T. Calliess 2D versus 3D templating in total knee arthroplasty The knee, 23(1) (2016), pp. 149-151 doi.org/10.1016/j.knee.2015.08.014 [14] C.D. Stickley, J.J. Wages, R.K. Hetzler, S.N. Andrews, C.K. Nakasone Standard radiographs are not sufficient for assessing knee mechanical axis in patients with advanced osteoarthritis The journal of arthroplasty, 32 (2017), pp. 1013-1017 doi.org/10.1016/j.arth.2016.09.024 [15] D.Y. Ponzio, J.H. Lonner Preoperative mapping in unicompartental knee arthroplasty using computed tomography scans is associated with radiation exposure and carries high cost The journal of arthroplasty, 30 (6) (2015), pp. 964-967 doi.org/10.1016/j.arth.2014.10.039 [16] O. Muharemovic, A. Troelsen, M.G. Thomsen, T. Kallemose, K.K. Gosvig 32 The effect of personalized versus standard patient protocols for radiostereometric analysis (RSA) Radiography, 24 (2) (2018), p. e31-e36 doi.org/10.1016/j.radi.2017.11.006 [17] E.H. Garling, B.L. Kaptein, K. Geleijns, R.G.H.H. Nelissen, E.R. Valstar Marker Configuration Model-Based Roentgen Fluoroscopic Analysis Journal of Biomechanics, 38(4) (2005), pp. 893-901 doi: 10.1016/j.jbiomech.2004.04.026 [18] E.H. Barry, H.W. Eves Introducción a las transformaciones geométricas Mexico: Compañía Editorial Continental (1976) [19] R. Madanat, N. Moritz, H.T. Haro Three-dimensional computer simulation of radiostereometric analysis (RSA) in distal radius fractures Journal of Biomechanics, 40(8) (2007), pp. 1855-1861 doi.org/10.1016/j.jbiomech.2006.07.004 [20] S. Attaway MATLAB: a practical introduction to programming and problem solving (2nd Ed) Massachussets (USA): Elsevier (2012) [21] M.B. Williams, E.A. Krupinski, K.J. Strauss, W.K. Breeden, M.S. Rzeszotarski, K. Applegate, M. Wyatt, S. Bjork, J.A. Seibert Digital radiography image quality: Image acquisition Journal of the American College of Radiology, 4(6) (2007), pp. 371-388 doi: 10.1016/j.jacr.2007.02.002 [22] P.W. De Bruin, B.L. Kaptein, B.C. Stoel, J.H.C. Reiber, P.M. Rozing, E.R. Valstar Image-based RSA: Roentgen stereophotogrammetric analysis based on 2D–3D image registration Journal of Biomechanics, 41(1) (2008), pp. 155-164 doi: 10.1016/j.jbiomech.2007.07.002 [23] M.S. Emami, K. Omar (2013) A low-cost method for reliable ownership identification of medical images using SVM and Lagrange duality Expert systems with applications, 40(18) (2013), pp. 7579-7587 doi: 10.1016/j.eswa.2013.07.062 [24] R.Y. Cai, X.H. Yuan, C. Rorabeck, R.B. Bourne, D.W. Holdsworth Development of an RSA calibration system with improved accuracy and precision 33 Journal of Biomechanics, 41(4) (2008), pp. 907-911 doi: 10.1016/j.jbiomech.2007.11.012 [25] J. Ioppolo, N. Borlin, C. Bragdon, M. Li, R. Price, D. Wood, H. Malchau, B. Nivbrant Validation of a low-dose hybrid RSA and fluoroscopy technique: Determination of accuracy, bias and precision Journal of Biomechanics, 40 (3) (2007), pp. 686-692. doi: 10.1016/j.jbiomech.2006.01.012. [26] M Gammunto, S. Martelli, C. Trozzi, L. Bragonzoni & A. Russo A simulation environment for estimation of the performance of RSA cages Computers in Biology and Medicine, 38(9) (2008), pp. 1000-1006 doi: 10.1016/j.compbiomed.2008.07.007 [27] V. Sarwahi, S. Ayan, T. Amaral, S. Wendolowski, R. Gecelter, Y. Lo, B. Thornhill Can postoperative radiographs accurately identify screw misplacements? Spine deformity, 5(2) (2017), pp. 109-116 doi.org/10.1016/j.jspd.2016.10.007 [28] W. Wojciechowski, A. Molka, Z. Tabor Automated measurement of parameters related to the deformities of lower limbs based on x-rays images Computers in Biology and Medicine, 70 (2017), pp. 1-11 doi.org/10.1016/j.compbiomed.2015.12.027 [29] S. Ci-Bin, L. Jiing-Yih, C. Ren-Yi, S. Kao-Shang, C. Kuo-Jen, L. Shang-Chih Automatic model-based roentgen stereophotogrammetric analysis (RSA) of total knee prostheses Journal of Biomechanics, 45(1) (2012), pp. 164-171 doi.org/10.1016/j.jbiomech.2011.09.011 [30] S. Imai, K. Higashijima, A. Ishida, Y. Fukuoka, A. Hoshino, H. Minamitani Determination of the position and orientation of artificial knee implants using markers embedded in a bone: preliminary in vitro experiments Medical Engineering & Physics, 25(5) (2003), pp. 419-424 doi: 10.1016/S1350-4533(03)00037-7 [31] A. Bjork Facial growth in man studied with the aid of metallic implants Acta Odontologica Scandinavica, 13 (1995), pp. 9-34 doi: 10.3109/00016355509028170 [32] L. Montagna, L. Bragonzoni, M.L. Zarnpagni, A. Russo, M. Motta, U. Albisinni, M. Marcacci 34 Investigation into the detection of marker movement by biplanar RSA Medical Engineering & Physics, 27(8) (2005), pp. 641-648 doi: 10.1016/j.medengphy.2004.12.004 [33] R. Madanat, N. Moritz, E. Vedel, E. Svedström, H.T. Aro Radio-opaque bioactive glass markers for radiostereometric analysis Acta Biomaterialia, 5(9) (2009), pp. 3497-3505 doi: 10.1016/j.actbio.2009.05.038 [34] A.M.T. Choo, T.R. Oxland Improve RSA accuracy with DLT and balanced calibration marker distributions with an assessment of initial-calibration Journal of Biomechanics, 36(2) (2003), pp. 259-264 doi: 10.1016/S0021-9290(02)00361-5 [35] O. Muharemovic, A. Troelsen, M.G. Thomsen, T. Kallemose, K.K. Gosvig A pilot study to determine the effect of radiographer training on radiostereometric analysis imaging technique Radiography, 24(2) (2018), pp. e37-e43 doi.org/10.1016/j.radi.2017.12.003 [36] O. Muharemovic, A. Troelsen, M.G. Thomsen, T. Kallemose, K.K. Gosvig Design and evaluation of learning strategies for a group of radiographers in radiostereometric analysis (RSA) Radiography, 23(4) (2017), pp. e80-e86 doi.org/10.1016/j.radi.2017.05.015 [37] A. Cazasnoves, S. Sevestre, F. Buyens, F. Peyrin Statistical content-adapted sampling (SCAS) for 3D computed tomography Computers in Biology and Medicine, 92(1) (2018), pp. 9-21 doi.org/10.1016/j.compbiomed.2016.11.001 [38] M.A. Haidekker, L. Dain-kelley Morrison, A. Sharma, E. Burke Enhanced dynamic range x-ray imaging Computers in Biology and Medicine, 82(1) (2017), pp. 40-48 doi.org/10.1016/j.compbiomed.2017.01.014 [39] J.N. Cooper, D.L. Lodwick, B. Adler, C. Lee, P.C. Minneci, K.J. Deans Patient characteristics associated with differences in radiation exposure from pediatric abdomen-pelvis CT scans: A quantile regression analysis Computers in Biology and Medicine, 85 (2017), pp. 7-12 doi.org/10.1016/j.compbiomed.2017.04.003