Does pixel/voxel-size limit the measurement of distances in CBCT-tomography?

The necessity to obtain relevant structural information from tomographic images is an all-pervasive step in a host of clinical and research-areas. Cone Beam Computed Tomography (CBCT) is the imaging modality often used among the many available. Currently approaches to extract structural properties from raw CBCT-images, involve some manual intervention by experts to measure their properties, such as size and displacements of their geometrical structures. Regarding the factors limiting the precision of these measurements, such as voxel-size and image contrast, we find conflicting statements in the literature. It is therefore useful to provide accurate data under well-defined experimental conditions. Here we present a method and associated software to measure displacements of geometrical structures. We also determined the minimum measureable displacement and minimum detectable defect in terms of voxel size. We select as our geometrical structure a sample of bovine bone and to provide a set of defects, we drilled a pattern of holes into it. We determined the hole’s three-dimensional structures using confocal spectroscopy. In order to obtain the minimum measurable displacement, we acquired CBCT-tomographies containing a stationary reference and micro-metrically cnc controlled displacements of the sample. We then process these images with our software to extract the distances and compare them with the cnc displacements. All our processing includes a computational interpolation from the voxel-size of 0.35 mm corresponding to our CBCT-tomographies, down to 0.05 mm. We find that sample-displacements can be measured with a precision of ~ 20μ, 17 times smaller than the voxel-size of 0.35 mm. To measure the size of the holes using our CBCT-tomographies, we first register the holes onto a hole-free region of the sample with our software, then overlay the result with the three-dimensional structure obtained from confocal spectroscopy. We find the minimum detectable hole-size to be 0. 7 mm, twice the voxel-size.

The extraction of meaningful geometrical structures from digital images is a 6 fundamental problem in many research areas with innumerable applications, specially in 7 the medical areas [1][2][3]. Recently we have seen enormous advances in computer-based 8 methods, due to, inter alia: the overall increase of computational power, the 9 development of new algorithms and imaging techniques. The analysis of biomedical 10 images, which has the bonus of immediate practical application, has generated a 11 particularly strong impetus. Here structural quantitative imaging analysis has been 12 used in disease diagnosis, treatment, monitoring and surgical planning. Among the 13 imaging modalities employed in those applications, Cone Beam Computer Tomography 14 (CBCT) has been one of the most frequently used. 15 CBCT has several advantages, such as: the greater amount of information obtained 16 in comparison with 2D tomographic methods, lower dose delivery to the patient [4,5] in 17 comparison with conventional Computer Tomography [6] and relatively low cost. Yet, in 18 comparison with Computer Tomography, CBCT images are noisier and have lower 19 contrast resolution [7,8]. 20 Almost all of the applications of CBCT images require quantitative analysis of 21 geometric structures, the most important being the measurement their sizes and the 22 measurement of the distance between different structures. 23 The main limitations in the detection of a particular structure are voxel-size and 24 image-contrast. Periodic patterns with differing spacing have been traditionally used to 25 compare the precision of different setups, with the conclusion that although voxel-size 26 certainly is a limitation to infer the minimum spacing detectable, an accuracy smaller 27 than voxel-size is obtainable [7]. Limitations due to image-contrast have been analyzed 28 recently by [9], who report that up to 30 % false positives can result even for a contrast 29 of 80%. 30 Furthermore current approaches to extract geometrical properties from raw data 31 usually involve some manual intervention by experts. For example to extract distances 32 from 2-or 3-dimensional CBCT-tomographies a linear measure using a digital caliper is 33 often used to compare the accuracy of different experts [10,11]. With regard to the 34 precision of different types of physical measurements [12][13][14][15][16][17][18], we find statements that 35 voxel-size is not important [11,19], whereas a review asserts that the improvement due 36 to better resolution is not well quantified [20]. Yet the size of the defects studied is 37 rather large ( ≥ 3× voxel-size) and their geometrical characterization lacks precision.

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Therefore the minimum size of detectable defects is unknown. We conclude that these 39 methods merit further discussions.

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In this paper we want to address two issues concerning images acquired by CBCT 41 tomography: what is the precision with which we are able to 42 measure distances between displaced structures. For this purpose we manufacture a 43 sample of bovine bone, measuring ∼ 2cm × 3cm × 0.5cm, shown in Fig.(1C). We 44 actively displace this macroscopic sample and compare its CBCT-tomographies, 45 which also contain a fixed reference, before and after displacement. 46 obtain the minimum size of defects, using CBCT images. To produce the defects, 47 we drilled a pattern of holes into the sample shown in Fig.(1F). We measured the 48 three-dimensional structure of the holes using confocal spectroscopy [21]. Since 49 the holes have a contrast better than the usually available in real life, as e.g. in 50 the detection of the condylus [9], our sample was immersed in water to 51 approximate these real life conditions.

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To obtain the minimum detectable displacement we use the setup shown in Fig.(1A). 53 We accurately displace the sample using a micrometer with 10µ precision. We obtain CBCT-tomographies ( voxel-size = 0.35 mm ) of the whole setup shown in Fig.(1B) and 55 extract the region corresponding to the bone-sample before and after displacements. We 56 describe the registration algorithm used to register the sample.

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To obtain the minimum size of defects discernible with our algorithm, we first 58 register the holes onto a hole-free region of the sample using our algorithm. The For data-acquisition we prepared a sample with rectangular cross-section of 2cm × 3cm 64 and a depth of 0.5cm from a piece of bovine tibia-cortex bone, sectioned with a 65 diamond-shaped cutting machine (ISOMET 1000, Buheler IL, USA). The anatomical 66 structure of this sample should be typical for many real life situations and we therefore 67 expect our results to offer a useful benchmark for many applications. We then polished 68 the sample under constant irrigation with a polishing machine (Buehler, Lake Bluff, IL, 69 USA), using a sequence of silicon carbide granules (150,320,400,600,1200,1600,3000) and 70 finished it with diamond paste and felts.  Table 1. The axis of the holes will be called Z-axis, which points vertically. In order to infer the size of the smallest holes, which we can detect using tomographic 77 images processed by our algorithm, we need a precise enough measure of their spatial 78 geometry. We therefore used confocal microscopy [21] (Lext 3D Laser Measuring, 10 µ per division. We fixed a vertical PVC rod on the static part of the translation stage 94 providing a reference to measure distances from. We mounted a PVC support to hold a 95 Petri-dish containing the sample (Fig 1 A, B)  We aim to provide a benchmark for the precision with which distances between well 105 defined geometrical structures can be measured. For this purpose we select as 106 geometrical structure our macroscopic sample (with cross-section of 2cm × 3cm ) shown 107 in Fig.(1 C), whose displacement is to be measured. The structure to be displaced, 108 shown in Fig.(1 B) contains not only our sample, but also the dish filled with water and 109 the vertical PVC rod, which acts as a static reference. 110 We segment the volumetric image of the whole setup Fig.(1 B), using the region 111 competition algorithm [22] implemented in the ITK-SNAP segmentation software [23], 112 before and after displacement to extract the regions of interest: bone sample, circular 113 dish and rod. Yet the borders of the bone sample, which will be used to measure 114 displacements, are not always well defined. To refine their positions, we expand the 115 region labeled bone sample via a level set contour propagation [24] followed by an AND 116 operation between the bone sample region before and after expansion. A binary 117 threshold filter [25] is then sufficient to obtain a precisely labeled bone sample. This 118 eliminates eventual water regions adjacent to the bone sample. 119 We first increased the image-resolution by a factor of seven reducing the voxel-size 120 from 0.35 mm to 0.05 mm, using cubic spline interpolation [26]. We then registered only 121 the regions corresponding to the bone sample to compute the Distance Transform [27] 122 between the original and the displaced images. We define the displacement to be the 123 difference between the centroids of the original and the transformed bone sample regions. 124 The pseudo-code 1 for these steps runs as follows: 125 Algorithm 1: Displacement calculation input : input images: I orig , I displ output : displacement D T 1 ← rigid registration of I orig , I displ ; I r orig ← T 1(I orig ) ; T 2 ← demons registration of I r orig , I displ ; T ← T2 • T1; C ← centroid of I orig sample region;

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Hole detection 127 In order to measure the size of the minimal hole detectable with our registration 128 process, we again increased the resolution sevenfold to a voxel size of 0.05 mm and 129 applied a curvature anisotropic diffusion filter [26,28] to reduce image noise. We then 130 registered a region of the sample without holes onto the region with holes using the 131 demons algorithm [29] 132 We then overlayed the registered images with the hole's profile obtained by confocal 133 spectroscopy as follows. Given a hole's image, we measured the coordinates of the hole's 134 center. Out of the 3-dimensional image we cut a plane containing the center and To estimate the minimum size of the detectable holes, we used the images of the holes, 153 which have been registered by our algorithm onto the hole-free bone sample. We then 154 compared these registered images ( color-coded ) with the hole's profile obtained by 155 confocal spectroscopy ( in black ) in Fig(3). 156 1 The demons registration step T 2 in the algorithm is actually the identity here, since the original and the displaced sources for the CBCT-images are identical. It is included, because it is part of our software package, which aims at comparing differently distorted sources.  Table I, mapped out by  We see that holes of sizes ≥ 0.7 mm - Fig(3 A,B) -are very well detected, but not 157 the smaller holes. We note however, that the limiting factor here is not the 158 detection-algorithm, but the voxel-size, as already noted by [20], since the DICE 159 coefficient [30] turns out to be 0.99. Two experts with several years experience using 160 the semi-automatic ITK-SNAP software [23] to segment the image were unable to label 161 even the 0.5mm diameter holes. 2. the minimum detectable size of bone defects using those images.

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In our displacement measurements we used a bovine bone sample with cross-section 168 of 2cm × 3cm and a depth of 0.5cm. The anatomical structure of this sample should be 169 typical for many real life situations and we therefore expect our results to offer a useful 170 benchmark for many applications. We take special care to precisely extract the sample's 171 borders, since their displacements are to be measured. Computationally increasing the 172 image-resolution seven times, we are able to measure displacements with a precision of 173 20µ, which is ∼ 17 times smaller than the voxel-size of 0.35 mm. We also obtain a 174 minimum detectable defect-size of ≥ 0.7 mm, which was twice the voxel-size.

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Although we used data acquired with one tomograph only, with images acquired 176 using just one set of configurations, we expect our results to be very robust relative to 177 the voxel-size.