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This paper describes a method based on an automatic segmentation process to coregister carpal bones of the same patient imaged at different time points. A rigid registration was chosen to avoid artificial bone deformations and to allow finding eventual differences in the bone shape due to erosion, disease regression, or other eventual pathological signs. The actual registration step is performed on the basis of principal inertial axes of each carpal bone volume, as estimated from the inertia matrix. In contrast to already published approaches, the proposed method suggests splitting the 3D rotation into successive rotations about one axis at a time (the so-called basic or elemental rotations). In such a way, singularity and ambiguity drawbacks affecting other classical methods, for instance, the Euler angles method, are addressed. The proposed method was quantitatively evaluated using a set of real magnetic resonance imaging (MRI) sequences acquired at two different times from healthy wrists and by choosing a direct volumetric comparison as a cost function. Both the segmentation and registration steps are not based on a priori models, and they are therefore able to obtain good results even in pathological cases, as proven by the visual evaluation of actual pathological cases.

Diseases related to the wrist, such as rheumatoid arthritis (RA), have a major negative impact on quality of life because they are the most common causes of severe long-term pain and physical disability [

Even though conventional radiography is considered the traditional gold standard for evaluating bone erosions, poor performances (i.e., overall sensitivity, specificity, and accuracy of 25.5%, 98.3%, and 70.1%, resp.) have been reported for detecting wrist erosions [

For the diagnosis and monitoring of disease progression, a longitudinal study is necessary where the physician conducts several observations of the same subject over a period of time. A major objective is then the definition of a simple but effective procedure for volume registration while preserving the original intensity values. To this end, a simplified method is here proposed whose main objective is the achievement of a limited effect of interpolation-related blurring. The registration result allows the detection of changes in the morphological bone shape, the localization of changes with respect to the bone volume and surface, and the direct grey-level comparison after registration.

Repeatability of the wrist tomographic examination is affected by difficulties in perfectly reproducing the location and orientation of the anatomical district. Even if translation or angular rotation differences are expected to be relatively small, changes more strongly affect the coronal plane than the sagittal and axial planes. With the aim of a direct shape comparison, a 3D registration is then required to align two volumes acquired at different times so that they overlap and a quantitative analysis of changes becomes possible.

In this context, this work proposes a segmentation-based registration method allowing the overlapping of two MRI tomographic spatial sequences so that the shape of a bone at two different instant times can be compared. Such a unimodal 3D-3D multitemporal registration makes it possible to monitor disease progression in the rheumatic wrist. A rigid registration was chosen to avoid artificial bone deformations and to allow finding eventual differences in the bone shape due to erosion, disease regression, or other eventual pathological signs.

The automatic segmentation of the bone of interest from both acquired volumes is the starting point for a good estimate of the location and orientation parameters, which is here performed through the computation of the inertia matrix. The subsequent steps of registration are performed on the basis of principal inertial axes of each volume.

In contrast to more classical approaches, the proposed method suggests splitting the 3D rotation into successive basic rotations about one axis at a time (also termed elemental rotations). In such a way, only a single plane is affected at one time (i.e., coronal, sagittal, and axial plane), starting from the most important one, as indicated by the major principal inertia axis. In addition, singularity and ambiguity drawbacks affecting other classical methods (e.g., the Euler angles method) are addressed.

Depending on the carpal bones, stopping at one basic rotation is sometimes better than performing two rotations or the full 3D transform. In fact, the errors introduced by the interpolation process can be bound to one plane at a time and can be avoided in those planes where the error is larger than the expected registration benefit.

Such a result has been achieved by applying and quantitatively evaluating the proposed method to a set of real tomographic sequences from healthy subjects and by choosing as a cost function a direct volumetric comparison of a wrist bone acquired at two different times. Starting from such empirical results, the suggestion for the best rotation steps for each bone is derived and can be applied also in the pathological situation. The method can also be applied to pathological bones because the segmentation process is based on the tomographic acquisition grey-level values and does not make any use of atlas or anatomical models. Furthermore, bone morphological changes do not affect the registration procedure. The direct application of the method to such pathological cases has been compared with the classical Euler method, showing that singularity and ambiguity drawbacks do not affect the proposed solution.

Due to the good registration result, 3D visualization of the registered bone surfaces could be displayed to assess eventual morphological changes related to pathology evolution, or multiplanar reformatted slice cuts of the two registered volumes could be displayed, along with the original grey levels, to visually compare corresponding slices of a given bone.

Some aspects led to better results for a few bones when applying only one elemental rotation instead of all three (or even the classical rotation matrix):

Because the first rotation refers to the eigenvector associated with the largest eigenvalue, its effect is the most relevant, it is associated with the most informative content, and it aligns the two bones along their first principal axis.

The second and the third rotations refer to the second and the third principal axes, which are more critical and might be affected by a larger estimation error.

Even though repeatability cannot be assured in two different tomographic acquisitions, the flat shape of the hand and wrist assures that differences in the position and orientation likely affect the coronal sections more than the axial or sagittal ones.

The rotation angle might be so small that the errors introduced by the transformation and interpolation steps are larger than the benefit of the rotation itself.

The two major registration strategies can be classified into intensity-based and feature-based methods. An intensity-based approach compares pixel/voxel intensity by means of correlation metrics. Feature-based methods find correspondence between significant features such as points, lines, and surfaces, and a geometrical transformation is then determined to map the two volumes [

Longitudinal volumetric image analysis is an important topic, especially investigated for brain structure studies. The review paper by Mills and Tamnes [

Indeed, from the imaging point of view, brain morphology is very complex and registration turns to be a very crucial task where intensity-based procedures are mainly required. As an example, the research carried out in [

In [

The notable brain size and the structured grey-level content assure a reliable statistical information basis for such intensity-based approaches. In contrast, given their unique shape, hand bones are better suited for applications of feature- and geometrical-based methods. In addition, their small size, the MRI low field, and the low spatial resolution make it harder to use intensity-based algorithms which cannot be based on enough statistically significant data.

As previously reported [

According to previous reports [

After the identification of the significant features, their correspondence can be estimated using their descriptors, preferably invariant to the expected image deformation. A group of methods devoted to register satellite images uses moment-based invariants [

The mapping function design step, subsequent to the feature correspondence step, is devoted to designing a transform algorithm so that the corresponding features of the two volumes would be as close as possible. At this level, the type of mapping function is chosen, and its parameters are estimated.

Volume registration might be a global or a local process. In the present application, we were interested in coregistration of single bones to monitor the eventual disease evolution. To this end, even though the spatial configuration of the various bones strongly depends on the position of the anatomical part, this problem is solved by using a local registration, where the attention is focused on single bones (volumes of interest) separately. Because each bone is a rigid body, affine transformation is sufficient to achieve a good result and allows for a final morphological comparison.

Finally, the mapping functions constructed at the previous step are applied during the transform step so that the volumes are actually registered.

In the next paragraph, a short overview of the most widely used methods is reported, referring to mapping function design and parametric transformation models addressing the rigid registration. They focus on extraction and correspondence analysis of location and orientation descriptors of the objects of interest and propose a few analytical formulations for the registration solution.

In space science, the “attitude” of a flying object refers to its orientation with respect to a fixed inertial frame. In robotics, the estimation of orientation and location parameters is a key aspect of rigid-body kinematics. The problem addressed in the present paper has some similarities with these concepts, as the same bone is acquired in a different position and orientation by means of two (nonregistered) tomographic acquisitions performed at some time distance, so that the imaged bone volumes have a different orientation in 3D space.

A very detailed and mathematically sound review of related concepts is given in [

Let

As defined in [

In practice, the most common orientation representations are the rotation matrix, Euler angles, and unit quaternion. These constitute the basis of most attitude and orientation estimation techniques, and their properties and group structures are of great importance [

The Euler angle parameterization is not global, and angles are not well-defined for

Both Euler angles and unit quaternions are proven to have a corresponding rotation matrix. Using Euler’s theorem, which implies that anybody orientation can be specified in terms of a rotation by some angle about an arbitrary axis, unit quaternions can assume a formulation that gives the corresponding rotation matrix through the Rodrigues formula.

The rotation matrix is a global and unique representation of orientation. By using this matrix, corresponding voxels in coordinates of inertial (

Similar to Euler angles, the rotation matrix can be obtained as a product of three different rotation matrices, each corresponding to an elemental rotation about one of the axes of the fixed coordinate system or about the rotating coordinate system. However, as it was experienced in the present work, for the given application, it is sometimes better to avoid one or even two rotations.

In medical applications, registration methods based on principal axes have been applied in different cases of study such as in [

Referring to the present application, several papers exist in the literature about the registration of the whole wrist. Most of these studies correlate the registration with the wrist kinematic [

In [

Unlike in [

The proposed method is not intended to register the whole wrist where kinematic studies are required, and neither does it address the matching with some idealized model or atlas. On the contrary, the purpose of our work is the registration of individual bones, acquired at different times, to detect eventual changes and support the presence of eventual erosion and evolution of the rheumatic disease in general.

The method is supported by a robust, automatic, and reliable segmentation [

Based on absolute and central moments of the obtained 3D bone volumes, the proposed method takes the principal axes as feature descriptors and computes them starting from the inertial matrix. The transformation model approach proposes to align both bone volumes to the inertial frame to give them the same location and orientation in 3D space and to allow the subsequent shape comparison. To this purpose, instead of the direct application of one roto-translation matrix, the procedure proposes successive rotations on each 2D plane (along with appropriate translations). The optimization problem is then related to the possible registration parameter configurations, which describe the execution order of the 2D rotation axis (in the maximum of three rotations), the number of such rotations, and the direction of each rotation. Such geometrical considerations, described at Section

As a feature-based approach, the estimate of the transformation parameters and the possible configurations are computed in a deterministic way. The cost function adopted is the volumetric comparison between the two aligned bone volumes, as computed by the confusion matrix, which will be described later. Such quantitative metrics have been applied to the available database of real non-pathological cases to drive the selection of the best parameter configuration for each wrist bone type, as explained in the following section.

The fixed inertial frame

The carpal bones in different visualizations: (a) a coronal section from the original spatial sequence, (b) an axial, and (c) a sagittal section from multiplanar reformatting.

A graph-based segmentation [

In other words, the segmentation process (

For the two obtained VOIs, the principal axes are extracted as descriptors to be used in the feature correspondence step. They are represented by (

The fixed inertial frame

The centroid of both VOIs is computed by applying the first absolute moments of

The purpose of the method is to geometrically register

The above metrics of (

Specifically, sensitivity (SENS) and precision (PR) parameters are computed:

Instead of sensitivity versus 1-specificity, the plot of sensitivity versus 1-precision allows a more significant receiver-operating-characteristic (ROC) scatterplot to be drawn and will be used in Results to prove the performances of the proposed method.

As already mentioned, the proposed approach is an iterative method that avoids arbitrary rotations in the 3D space but performs sequential basic rotations. At each step, the central inertia matrix of the rigid body is computed together with the eigenvectors. From here, the rotation angle and direction are estimated. After each rotation, the volumes are changed, and then new parameters must be recomputed by means of a new inertia matrix.

From among the possible parametric configurations, the most appropriate one can be evaluated for each carpal bone through application to a dataset of real non-pathological volumes where no changes are expected.

After obtaining the eigenvectors of VOI

Let the projection of the first principal axis

Starting from the newly computed eigenvectors (as described at (

It can be experimentally demonstrated that the best result is sometimes obtained with less than three rotations. In any case, according to (

The proposed method has been implemented in Matlab and the processing time on a i7-7500U, CPU 2.70 GHz, RAM 8 GB computer is approximately 1 minute.

The proposed registration technique has been tuned and characterized by utilizing a set of synthetic volumes. After a

(a) One slice of the synthetic volume and (b) one slice of the rotated synthetic volume.

In addition, because the applied rotations are known, the error between the computed and actual angles can be evaluated, which is approximately

After evaluation of the synthetic volumes, the method was applied to real cases. These refer to wrist bones acquired in T1-weighted modality using a low-field ESAOTE Artoscan at 0.2 Tesla. All volumes were acquired in the coronal plane, with a resolution of 0.5 × 0.5 mm^{2} and image size of 256 × 256 pixels. Six patients were analyzed, all having an initial and a follow-up tomographic acquisition. Focus of the study was the following eight wrist bones: capitate, scaphoid, lunate, hamate, pisiform, trapezoid, triquetrum, and trapezium. The number of healthy bones on which the method was quantitatively evaluated was 35. A total of 13 pathological bones were also registered, but no quantitative measures were possible in such cases.

In Figure

A sample coronal slice from the initial MRI spatial sequence (a) and the follow-up spatial sequence (b).

When required, to make the volume isotropic, an interpolation is applied, thus making the voxel size equal to 0.5 × 0.5 × 0.5 mm^{3}. Once the isotropic volumes are obtained, the segmentation is applied with the purpose of extracting individual bones volumes

Bone configuration: two sample coronal slices from the first acquisition, along with the contours of the segmented bones.

As an example, a coronal section of the capitate bone as segmented from the two original volumes is shown in Figure

(a) A sample coronal slice of the capitate bone volume segmented from the first acquisition

For each bone, the proposed registration procedure is applied to volumes

3D surface visualization of the registered capitate bone (initial volume in green, follow-up volume in blue): from (a) first rotation in the coronal plane, (b) second rotation in the axial plane, and (c) third rotation in the sagittal plane.

Corresponding sample slices from the first initial MRI sequence (a) and the follow-up sequence (b) after registration of the capitate bone.

The metrics proposed in Section

Figure

ROC plot, where triangles correspond to one rotation, squares to two rotations, and circles to three rotations.

For each bone type, a triangle corresponds to the performance of the first basic rotation, a square to two rotations, and a circle to three rotations. As one can notice, all of the bones, except the trapezoid and triquetrum, obtained the best registration result with three rotations. For the other two bones, one rotation was sufficient.

The trapezium showed the most remarkable improvement when comparing the first, second, and third rotations. The same trend, although a little less considerable, characterized the hamate and capitate bones. For this last bone, the result of the quantitative assessment was in line with the visual evaluation. In fact, as shown in Figure

In general, the obtained results were satisfactory, with all showing sensitivity values larger than 0.89 and precision values larger than 0.77 (the worst result being reported for the triquetrum). The best registration result was obtained for the scaphoid, with a sensitivity value of 0.95 and precision value of 0.89.

As an argumentation with respect to the more classical Euler’s angles method, the same healthy cases were registered by applying the single classical rotation matrix (Direction Cosine Matrix), and a subsequent quantitative evaluation was carried out. Additionally, in this case, the Euler angles were estimated from the analysis of the principal inertia axes of the segmented bone volumes. As an example of a quantitative comparison, Table

Evaluation of Euler’s method and our method.

Scaphoid | Sensitivity | Precision |
---|---|---|

Euler’s method | 0.737 | 0.672 |

Our method | 0.949 | 0.891 |

The advantage of the proposed method is due to two main reasons. In fact, the Euler method suffers from well-known problems such as orientation ambiguity and non-singularity, which do not affect the proposed approach. In addition, with the proposed method, the best result could be obtained after evaluating one, two, or three rotations based on the specific bone (indeed, in the scaphoid case, three rotations were considered the best solution, as shown in Figure

The lower values for the sensitivity and precision of the Euler method are actually due to the fact that, for some scaphoid cases, the two volumes have been registered with a correct direction but a wrong orientation of one axis. In other words, the bottom and top of the two bones are not correctly overlapping.

After confirming that the registration results were satisfactory and accurate when dealing with non-pathological bones, the method was applied to pathological cases to verify its applicability for the assessment of disease evolution. As mentioned previously, rheumatologists affirmed the presence of rheumatoid arthritis on 13 different bones in the available database. The pathological signs only locally affected bone morphology, and the preserved similarity between the initial and follow-up bone volumes enabled a sufficiently good registration. In addition, the proposed segmentation method, which does not make use of a priori knowledge or an anatomical model, was able to extract in a correct way the bone volumes, despite erosion or other pathological signs.

As an example, a pathological trapezoid bone is shown in Figure

A pathological trapezoid bone: fusion of initial and follow-up volumes.

As a consequence, red pixels correspond to the erosion process, while green pixels indicate more probable errors of the registration processing chain (i.e., actual registration errors along with segmentation errors). Even in this given case, where significant erosion was shown to affect the bone, the registration result was very accurate, proving that the pathological sign does not invalidate the proposed method.

The proposed method, starting from intra-patient tomographic volumes acquired at different time points, proved to reliably coregister carpal bones, in order to detect eventual morphological changes. The approach operates directly on MRI spatial sequences, without any atlas or anatomical model and was also shown to be applicable to pathological cases. The retrospective analysis of given cases represents a feasibility study proving the effectiveness of the proposed method for tracking the progression of erosion or edema related to rheumatoid arthritis.

In contrast to other 3D model registration approaches, which are based on surface feature representations and matching, the present method is volume-based. Compared with the more classical Euler angles, it does not suffer from problems such as the ambiguity in axis direction and the fact that an infinite number of rotations cannot be parametrized. In addition, it is able to reduce the interpolation blurring effect.

The assessment of the method’s performance was based on healthy bones where no significant changes were expected from the initial to the follow-up study, and a quantitative evaluation was carried out, thus objectively demonstrating the correctness and validity of the method. In these cases, the registration purpose is a perfect result, where deviations are only due to errors or approximations associated with the phases of parameter estimation and interpolation, in addition to eventual segmentation errors. For the specific registration procedure, no phantom or other ground-truth standard data are required because the direct comparison of corresponding bones that are not changed over time is possible.

Indeed, the proposed method was also applied to pathological studies, the location and size of the pathology being identified by specialist physicians. Bone erosion only locally affected the bone volume of interest, and changes were minimal and limited, which showed that the method can produce satisfactory results in pathological cases as well. This approach allows tracking changes in bone morphology without these changes affecting the registration process.

When comparing with published literature, quantitative evaluation of individual cases is very seldom available, especially when dealing with longitudinal wrist bone analysis. In some cases, such as in [

In the statistical study presented in [^{3}) in that study, while it is 0.5 × 0.5 × 0.5 (mm^{3}) in the current one, the present performances compare favorably.

In conclusion, the present feasibility study, starting from low-field MRI spatial sequences, proved that it is possible to automatically quantify bone erosion by simple image processing algorithms, without a priori knowledge or models. Innovative aspects rely on the fact that the proposed method is an automatic, fast, and simple image processing procedure for quantitative analysis in longitudinal wrist studies. Being based on the only original MRI grey levels, no use of a priori knowledge or models is required.

After this preliminary study, a rigorous clinical validation of the method in pathological cases is planned as a future activity.

The authors declare that there are no conflicts of interest regarding the publication of this paper.

This work was supported by the Project MEDIARE: “New Methodologies of Diagnostic Imaging for Rheumatic Diseases”; PAR FAS 2007–2013 Program 4, Pos. no. 14.