MORPHOLOGY ANALYSIS OF HUMAN KNEE USING MR IMAGERY

MORPHOLOGY ANALYSIS OF HUMAN KNEE USING MR IMAGERY D. Chetverikov 1,2, G. Renner 1 1 Computer and Automation Research Institute, Budapest, Hungary; 2 Eötvös Loránd University, Budapest, Hungary We present a novel system for building a 3D model of human knee based on a sequence MR images. The system applies sophisticated modern image analysis and geometric modelling methods as well as graphical tools developed to investigate the morphology and functionality of human knee joint. The process of building a 3D model is described and illustrated by results for real measured data, followed by a discussion of system applications. Introduction The knee joint plays an essential role in the human motion. During human phylogeny, the anatomy of the knee has accommodated to the requirements of every day life activities, according to the interaction of form and function. The description of the kinematical geometry of the joint and the geometry of the contacting parts during motion has a primary importance from the point of view of the function of the knee. Deficiencies, deformations and deterioration in the human knee can be investigated by medical imaging techniques, such as X-ray, CT, MRI. In everyday clinical praxis X-ray is used, which shows anatomical structures in well defined projections. The knee is a complicated structure in space, therefore MRI or CT image sequences provide more detailed spatial information on the internal structures (bone, cartilage, ligament) than X-ray. By making visible the shape and morphology of some of the joint diseases these methods offer invaluable support to the practising physician and also to the orthopaedic surgeon. The same tools can be applied to the analysis of the kinematic behaviour of the knee. In the medical practice, MR and CT images are usually evaluated by a human evaluator. In the case of the human knee, however, this can be very difficult due to the low contrast of the anatomical components. In addition, the synovial fluid between the opposite cartilage surfaces of femur and tibia (upper and lower leg) blurs the images. Efficient and sensitive computer methods for contour detection and segmentation can improve the quality of evaluation. Based on the contours of the knee surfaces, accurate computation of distances, directions and angles within the joint is possible, which provides solid basis for orthopaedic handling and surgery. Many important properties of the healthy or pathological knee joint can only be evaluated in three dimensions. Pre-operative computer planning of surgical interventions also demand three dimensional representation of the knee. Consequently, a 3D computer model must be built using information extracted from the 2D images. Boundary surfaces of different anatomical structures appear as contours in MR images of the knee. The most important contours are that of the bone and the cartilage. These contours can be delineated manually by using the interactive facilities of the computer, or by an automatic process. Both approaches are based on the fact that different anatomical structures have different intensity values. Contour detection algorithms extract region boundaries as the lines where the intensity gradient is large. Most of the investigators in the literature of medical imaging [2, 4] use manually specified contour points and constructs a contour line by connecting and smoothing them. Patel et al. [6] reconstruct complete joint surfaces from MR images, but the emphasis is on recovering the knee motion by subsequent registrations, so accurate extraction of the contours is not needed. We have developed computer tools to extract bone and cartilage contours from a series of MR or CT slice images of the knee. They are carefully tailored to the specific contrast and shape features of knee images. Our aim was twofold: to provide a robust, easy to use tool for the physician to evaluate anatomical properties of the knee (shape defects, contact regions of

the joint, etc.) based on the contours; and to use them to build three dimensional computer models. Sequences of three-dimensional contours were used to reconstruct the geometry of tibia and femur. Because of the large number of the contours and the satisfactory contrast along the contours, automatic or semi-automatic contour detection is feasible. In addition to the gradients, geometric properties (continuity, smoothness) were taken into consideration to improve the quality of the contour. Contour detection proceeds slice by slice, and the procedure uses the result obtained for the previous slice. The deviation is measured and minimised, in order to maintain smooth transition between consecutive contours. The algorithm provides discrete 3D data points. Then continuous contour curves are created, which improves the quality of graphical representations and facilitates the mathematical analysis of the shape. The shape of a contour curve is influenced by the anatomical expectations of the physician. This paper presents the major image processing, segmentation and geometric modelling tools of our system. 1. Data acquisition and image analysis CT and/or MR image sequences of the knee are read in DICOM format A computer program was built for visualising the acquired data of the scans, using commercial programs [10]), assembled and modified according to the requirements of the knee investigations. The image sequences are stored on a 3D grid as a volumetric model, allowing to create images in the main anatomical directions (sagital, coronal, horizontal). Different image analysis and graphical tools have been developed to analyse the morphology of the knee. The MR images the system receives as input are of relatively low quality. Contrast stretching, adaptive nonlinear intensity mapping and noise removal are needed to enhance the slice images, improve their visibility and make them suitable for segmentation. While standard techniques can be used to enhance intensity, care should be taken not to blur the contours during noise removal. Adaptive histogram equalisation [7] helps enhance contrast and details; however, it may amplify noise as well. To remove noise without blurring the contours, we apply an adaptive filter that operates as follows. In a sliding window, the algorithm considers all pairs of pixels that are symmetric with respect to the central pixel of the window. From each pair, the pixel is selected whose value is closer to that of the central one. The selected pixels together with the central pixel form a set from which the output value, the mean of the set, is calculated. The selected pixel is likely to belong to the same class, either object or background, as the central one. This solution allows us to avoid averaging across contour that leads to contour blur. Figure 1 provides an example of blur-free MR image enhancement using the symmetric adaptive filter. original enhanced Fig. 1. Illustration of blur-free MR image enhancement using symmetric adaptive filter.

Fig. 2. Example of segmentation by contour expansion with the Fast Marching method. After enhancing all images of the MRI sequence, we proceed with image segmentation which is a critical step of system operation: the stability and accuracy of all further steps depend heavily on the bone and cartilage contours extracted during the segmentation. We have implemented and compared two different segmentation methods: Fast Marching and Active Contours. Below, we present the two methods separately. 1.1. Segmentation using Fast Marching and Level Set methods The Fast Marching method [8, 9] is an efficient segmentation algorithm. The method is applicable to bone segmentation if the intensity of the bone differs significantly from that of the adjacent tissue, that is, the contour of the bone is clearly visible. This condition holds for the noise-filtered MR images. The basic idea of Fast Marching is propagating a contour with a speed depending on local contrast. An initial closed contour must be given. In our case, this is a circle of small radius automatically selected within the bone area. A point (x, y) of the contour is moving outwards orthogonally to the contour with the speed determined by the intensity gradient: 1 v(x, y) = ɛ + I(x, y), where I(x, y) is the intensity value at (x, y), and ɛ = 1 is added for numerical stability. An example of propagation is shown in figure 2. Regions produced by the Fast Marching method are usually good, but sometimes they may contain outflows due to the occasional low contrast of the bone region contours. To detect an outflow, we calculate the area of the segmented region in each slice. A defective region is indicated if the relative absolute area variation with respect to the previous slice exceeds a certain limit. To correct segmentation errors due to outflow, we calculate the union of the regions in the slices adjacent to the erroneous one, then obtain the corrected region as the intersection of this union and the defective region. Figure 3 exemplifies the process of outflow correction. original corrected Fig. 3. Outflow correction: original and corrected 3D surfaces. The segmented regions obtained by the Fast Marching are noisy. To smooth them we use the Min/Max Flow which is a Level Set Method [9]. The method is based on the partial

Fig. 4. Region smoothing by Min/Max Flow. Top row: Original regions. Bottom row: Smoothed regions. differential equation I t = F min/max I, where I t is the temporal derivative of the time-varying image function I(x, y, t), the spatial gradient, F min/max the propagation speed. The latter is calculated as F min/max = { max( κ, 0) min( κ, 0) if average r (x, y) < 0 otherwise Here κ is the curvature, average r (x, y) the average intensity value within radius r around a point (x, y). The curvature is calculated as κ = I xxi 2 y 2I x I y I xx + I yy I x (I 2 x + I 2 y ) 3/2 An example of region smoothing with the Min/Max Flow is shown in figure 4. For very noisy images, the efficiency of the Min/Max Flow algorithm can be improved by replacing average r (x, y) by the median. The Fast Marching method can be extended to volumetric images. However, we do not use 3D Fast Marching because the outflow problem is more severe in the 3D case: an outflow in a slice can propagate to many voxels of the 3D volume. In 2D, an outflow can only affect the pixels of the processed slice. 1.2. Segmentation using Active Contours An alternative to the Fast Marching is the Active Contour method [1]. To initialise the Active Contour based segmentation, we apply manual initialisation of the contour following process. For a sequence of MRI images, the manual initialisation is done only once: for the slice image in the middle of the sequence. The time needed for the initialisation is negligible compared to the overall time required by data acquisition and processing. The semi-automatic approach improves robustness without significantly increasing the overall time. Figure 5 shows a sagital slice image with manually specified initial bone contours overlaid. Note that in the case of tibia (the lower bone) the initial contour is imprecise and not smooth enough. The contour is refined and made smoother by an iterative active contour algorithm [1]. Assuming the continuity of most of the data volume, the refined initial contour is then extended to a neighbouring slice by adapting it to the slice data. This is done by a similar active contour procedure. In this way, the algorithm propagates through the slices of the data until a

manual refined Fig. 5. Segmentation by Active Contours. Left: Manually specified contour. Right: Refined active contour. drastic change in the contour geometry occurs due to a significant change of the bone shape in that view. The cost function of the active contour includes a number of weighted components such as contour smoothness, intensity changes along and across the contour and the distance between neighbouring points of the contour. Adding a shape prior is planned to further improve the robustness of the procedure. We have compared the Active Contour based segmentation procedure to the Fast Marching method on numerous real data. Currently, we prefer using the Active Contour approach since it is usually more robust and precise. However, the Fast Marching method is also applicable, and it has the advantage of being completely automatic. 2. Building 3D model of the knee For detailed investigations of the knee, especially for motion analysis and designing surgical interventions, 3D modelling and visualisation of bone and cartilage surfaces are needed. Isosurface representation of spatial anatomical structures is provided by most medical imaging software. Using it physician get can an idea of the 3D morphology of the knee; however, it is suitable neither for exact representation of the shape nor for numerical evaluations. We have developed different types of surface reconstructions, triangulated surfaces and continuous surfaces, based on the information extracted from MR images. Triangulated representations of knee surfaces are sufficient for most medical investigations. High degree, smooth surfaces are needed for analysis of fine details and motion Relevant geometric properties (normal vectors, tangent planes, curvatures, plane intersections, etc.) can be computed for both surface representations, but the accuracies are considerably different. Figure 6 demonstrates examples of a measured point set and a triangulated set. A triangulated surface consists of very small triangles between data points that are consecutively connected to each other. The data points come from MRI scans, as the points of the corresponding contours. The sequence of contour points is positioned in 3D according to the spatial position of a slice containing the contour. In the case of cadaver studies, data points from internal anatomical structures can be collected by a (laser) scanning process. A good triangulation must meet several requirements: it must be topologically correct (no holes or flying triangles), must eliminate outlier points, triangles must have comparable side lengths and angles, their size must reflect the curvatures of the surface, etc. We have developed algorithms and computer programs to triangulate and decimate point sets of anatomical structures, which are able to handle imperfections coming from contour detection and measurements [11]. If necessary, topological corrections are performed, or the triangulation is decimated to reduce the

points triangles Fig. 6. A measured point set and a triangulated set. size of the data set. Figure 7 shows the femur and tibia as triangulated surfaces. Although they look quite smooth, the surfaces are not tangential continuous because of triangulation. femur tibia Fig. 7. Triangulated surfaces of the femur and tibia. For detailed biological or medical investigations, especially for studying the size and shape of the contacting regions of cartilage and motion of the knee, accurate and smooth representations of the functional surfaces of the knee are needed. Input data for creating high quality smooth surfaces is a point set coming from the contours of MR images, or from 3D scanning. The point set must be noise-filtered and decimated. First data points must be topological ordered, which is usually done by creating a triangulation over the surface points. Triangulation provides neighbourhood relations, which is important for fitting surfaces and for geometric calculations. Because we fit parametric surfaces commonly used in computer graphics and CAD (e.g. Bézier, B-spline, NURBS) over the point set, parameter values must be attached to the data points. This is performed by projecting the data points to a simple and rough approximating surface, which is usually the bicubic Coons surface defined by the boundary points of the point set [5]. The shape defining data of the surface are computed by minimising the functional F (C ij ) = N ( ) 2 S(uk, v k ) P k + λ k=1 S(u, v) = i,j B ij (u, v)c ij, S ( S 2 uu + 2S 2 uv + S 2 ) vv du dv where the continuous surface S(u, v) is fitted to N data points P k. The surface is defined by the basis functions B ij (u, v) and control points as shape parameters C ij. The functional contains

Fig. 8. Active surfaces of the femur and tibia. two terms. The first one is the squared distances of data points to the surface points with the same parameter value; it reflects the accuracy of the surface. The other one describes some geometric quantity responsible for smoothness, here the approximation of the surface curvatures (lower indices denote derivatives). Parameter λ (set by the user) defines the ratio between accuracy and smoothness. Minimisation of the functional defines control points of the surface and thus its unique mathematical representation. We have developed methods for parametrising point sets, solving the functional minimisation problem, and improving parameterisation during the fitting process; the details can be found in [11]. In figure 8 active surfaces of femur and tibia (where motion occurs) are shown, together with the underlying point sets. 3. Geometric evaluation There are numerous ways to analyse the shape and metric properties of knee surfaces, based on a precise geometric model. Various geometric properties such as tangent planes, normal vectors, extreme points, lines of intersections, distribution of mean and Gaussian curvatures, feature points, characteristic lines, etc. can be calculated, visualised and evaluated. Combining these procedures, we have developed computer programs that perform knee specific evaluations; either for clinical praxis or for preoperative surgical planning. As examples for both of them we demonstrate determination of the locus of contact points on the femur, and finding fixing points for crucial ligament replacement surgery. Fig. 9. Paths of contact points on the femur and tibia surfaces. Contact points (more precisely gravity centre of contact areas) between femur and tibia are important in detecting deficiencies in knee motion. They can be determined by analysing contour curves (looking for common points) of tibia and femur on images containing the contact point. When MR image sequences are acquired in different flexion angles of the knee, the path of the contact point on knee surfaces can be evaluated. Its location and different geometric properties (curvatures, torsions) are good indicators for different kinds of pathological states. Figure 9 shows paths of contact points on the femur and tibia surfaces; in the latter case, the top

view is given. The big difference in the lengths of the tibia and femur contact paths indicates the considerable sliding of the parts during flexion. Finding the motion path on the tibia surface is difficult. The motion path here is short compared to the femoral side, and the measured contact points are scattered, although in reality they probably form a single point or a very small area. A small side motion of in the knee could also contribute to the dispersion of the contact points. MRI sequences acquired for different knee flexions also facilitate detailed study of the knee kinematics, including the internal joint motions. First, precise geometric models of the femur and the tibia the cartilage surfaces are constructed. Using the robust registration algorithm [3], the transformation is then computed that brings the tibia surfaces in each flexion position into the starting position. The same transformation is applied to the femur surfaces, which gives the femur positions relative to the tibia. In this way three-dimensional motion of the knee can be investigated, and the curves describing the knee motion can be derived. Most important of such curves are the rotation and abduction as a function of the flexion. Sample rotation and abduction curves computed from the MRI sequence of a knee are shown in figure 10. Fig. 10. Rotation and abduction as a function of flexion. Important parameters of knee motion are the position, size and shape of the contact region between the tibia and femur during motion, investigated mainly on the tibia surface. These parameters can also be evaluated applying the above technique. Having an accurate computer model of the femur and tibia in a given position in space, distances between them can easily be computed and contact properties can be analysed. This is illustrated in figure 11 where the varying distances are shown by colour coding. Pictures for different flexions provide information on the variation of the contact region on the tibia surface during knee motion. Fig. 11. Contact region on the tibia surface during knee motion. Replacing seceded crucial ligaments (Anterior Crucial Ligament, ACL surgery) is an operation frequently performed in orthopaedic practice. The success of the operation is greatly affected by finding appropriate fixing points for the ligament on the tibia and femur surfaces. These points must maintain approximately constant distance during motion of the knee and

satisfy certain anatomical constraints. It is highly advantageous to determine them before operation and perform the surgical intervention using their position on the knee surfaces. Whenever MR image sequences of the patient s knee in different flexion angles are available before operation, it is possible to localise optimal fixing points for the crucial ligament. An accurate 3D model of the knee surfaces can be constructed using MR images and the above methods. With the spatial model all necessary geometric computations can be performed. Sample results of the distance evaluation are shown in figure 12. On the left-hand side the femur surface is coloured according to distances between a fixed point on the tibia and moving points on the femur; on the right-hand side ligament distances between the two points are shown as a function of the flexion angle. distance map ligament distance plot Fig. 12. Distance evaluation for ACL surgery. Conclusion Analysis of the geometric properties of the knee is important from many points of view. Based on shape information physicians can draw conclusions on the healthy and pathological state of the knee. Surgeons can design surgical intervention using geometric data of the knee. Better understanding of the morphology and functionality of the knee may lead to improvement of existing prostheses. Accurate geometric information facilitates preoperative design of knee surgery and computer control during surgery. We have developed efficient and robust tools to analyse MR and CT images, perform geometric calculations in 2D, reconstruct medical/biological surfaces from images and scans, register and merge them and evaluate them in 3D. Although the methods and programs were developed to satisfy specific aims and requirements of knee studies, many of the elements can be efficiently used to investigate similar biological structures. Our aim in the future is to develop a complete computer aided knee surgery and navigational system based on the image processing, geometric and graphic components discussed above. Acknowledments The medical and surgical expertise of this study was provided by Prof. G. Krakovits. The authors also acknowledge the valuable contributions of L. Szobonya, L. Hajder, F. Pongrácz and I. Kardos. References 1. A. Blake and M. Isard. Active Contours. Springer, 1998.

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