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We proposed the use of a hybrid deformable image registration approach that combines compact-support radial basis functions (CSRBF) spline registration with intensity-based image registration. The proposed method first uses the pre-viously developed image intensity-based method to achieve voxel-by-voxel correspondences over the entire image re-gion. Next, for those areas of inaccurate registration, a sparse set of landmark correspondences was defined for local deformable image registration using a multi-step CSRBF approach. This hybrid registration takes advantage of both intensity-based method for automatic processing of entire images and the CSRBF spline method for fine adjustment over specific regions. The goal of using this hybrid registration is to locally control the quality of registration results in specific regions of interest with minimal human intervention. The major applications of this approach in radiation ther-apy are for the corrections of registration failures caused by various imaging artifacts resulting in, low image contrast, and non-correspondence situations where an object may not be imaged in both target and source images. Both synthetic and real patient data have been used to evaluate this hybrid method. We used contours mapping to validate the accuracy of this method on real patient image. Our studies demonstrated that this hybrid method could improve overall registra-tion accuracy with moderate overhead. In addition, we have also shown that the multi-step CSRBF registration proved to be more effective in handling large deformations while maintaining the smoothness of the transformation than origi-nal CSRBF.

Deformable image registration is becoming an important tool for tracking changes in a patient’s anatomy during radiotherapy^{ }[1-9]. The image-intensity method [

To improve the robustness and accuracy of deformable image registration in certain clinical situations, we have proposed a hybrid registration approach that combines the landmark-based method, using compact-support radial basis functions (CSRBFs) [

Though some researchers have proposed combination of image intensity and landmarks for registration [14-18], the major difference is that they applied both image intensityand landmark-based registrations to the entire image and calculated the registration by optimization of global cost function. In our proposed method, the image intensity-based registration applies to the entire image; while the landmark-based registration applies only to a local region where accurate registration results cannot be achieved using intensity-based registration alone. We also proposed a multi-step CSRBF registration method that has proven to be robust in handling large deformations and insensitive to the registration parameters used.

Image intensity-based registration is also called voxel similarity-based registration because the registration metric and the cost function is calculated directly using the voxel-intensity values of the two images to be registered, i.e., source S(x) image and target T(x) image [

We previously implemented a variational diffusion registration [_{k} (k = x, y, z in the Cartesian coordinate system) can be determined by solving the registration equation [2,3,17].

where a is the smooth parameter; S(x_{k}) and T(x_{k}) are the source and target image intensities, respectively; and . The displacement fields are the solutions of the nonlinear partial differential equations in Equation (1), and these equations can be solved numerically using a multigrid method [

CSRBF spline registration requires user intervention to establish the correspondences between the same object in both source and target images. The main steps in landmark based registration algorithm consist of identifying pairs of corresponding landmarks in images, computing the appropriate transformation parameters, and then interpolating transformation fields for positions not on the landmarks.

The properties of radial basis functions (RBFs) are well defined and have proven to be efficient for interpolation [

where

This function is positively defined in a 3D space and has C^{2} continuity (second derivative continuity). An RBF is based on a scalar radius r and is symmetric around a control point. The smaller the value of a, the more local the confinement of for at the landmark control points. To handle large deformations using CSRBF spline registration, a large a value may be needed [

CSRBF spline registration initially identifies the corresponding coordinate locations (control points) in both source and target images. The control-point pairs are then used to determine a coordinate transformation (mapping) between two images. The set of control points in the source image can be denoted as, and the corresponding control-point set in the target image can be denoted as. A successful registration should map every point in to the corresponding. Transformation of the mapping function can be represented as a linear combination of the CSRBFs of the distance between two control-point sets:

where i, j = 1, ···, N, and N is the number of control points. For a given set of control points, the transformation coefficients are the solutions for this linear system:

where and.

Once the transformation coefficients are obtained as depicted in Equation (5), the mapping between any points x other than the control points is given as follows:

where x' is the corresponding point of x, and

is the displacement field for point x. The displacement fields generally depend on both the number and spatial distribution of control points. For a given set of controlpoint pairs, the displacement fields depend on the value of the compact support parameter a. Usually, a small a value makes the transformation more local to the control points, and the deformation of the object is relatively small. In certain clinical situations, a large a value is required to handle large deformations in a small region. When a large a value is used, the CSRBF spline transformation may influence the outer regions away from the area defined by the control points.

With large deformations, landmark-based registration may produce “cross-over” of the grid, as shown in

After selection of the control-point pairs, the distances between the corresponding points roughly represent the deformation of the subject from the source image to the target image. Using one-step registration would necessitate a large compact support parameter a for large deformation, which could violate the locality [

The multi-step registration initially registers the source control points (real points) to the virtual control points

, which are closest to the source control points. Next, the previous virtual control point set is used as the source set and registered to the next virtual control point set. This procedure is repeated until the final step, which registers the last set of virtual control points to the target control point set. The number of registration steps is n+1. In each step of the registration process, the same compact support parameter value a is used. The transformation coefficients are obtained for each step by repeatedly solving Equation (5), and these coefficients are used sequentially to calculate the displacement fields of every point using Equation (6) in the region of interest. The final displacement field between the source and target images is the cumulative displacement field obtained in the multiple steps.

The image intensity-based deformable registration algorithm [2,9] works well for many clinical applications when the image quality is good and source and target images are relatively similar. The hybrid registration algorithm is designed for situations when the image quality and / or source and target similarity are inadequate for successful registration using the image intensity-based algorithm alone. Therefore, prior to the use of the hybrid registration, we first registered the image pair using intensity-based diffusion registration algorithm. We used the hybrid registration only when the diffusion registration could not produce satisfactory results.

As illustrated in the flowchart in

The combination of two displacement fields should satisfy the smooth constraints for the two registration methods, and the local registration should not affect global registration of the entire region. In our implementation, we first used a spline function to connect all of the landmark points to calculate a closed 3D volume. We marked the interior of this 3D volume as the “in-field” volume, where the displacement field was determined solely using the CSRBF spline registration method. We then expanded the 3D surface of this in-field volume to create a “transition zone” with the width of the expansion equal to the compact support parameter a. Within the transition zone, the displacement field was determined according to the weighted average of both of the displacement fields calculated using the intensity-based method and the landmark-based method at each position:

where the weighting function can be determined using a linear function of Euclidean distance transformation [25,26] between the point of interest x and the 3D surface of the in-field volume:

where the compact support parameter a normalizes the weighting function to a value within [0,1]. This is illustrated in the two-dimensional (2D) diagram in

Computed tomography (CT) images for four patients with prostate cancer, two patients with lung cancer, and one patient with head and neck cancer were used to test the accuracy and efficiency of the hybrid deformable image registration method. For the prostate cancer patients, daily (target) CT images were acquired using a CT-on-rails system (EXaCT; Varian Medical Systems, Palo Alto, CA). For the lung cancer patients, four-dimensional (4D) CT images were used to test the registration of tumor sliding along chest wall. The 4D CT data were reconstructed using CT scans acquired synchronously with the respiratory signals representing 3D CT data sets at various breathing phases (total, 10 phases). In 4D CT images, the 0% phase CT images (denoted by T0) and 50% phase CT images (denoted by T5), corresponding to the end of inspiration and expiration scans, respectively, and the largest range of tumor motion owing to respiration appears in the CT images in these two phases.

We used the results of the deformable image registration to determine whether this transformation can correctly auto-delineate the daily target volume. In this procedure the manually drawn contours in the source CT image were transformed to the contour in target CT image using the displacement field calculated from deformable image registration. The deformed contours in the target image automatically segmented the anatomic structures in the target image. Precise registration led to the deformed contours correctly fitting the corresponding structures in the target image. Therefore, the deformed contours facilitated evaluation of the image registration algorithm. The deformable image registration methods were implemented using the C++ programming language. All tests were run on an Intel Pentium IV (2.8 GHz)-equipped personal computer.

In the multi-step CSRBF registration algorithm, the number of steps is specified by the user and depends mainly on the distances between source and target control points. Our experience showed that a four-step procedure is reasonable for challenging cases with very large deformations.