Three-dimensional elastic image registration based on strain energy minimization: application to prostate magnetic resonance imaging

Abstract

The use of magnetic resonance (MR) imaging in conjunction with an endorectal coil is currently the clinical standard for the diagnosis of prostate cancer because of the increased sensitivity and specificity of this approach. However, imaging in this manner provides images and spectra of the prostate in the deformed state because of the insertion of the endorectal coil. Such deformation may lead to uncertainties in the localization of prostate cancer during therapy. We propose a novel 3-D elastic registration procedure that is based on the minimization of a physically motivated strain energy function that requires the identification of similar features (points, curves, or surfaces) in the source and target images. The Gauss-Seidel method was used in the numerical implementation of the registration algorithm. The registration procedure was validated on synthetic digital images, MR images from prostate phantom, and MR images obtained on patients. The registration error, assessed by averaging the displacement of a fiducial landmark in the target to its corresponding point in the registered image, was 0.2 ± 0.1 pixels on synthetic images. On the prostate phantom and patient data, the registration errors were 1.0 ± 0.6 pixels (0.6 ± 0.4 mm) and 1.8 ± 0.7 pixels (1.1 ± 0.4 mm), respectively. Registration also improved image similarity (normalized cross-correlation) from 0.72 ± 0.10 to 0.96 ± 0.03 on patient data. Registration results on digital images, phantom, and prostate data in vivo demonstrate that the registration procedure can be used to significantly improve both the accuracy of localized therapies such as brachytherapy or external beam therapy and can be valuable in the longitudinal follow-up of patients after therapy.

Fig 1

Schematic of the registration model. Voxels in the source image are denoted p(x,y,z). Under deformation they evolve into P(X,Y,Z) or P’(X’,Y’,Z’) if the source voxel belongs to the set Σ. In this figure, Σ is a voxel set of interest as denoted by the red square. Also, Ω denotes the image volume, and ∂Ω denotes the surfaces of volume Ω.

Fig 2

Flowchart of the registration procedure and the SEMT algorithm using Gauss–Seidel method.

Fig 3

Semi-log plot depicting the number of iterations required for a given maximum difference.

Fig 4

The 3D open look of a the source sphere, b the target ellipsoid, and c the registered volume with a character ’F’ inside. The Cartesian coordinate system is shown for later reference in Figure 6.

Fig 5

Prostate phantom: a picture of the prostate phantom, and b representative MR image. Sesame seeds are shown as dark dots in the gland.

Fig 6

Registration results on the synthesized digital phantom. Each row shows images in each of the orthogonal planes. Columns represent image type. The source image column is overlaid by the displacement vectors in three orthogonal planes. Also shown in target image (xy plane) are the landmarks (marked by red crosses) used to measure the accuracy of registration. NCC improved from 0.91 to 1.0.

Fig 7

Registration results of the prostate phantom in the xy plane: a target image, b source image overlaid with the displacement, c registered image, d intensity difference image (a–b), and e intensity difference image (a–c). NCC improved from 0.84 to 0.99.

Fig 8

Representative registration results for a prostate patient: a target image with five feature landmarks marked by red crosses, b source image overlaid with the displacement, c registered image, d intensity difference image (a–b), and e intensity difference image (a–c). NCC improved from 0.69 to 0.97.

Fig 9

Graph depicting the performance of SEMT, affine, and B-spline registration algorithms on prostate images. Minimum registration errors are realized with the SEMT algorithm.