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.

Appendix 1

The relationship between the two constants λ and μ is:

$$ \lambda = \frac{{2\upsilon \mu }}{{1 - 2\upsilon }} $$

(A. 1)

And the relationship between the unit volume change e and normal strain ε _x is:

$$ e = \left( {1 - 2\upsilon } \right){\varepsilon_x} $$

(A. 2)

So the first term λe ² of Eq. 1 can be rewritten as:

$$ \lambda {e^2} = \frac{{2\upsilon \mu }}{{1 - 2\upsilon }}{\left( {\left( {1 - 2\upsilon } \right){\varepsilon_x}} \right)^2} = 2\upsilon \mu \left( {1 - 2\upsilon } \right)\varepsilon_x^2 $$

(A. 3)

Given Poisson’s ratio v = 0.495,23 the above expression is approximately

$$ \lambda {e^2} = 0.01\mu \varepsilon_x^2 $$

(A. 4)

which makes it negligible as compared to the other two terms in Eq. 1.