- Open Access
Non-rigid registration of breast surfaces using the laplace and diffusion equations
© Ong et al; licensee BioMed Central Ltd. 2010
- Received: 25 June 2009
- Accepted: 12 February 2010
- Published: 12 February 2010
A semi-automated, non-rigid breast surface registration method is presented that involves solving the Laplace or diffusion equations over undeformed and deformed breast surfaces. The resulting potential energy fields and isocontours are used to establish surface correspondence. This novel surface-based method, which does not require intensity images, anatomical landmarks, or fiducials, is compared to a gold standard of thin-plate spline (TPS) interpolation. Realistic finite element simulations of breast compression and further testing against a tissue-mimicking phantom demonstrate that this method is capable of registering surfaces experiencing 6 - 36 mm compression to within a mean error of 0.5 - 5.7 mm.
- Electrical Impedance Tomography
- Registration Method
- Magnetic Resonance Elastography
- Target Registration Error
As breast cancer is estimated to kill over 40,600 people and be diagnosed in more than 194,000 in 2009 , its detection and treatment is an important area of scientific research. Many novel techniques to aid in tumor detection are being developed that exploit the difference in physical properties between healthy and cancerous tissue. Some of these techniques measure the optical, electrical, or elastic properties of tissue, such as near-infrared tomography , electrical impedance tomography , ultrasound elastography , magnetic resonance elastography [5, 6], and modality-independent elastography (MIE) [7–9]. Regardless of the means of data acquisition, it is important to recognize and account for the soft-tissue deformation mechanics of the breast during any analysis.
Previous work in non-rigid registration methods can be broadly categorized as being feature-based or intensity-based. Feature-based methods use only the geometric information extracted from an image, such as a polygonal mesh constructed from a segmented image. Examples of feature-based methods include symmetric closest point , robust point matching , methods involving implicit functions , and finite element modeling . One type of feature-based registration involves the use of splines to interpolate the displacements between tracked control points. Polynomial splines, B-splines, and thin-plate splines (TPS) are among the most commonly used. However, the difficulty with using any type of spline is determining accurate displacements at the control points; the displacements must either be tracked with fiducial markers or estimated using another method such as in .
In contrast, intensity-based methods utilize the intensities in the image volume, sometimes in addition to the geometric information, to register two images. Rueckert [15, 16] proposed a method to maximize image similarity and preserve smoothness, while a similar volume-preserving optimization method was developed by Rohfing . Tanner  employed a free-form b-spline deformation to maximize image similarity in a volume-preserving cost function, similar to Rueckert. The registration algorithm was validated by taking clinical dynamic contrast enhanced MR images, deforming them using a biomechanical FEM model and then calculating a TRE. Optical flow  and fluid flow  techniques have also been used for breast image registration. Unfortunately, like all optimization schemes, intensity-based methods are subject to the need for good initialization and vulnerable to local minima. In addition, intensity information may not be readily available for some applications such as near-infrared breast tomography, electrical impedance tomography, microwave tomography, or mechanical imaging. Even if intensity images are available, they may be subject to geometric distortions or contrast changes.
For the particular application of registering breast surfaces having undergone a quasi-static mechnical compression, our prior work has indicated that fiducial-based spline interpolations, while powerful, do not translate well with experimentation on tissue and substances not amenable to the fixation of physical markers. Likewise, intensity-based methods may not be desirable due to computational expense or the unavailability of suitable intensity images. Therefore, in an attempt to balance the best attributes of these classes of methods, we present a semi-automated method that does not rely on either control points or explicit knowledge of the internal image intensity pattern. This is accomplished by using the Laplace or diffusion equations to calculate equivalent surface energy distributions in order to estimate a generalized displacement field. The accuracy of the method was evaluated by comparison to our internal gold standard of a thin-plate spline interpolation method  in both biomechanical simulations and experimental deformations on breast phantoms.
The basic premise of this work was to evaluate whether the equivalent potential energy distributions modeled by a partial differential equation (PDE) over an undeformed ("source") surface and a deformed ("target") surface could be used to determine correspondence between the two surfaces. For our specific method, finite element models of two classic PDEs (Laplace's equation and the diffusion equation) were solved independently over the source and target surfaces using the Galerkin method of weighted residuals with Lagrange polynomial interpolation . Equivalent potential energy isocontours were calculated and matched on a closest-point basis, and from this correspondence the final displacements at all mesh nodes were interpolated.
where Φ represents the potential and α is the diffusion coefficient.
Obtain the undeformed source mesh and deformed target mesh that define a breast surface before and after deformation.
Assign boundary conditions at nipple and/or chest wall nodes
Solve the PDE (diffusion or Laplace) over the source and target meshes using FEM.
Extract isocontours on the source and target surfaces.
Determine point correspondence between source and target isocontours using SCP (Figure 3).
Interpolate displacements at source isocontours to all source nodes.
where X i , Y i , Z i are the coordinates of the control points, N is the number of control points, and a, b, c, and F are scalar weighting factors.
To assess the accuracy of the PDE-based and TPS methods described above, breast surfaces deformed using a biomechanical model were registered and the simulated displacements were used to calculate the registration error. A CT image volume of a pendant human breast (256 × 256 × 130, voxel size 0.6 mm3) and an MR image volume of a pendant breast from a different patient (256 × 256 × 98, voxel size 1.0 mm3, 3D T1-weighted fat-nulling inversion pulse sequence) were obtained. Both image volumes were segmented using ANALYZE 6.0 (Mayo Clinic, Rochester, MN), and triangular source meshes consisting of 6,313 and 3,942 nodes, respectively, were constructed.
Target surfaces were created by deforming the source surfaces using a finite element model of the breast as a linear elastic, Hookean solid under localized compression. The nodes along the chest wall were made to be fixed (Type I) and a Gaussian-shaped stress distribution (Type II) applied to the lateral surface of the breast. To provide a challenge to our registration method, the shape and magnitude of the applied stress was varied. First, the CT source surface was deformed using a circular contact area with a maximum displacement of approximately 33 mm. In the second simulation, a rectangular contact area was used with a maximum displacement of 13 mm. Finally, in the third simulation, the MR source surface was deformed assuming a rectangular contact area and a maximum displacement of 6 mm.
In addition to running the Laplace and diffusion registrations, TPS registration was performed for further comparison. Because the accuracy of the TPS method varies based on the number and distribution of control points, two different sets of registrations were performed. In the first analysis, a uniform distribution of control points over the breast surface was selected using a k-means algorithm. The number of control points was varied, and for a particular number of control points desired, 20 different configurations were selected to account for the initial random seeding of the k-means clustering algorithm. In the second analysis, a high number of control points in the deformed region (the part of the surface in contact with the simulated inflation bladder) and a lower number over the rest of the surface was used. Similarly, the error was calculated for a varying number of control points and configurations.
To assess the accuracy of the registration methods, the target registration error (TRE) was calculated as the Euclidean distance between the coordinates determined by non-rigid registration and the true target points. Because of the controlled model-based deformation, every node on the surface had a known correspondence from source to target surfaces, allowing for individual TRE determinations along with evaluation of mean and max error for the surface as a whole.
CT images (512 × 512 × 174, 0.54 × 0.54 × 1 mm voxel size) were acquired with the phantom subjected to three different states of mechanical deformation (undeformed, 50% of maximum bladder pressure, and full inflation of approximately 200 mm Hg). Triangular surface meshes were obtained by segmentation of the image volumes using the surface extraction tools in ANALYZE, and the coordinates of the fiducial centroids localized. These meshes contained 8,127, 6,777, and 8,260 nodes, respectively. The Laplace, diffusion, and TPS methods were then used to register the phantom surface meshes as described above. Accuracy was assessed by calculating the TRE at the embedded fiducials. For the TPS method, 33 of the fiducials were used as control points in the interpolation and the remaining fiducial was reserved for calculating the TRE. To assess the error over the entire surface, the TPS registration was repeated in a "leave one out" scheme, each time using a different fiducial to calculate the error, with the final TRE representing the average over all trials.
Simulation registration error
Simulation 1 (CT) (33 mm displacement)
Simulation 2 (CT)
(13 mm displacement)
Simulation 3 (MR)
(6 mm displacement)
Max TRE (mm)
Mean TRE (mm)
Max TRE (mm)
Mean TRE (mm)
Max TRE (mm)
Mean TRE (mm)
Phantom registration error
Phantom: 50% compression
(20 mm displacement)
Phantom: 100% compression
(36 mm displacement)
Max TRE (mm)
Mean TRE (mm)
Max TRE (mm)
Mean TRE (mm)
While neither the Laplace nor diffusion PDE-based methods were able to surpass the performance of a well-executed TPS-based interpolation, the results are encouraging overall as a first attempt of a method that does not require fiducials.
One area of further development with regards to the Laplace equation method is in determining specific regions to which boundary conditions are assigned. For the geometry of the breast, the nipple and chest wall areas were relatively evident and easily set; however, proper selection of these regions is important because the implicit correspondence between these regions determines the potential energy distribution and therefore correspondence for the rest of the surface.
While we do not perform a direct sensitivity analysis for boundary condition selection, previous work indicates that as long as the segmentation and resulting boundary condition error is below a certain threshold, reasonable results may be obtained. In , the PDE registration methods were used in a breast elastography application, and a boundary condition sensitivity analysis was performed. Reasonable results were obtained as long as the average error over all boundary condition nodes was below 0.5 voxels. Although a direct sensitivity analysis may be desirable, we feel this threshold may indicate the acceptable error for our PDE registration methods.
Some factors to consider in the diffusion method are the parameters controlling the behavior of the diffusion front over the breast surface. Careful selection of these parameters therefore enabled the diffusion-based method to outperform the Laplace method in certain cases. We believe this advantage may be related to the fact that the diffusion method requires fewer initial selections of boundary conditions to generate potential flow as compared to the Laplace method.
Although the gold standard TPS method outperformed the Laplace and diffusion methods, there are several factors to consider. The gold standard TPS interpolation method requires multiple points of constraint and is highly dependent on their number and placement. When a uniform distribution is used, the error decreases as the number of fiducials is increased but with diminishing returns. It should also be noted that to attain comparable TRE values, the non-uniform fiducial distribution requires fewer control points. The analysis of this behavior is illuminating in providing a set of benchmarks for future development of the PDE-based methods.
The literature is replete with registration methods developed for 2D and 3D mammographic applications, with a relative preponderance towards intensity-based methods. However, conventional intensity images may not be available for some applications, and even if available, they may difficult to utilize due to geometric distortion or contrast changes. The novel surface-based method we have presented may provide a suitable alternative for situations where intensity analysis is not amenable due to unavailability, computational complexity, unsuitable contrast, or imaging artifacts.
In further consideration of the accuracy of our method, we reviewed the excellent recent work presented in . In this work, twelve different types of deformations were modeled ranging from 5 - 10 mm. For the optimal registration parameters, the average TRE was reported to be 0.45 mm with a maximum TRE across a series of data sets of approximately 4 - 6 mm. While our results are not directly comparable, we note that the simulations with the closest representative applied deformations of 6 and 13 mm in magnitude had a mean TRE of 0.48 to 0.53 and max TRE < 3 mm. We feel that this is reassuring that our method can perform at this level of accuracy without the prescribed use of either fiducials or image intensity.
A novel surface-based non-rigid registration method has been developed for this work and compared to a relative gold standard of thin-plate spline interpolation. The results indicate that the Laplace and diffusion methods can accurately register breast surfaces that have experienced a wide range of physical deformation to within mean errors ranging from 0.5 - 5.7 mm. Although these PDE-based methods did not perform as accurately as the control, they may be viable registration techniques when fiducials are not available and image intensity comparison is not required.
The authors would like to thank Thomas Yankeelov, PhD (Vanderbilt University Institute of Imaging Sciences, Vanderbilt University, Nashville, TN) and John Boone, PhD (Dept. of Radiology, University of California-Davis) for the MR and CT breast volumes, respectively, as described in our simulation studies. This work was funded in part by a Whitaker Young Investigator Award and a Breast Cancer Research Program Predoctoral Traineeship Award (BC043661) of the Congressionally Directed Medical Research Program.
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