Motivation

Purely intensity based registration can achieve high similarity between the transformed deformable image and the reference image. But the transformation lacks the bio-mechanical consistency since the material properties of the tissues are not taken into account. Results may be unphysiological. Typical examples include deformations of bones, distorted soft tissue structures or incomplete vector fields because of insufficient image information (see fig. 1 for some extreme cases). We use different bio-mechanical modeling methods to include tissue properties in the registration and restrict the transformation to physically meaningful transformations.

Fig. 1: Examples of unphysiological registration results (transversal images). Images a, c and d are overlays that show the reference image in green and the deformable image in red. In d overlays before (left) and after (right) the registration are shown. White arrows indicate sites of bone deformation. In d the patient contour is distorted. In b reference image (left), deformable image (middle) and registration result (right) are shown separately. As a result of the registration the trachea has been deformed so that it seems to grow into the oesophagus
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Combination of the Fluid Flow method and Finite Element Simulation

Concept

We use a bio-mechanical model to correct the displacement vector field (DVF) calculated by an intensity based deformable image registration. The workflow of our method can be split into four procedures:

  1. Image registration: An intensity based deformable image registration calculates a DVF.
  2. Geometric Modeling: A volume mesh is created for each volume of interest in the deformable image.
  3. Interpolate Surface Motion: We interpolate surface motions from the DVF.
  4. Simulation: The surface motions are used as boundary conditions for the finite element simulation to calculate a new bio-mechanically consistent DVF.

Step 1: Image Registration
A DVF is calculated by an intensity based deformable image approach. It describes the displacement of each voxel in the deformable image to the corresponding position in the reference image. We use a fluid flow model [1] to smooth the DVF. This allows large deformations and sliding behavior like at the pleural cavity but also in areas where this behavior is not desired and needs to be corrected.

Step 2: Geometric Modeling
For the mechanical simulation we segment the volumes of interest in the source image and extract their surfaces using a marching cube approach. The number of triangles is reduced and the mesh quality is optimized to increase numerical stability in the Finite Element Simulation. The volume inside of each surface in the deformable image is meshed using tetrahedrons.

Step 3: Interpolate Surface Motion
We interpolate the displacement vector for each surface node using the DVF from step 1. Fig. 3 shows the meshed surface with the displacement vectors.

Fig. 2: Workflow for the registration of a lung
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Step 4: Simulation
We apply the displacement vectors of the surface motions as Dirichlet boundary conditions [2] to a 3D finite element simulation. The new DVF is calculated incrementally in several steps, the displacement of every surface point is increased in each step.

Fig. 3: . Left: the 3D surface mesh of the right lung. Right: surface mesh (gray mesh) and displacement vectors (black).
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Application examples

Model based registration of the Lung
Fig. 4 compares the DVF of the initial intensity based deformable registration using a fluid flow regularization with DVF corrected by the bio-mechanical model. Fig. 5 shows the transformed images. Unrealistic deformations, such as curl-shaped motion, have been removed by the the finite element simulation.

Fig. 4: Left: DVF after fluid flow registration. Big white arrows show inconsistent motions. Right: Consistent DVF after the finite element simulation
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Fig. 5: Left to right: Reference image, deformable image after fluid flow registration, deformable image after correction by the finite element simulation.
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Nonrigid registration by approximated biomechanical simulation

Concept

Our algorithm consists of four components: A geometrical model of the deformable VOI, a physical model that describes the stress-strain relation for the different tissue types present in the geometrical model, deforming forces that are exerted on the geometrical model and a solution strategy in which the actual deformation is calculated.

Mechanical model (includes geometry and physics):  
For its advantages regarding performance and memory requirements our mechanical model is based on the mass-spring concept. Models are defined at a high resolution (that is on the voxel grid) so that complex heterogeneous models can be created and complex vector fields can be represented. Each voxel is considered a node and assigned a mass. Structural, face diagonal and volume diagonal connections are established between nodes to allow for transmission of forces of arbitrary directions. A nonlinear stress-strain model has been developed in order to approximate the biomechanical properties of soft tissues.

Fig. 6: Components of a hexahedral volume element as found in the geometrical model. Eight nodes (black dots) are connected to each other by structural (left), face diagonal (middle) and volume diagonal (right) connections.
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Fig. 7: Comparison of stress-strain relations. The diagram shows the relative strain of a connection depending on its relative length. At a relative length of 100% (=1.0) the connection is neither compressed nor elongated and thus not deformed. Under compression (rel. length < 1.0) or elongation (rel. length > 1.0) nonlinear material (red curve) stiffens in a tissue specific way whereas the stiffness of linear material (blue curve) does not depend on its deformation. It is evident that consideration of nonlinear material properties is mandatory if models are to undergo large deformations.
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Deforming forces:  
The registration is performed by simulation, that is by exerting forces on the model and calculating the resulting deformation. There are two basic types of forces: Surface forces and internal forces. Surface forces pull the surface of a segmented structure from its position and shape in the deformable image to its position and shape in the reference image. Internal forces are acquired by establishing landmark correspondences between reference and deformable image, usually by an intensity based registration method such as Template Matching. The main distinction between these two types of forces is that the direction of an internal force is fixed towards its respective landmark coordinates in the reference image while the direction of a surface force is defined as towards the segmented structure in the reference image.

Solution:  
Deforming forces exerted on nodes as well as forces transmitted between nodes lead to accelerations and thus displacements of these nodes. Displacements in return influence these forces. That way the deformation caused by the deforming forces is propagated into the entire model. The model in return reacts to the deforming forces according to its material properties. Damping is applied to the nodes to remove kinetic energy from the model so that it converges towards an equilibrium state in which it has reached its final deformation. Once the sum of displacements inside the model is below a threshold, the registration is considered complete.

Fig. 8: Iterative calculation of the simulation.
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Application examples

Model based registration of a head-neck CT

Fig. 9: Construction of a three tissue head-neck model based on a planning CT. By means of segmentation voxels have been classified as either bone, regular soft tissue or lung tissue and assigned their specific material properties. Since the surface forces have not yet been included into our algorithm when this registration was performed, only internal forces were used as deforming forces.
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Fig. 10: Result of the head-neck registration. Frontal, sagittal and transversal overlay images show the overlay of the deformable image (red) and the reference image (green) before and after the registration. Deformation of bones is suppressed by defining them as very hard material in the model. The transversal image after registration depicts a residual deviation of the patient contour in the region of the left clavicle. That indicates the need for surface forces which have not been applied in this registration. The sagittal image after registration shows an incompleteness of the registration around the larynx. A more complex model of that region and perhaps other sources of deforming forces would be required for improved results if desired.
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Model based registration of the Lung

Fig. 11: Construction of two tissue lung models based on 4D-CT images. Lungs are segmented in the reference image (100% inhale) and deformable image (0% inhale) and considered the volume of interest for the registration. Voxels inside the lung are classified as either alveolar or bronchial tissue by a grey value threshold and assigned their specific mechanical properties. The picture on the right depicts two types of deforming forces: Surface forces (white arrows) that are directed from the surface of the deformable image (green) to the surface of the reference image (red) and internal forces (blue arrows) that are obtained by template matching of bronchial tree structures.
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Fig. 12: Result of a lung registration. Frontal, sagittal and transversal overlay images show the overlay of the deformable image (green) and the reference image (red) before and after the registration. As this result is obtained by simulation, it only contains physiological deformations according to the biomechanical model.
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References

[1] Modersitzki J (2004) Numerical Methods for Image Registration, Oxford University Press
[2] Bathe K J (1996) Finite Element Procedures, Prentice-Hall

Publications

Li P, Malsch U, Bendl R (2008) Combination of intensity based image registration with FEM simulation in Radiation Therapy, Physics in Medicine and Biology

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