This document proposes and implements an algorithm to optimize the placement of spatial saturation planes in magnetic resonance spectroscopy to minimize contamination from healthy tissue. It involves:
1) Segmenting tumor tissue from MR images
2) Defining an initial excitation region using principal component analysis on the tumor shape
3) Reconstructing the tumor as a 3D mesh and progressively refining it to obtain a set of conformal saturation planes
The algorithm was tested on 6 tumor datasets and achieved an average ratio of 0.87 for tumor volume to the optimized polyhedral volume, demonstrating improved conformity over traditional methods.
2. and erosion. Dilation is the processing of growing each As indicated in columns 2 and 3, in Table 1, the ratio of
pixel of the image by the dimension and value specified by volume of the tumor to the volume of the oriented bounding
the structuring element. Image erosion is the process of box is greater than the ratio of the volume of the tumor to
shrinking the image by an amount specified by the the volume of the axis aligned bounding box in 5 out of 6
structuring element. These images are then segmented based cases. If the tumor dataset in X, Y and Z directions are
on a standard deviation based, region grow technique to uncorrelated, the axis aligned bounding box would serve as
define regions of interest (ROI). A manual region grow tool a good approximation for the initial cuboid. If however, the
is also provided to segment images which could not be tumor data in X, Y and Z direction are correlated (as
successfully classified automatically. We implemented indicated by a high ratio of eigen values in X, Y and Z
segmentation successfully on 6 tumor datasets using the directions), performing principal component analysis fully
pulse sequence specified in Section 2.1. uncorrelated the data, thereby resulting in an oriented box
with a better packing factor between the tumor and the box.
2.2. Defining the initial excitation voxel This marginal improvement by adopting the PCA based
bounding box can be improved using the optimization
A good starting point for the initial excitation voxel is the technique described in section 3.
axis-aligned bounding box around the tumor. The
orthogonal axes for the tumor itself are coincident with X 3. ANALYSIS
and Y-axes in the plane of the slice and the Z-axis
perpendicular to the slice stack as shown in Figure 1. An It is clear from the problem statement that particular
axis aligned bounding box results in large intermittent attention has to be given to volumetric tumor shape. Several
volume between the box and the tumor, due to irregular techniques in advanced computer graphics exist to represent
tumor shape. The bounding box could be oriented with the tumor in 3 D such as isosurface rendering, higher order
regards to the original axis to maximize the fit of the tumor interpolation techniques and delaunay triangulation. Finite
within the box. The axes of maximum variation are element methods used for structural analysis consist of
identified using principal component analysis [6][7]. The creating a three-dimensional mesh representation of the
bounding box is oriented along the principal axes as shown body under appropriate loading and boundary conditions.
in Figure 1. The number of elements in the mesh, size of each element
Tumor and degree of interpolation used in mesh generation can be
adjusted to obtain different levels of granularity for whole
body deformation. Since computational cost is directly
related to the size of the mesh, efforts have been made to
Oriented simplify the surface of the mesh by collapsing edges [8][9].
bounding An important criterion in using a mesh decimation
box technique for medical imaging is to preserve object
topology during the mesh refinement process. Garland et. al.
Axis aligned have implemented an efficient algorithm which retains the
bounding box features of the original mesh even after decimation using
quadric error metrics [10]. Particularly attractive is the use
of a quadric error metric to identify the cost of contraction
at a given vertex.
Figure 1. Tumor dataset with PCA based bounding box
2D ROIs obtained from segmented images using techniques
described in section 2 are reconstructed to a three-
PATIENT- VOLUME VOLUME VOLUME RATIO OF EIGEN
SERIES OF TUMOR OF TUMOR OF PCA VALUES dimensional mesh using delaunay triangulation. The
/ VOLUME / VOLUME BASED
OF PCA OF BOX / X/Y Y/Z Z/X
resulting mesh is simplified by collapsing adjacent vertices
BASED CUBOID. VOLUME based on the quadric error metric to obtain an optimal set of
BOX OF CUBOID
1. 0.57 0.55 0.97 1.21 1.16 0.71 planes. Mesh decimation beyond 10% is accompanied with
2. 0.55 0.405 0.74 2.21 1.24 0.36 a significant loss in geometric shape and structural stability.
3. 0.538 0.538 0.99 1.04 1.07 0.89 The tumor data set in two dimensions can be smoothed
4. 0.58 0.56 0.97 1.28 1.13 0.7 using polynomial regression to improve the performance of
5. 0.6 0.577 0.97 1.18 1.19 0.70
the mesh decimation algorithm. Care must be taken to
6. 0.74 0.508 0.687 1.66 2.12 0.28
Table 1. Results of principal component analysis on the tumor prevent over-smoothing to preserve the tumor outline. The
dataset. polygonal fitting process yields better results in two
2.2.1 Results dimensions. The resultant mesh representation might still
show spikes in 3D posing a problem during the decimation
2
3. process. Convex representations in 2D might still be highly of planes. We therefore choose to regenerate the 3 D surface
non-convex in 3D. before the final mesh decimation step. This ensures that the
final mesh is suitably convex.
A lower dimensional unstructured meshing scheme such as
triangulation thus results in planes with disproportionate The original mesh can be decimated to 1% of its original
sizes resulting in an infeasible solution. We therefore opt for volume yielding 122 planes from the original 11600 planes.
using higher order tetrahedral meshes to preserve overall However any further reduction in the number of planes
mesh topology. After the initial triangulated mesh is causes loss of convexity and potentially larger values for
tissue contamination. To prevent this from happening, we
Figure 3. Progressive mesh decimation at 10% (1159 planes), 1% redefine the connectivity of the mesh at an intermediate
(122 planes) and 0.15 %. (18 planes) level of decimation by recomputing the convex hull of the
reduced mesh. The resultant mesh is further iteratively
decimated until the number of planes is less than 20. A
reasonable maximum number of spatial saturation planes
must be defined. The saturation planes are physically
realized on the MRI scanner sequentially in time, with the
total time available being limited due to regrowth of
saturated signal if the total time for saturation is too long.
Voxel The reader is referred to [10] for details of the mesh
decimation algorithm.
The adopted methods provide an intuitive approach to
Tumor
optimize plane placement around the three dimensional
tumour. A numerical solution for obtaining planar
configuration using Powell's method has been implemented
by Ryner, et al. [1]. Efforts to compute the performance of
these methods is currently underway at the National
Research Council of Canada.
4. RESULTS
To test the efficacy of this algorithm, we took 6 tumor
datasets and segmented them using the procedure described
in section 2. After segmentation, the tumor was
reconstructed and progressively decimated as per the
algorithm describe. We define the degree of contamination
as the measure of volume of healthy tissue relative to the
volume of the tumor tissue inside the voxel. The following
results were obtained on Macintosh G5 dual processor
machine with 2 GB of RAM at 2 GHZ. The results for each
of these cases are as listed in the Table 2.
generated and the points are smoothed, a 3D quick convex SR. NO. TUMOR TUMOR / TUMOR / NON TUMOR / TIME NUMBER
VOLUME CUBOID ORTHOGONAL CONFORMAL TAKEN OF
hull procedure [11] with tetrahedral connectivity for meshes (MM3) CUBOID VOXEL (S) PLANES
vertices is implemented. The points obtained are then 1. 7664.6 0.55 0.57 0.84 12.82 18
interpolated bilinearly over a uniform grid to obtain a finer
2. 5602.17 0.405 0.55 0.99 3.1 18
structured grid. Using isosurface rendering [12], we obtain
3. 7414.63 0.538 0.538 0.96 1.22 18
a volume representation of the tumor dataset. The resultant
4. 7739.93 0.56 0.58 0.73 1.2 16
surface is then progressively decimated to obtain a set of
5. 7980.68 0.577 0.6 0.88 1.22 18
spatial saturation planes conforming closely to the tumor
6. 7020.47 0.508 0.74 0.83 10.2 18
surface. The procedure adopted for progressive mesh
Average 7237.08 0.52 0.6 0.87 4.96 18
decimation
results in deteriorating the topology of the mesh. Table 2: Results for 6 tumor datasets
Consequently, measures to preserve topology take Optimization is implicit, fast and preserves overall tumor
precedence over obtaining the exact optimal configuration topology. For a 11,600 face object, decimation to 0.15%
3
4. takes 2.5 seconds on a Macintosh G5 dual processor [5] Haralick, Sternberg, and Zhuang, “Image Analysis Using
Mathematical Morphology,” IEEE Transactions on Pattern
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the mean values in the above table, the ratio of tumor 1987, pp. 532-550.
volume to the polyhedral volume increases from 0.52 in [6] I.T. Jolliffe Principal Component Analysis Springer-Verlag, 1986
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the tumor using the plane manipulation tools provided in the meshes. Computer Graphics (SIGGRAPH ’92 Proc.), 26(2): 65-
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5. CONCLUSION AND FUTURE WORK
We have proposed and implemented a computational
geometry based approach, in defining spatial saturation
planes for magnetic resonance spectroscopy applications.
Table 2 indicates a high degree of conformance between the
specified voxel and the surface of the tumor. The
advantages of prescribing saturation planes using this
technique can be realized by measuring the corresponding
signal obtained from NMR spectra on any MR scanner.
Because of the convex hull approximation, it is inevitable
that it would be difficult to prescribe saturation planes for
tumors that are highly nonconvex or without a closed shape.
It would be interesting to explore the use of an unstructured
grid for such tumor shapes with localized curvature based
mesh reduction. Other cases where this technique might be
inapplicable would be the presence of multiple lesions in the
same dataset. One could treat these as independent lesions
and investigate the above method for such tumors. The final
planar configuration obtained is an approximation obtained
based on the initial nonconvex tumor surface. A quantitative
measure of nonconvexity can be obtained by creating
regular nonconvex stereolithography (STL) phantoms
which resemble the tumor shape. Studies on the utility of
the algorithm to scan these STL models and obtain planes
are currently underway at the Health Sciences Centre at
Cancer Care Manitoba.
5. REFERENCES
[1] L. Ryner, G. Westmacott, N. Davidson, P. Latta.Automated
Positioning of Multiple Spatial Saturation Planes for Non-
Cuboidal Voxel Prescription in MR Spectroscopy. ISMRM 2005
[2] E. R. Danielsen, B. Ross. Magnetic Resonance Spectroscopy
Diagnosis of Neurological Diseases.
[3] Kreis R., Ernst T., and Ross B.D., Development of the Human
Brain In vivo quantification of metabolites and water content
using Proton Magnetic Resonance Spectroscopy. Magnetic
Resonance in Medicine 1993; 30: 424-437
[4] Serra J., ‘Image Analysis and Mathematical Morphology’,
Academic Press
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