A Hybrid Data Analysis And Mesh Refinement Paradigm For Conformal Voxel

A Hybrid Data Analysis and Mesh Refinement Paradigm for Conformal Voxel
Spectroscopy.
C. Sharma, L. Bolinger, L. Ryner

National Research Council of Canada, Institute of Biodiagnostics
ABSTRACT size of the parallelepiped reduces the amount of healthy
tissue inside the parallelepiped, but introduces error in
Recent advances in magnetic resonance spectroscopy have excluding tumor tissue outside the parallelepiped. Spatial
involved defining a conformal polyhedral shape around the saturation planes can be manually specified on the scanner
tissue of interest (TOI) [1]. Up until now, magnetic so that a higher proportion of the signal from the tumor
resonance spectroscopy involved specifying a cuboid tissue is obtained. This process is time consuming and is
encompassing the volumetric shape of the TOI. The signal based on the discretion of the spectroscopists.
obtained from the cuboid includes signal from healthy tissue
in addition to TOI, resulting in “contamination” of the total We have developed and implemented an automated
signal with non-TOI signal. Additional, a set of planes technique for optimal placement of saturation planes
known as spatial saturation planes may be prescribed conforming to the surface of the tumor. We seek an
around the TOI so that a larger proportion of TOI to non- improvement over the existing technique for manually
TOI signal is obtained. In this paper, we propose and prescribing saturation planes. The goal is to minimize any
implement an algorithm to optimize spatial saturation plane intermittent volume between the tumor surface and the
placement, resulting in greater proportion of signal from the prescribed set of planes, thereby minimizing contamination
TOI. This involves a better definition of the initial region of due to healthy tissue.
excitation using principal component analysis, followed by
a mesh-based approach for plane placement. 2. METHODS

Keywords: Magnetic Resonance Spectroscopy, Principal 2.1 Segmentation
component analysis
Tumors can be differentiated from surrounding healthy
1. INTRODUCTION tissue based on a comparison of difference in intensity
between healthy and surrounding tissue. A suitable contrast
Magnetic resonance spectroscopic (MRS) techniques have agent such as gadolinium can be injected to highlight the
been used to evaluate the chemical composition of healthy tumor tissue. Patient images are acquired using a General
and tumor tissue based on NMR spectra. In 1995, the FDA Electric 1.5 Tesla LX Signa Scanner using a Spoiled Echo
approved a fully automated MRS sequence for neuro Gradient Recalled Echo sequence with a 30 degree flip
magnetic resonance spectroscopy called the PROton Brain angle, an echo time of 7 ms and a recovery time of 24 ms.
Examination sequence [2]. (PROBE) This involves defining Total image acquisition time is approximately 10 minutes.
a single volume of interest in the brain using three The software for this project is written using Interactive
orthogonal planes and analyzing the spectra obtained from Data Language (Research Systems Inc.) on a Macintosh G5
the volume of interest to obtain concentration and nature of dual processor machine with 2 GB of RAM at 2 GHZ
metabolites [3]. The PROBE magnetic resonance sequence processor speed. All images are anonymized to maintain
has since given radiologists an efficient technique for patient privacy.
diagnosing cancerous tumors in the brain.
Patient images are loaded from the image database using the
The definition of orthogonal slice planes to define the single graphical user interface. The raw images may be noisy and
volume of interest is not without its drawbacks. Tumors are require some amount of preprocessing prior to actual
rarely rectangular in shape. The cuboid formed by segmentation. Tumor tissue might contain “gaps” and
orthogonal planes, leads to a large loss of signal, because “holes” without any clear demarcation from surrounding
any orthogonal cuboid can only prescribe the extents of a healthy tissue. Mathematical morphology tools provide a
tumor ignoring its irregular shape. Defining a rectangular wide variety of operators applied to fill in gaps within the
parallelepiped around the tumor results in obtaining signal tumor and define sharper boundaries around the tumor
from healthy tissue as well as tumor tissue. Decreasing the [4][5]. The most common among them are image dilation

0-7803-9577-8/06/$20.00 ©2006 IEEE 1 ISBI 2006

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

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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

takes 2.5 seconds on a Macintosh G5 dual processor [5] Haralick, Sternberg, and Zhuang, “Image Analysis Using
Mathematical Morphology,” IEEE Transactions on Pattern
machine with 2.0 GB of RAM at 2.0 GHZ. As indicated by Analysis and Machine Intelligence, Vol. PAMI-9, No. 4, July,
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
column 3 to 0.87 in column 5 using the mesh decimation [7] Pearson, K. (1901) On lines and planes of closest fit to systems
of points in space. Philosophical Magazine, 2, 559-572.
algorithm. [8] Hoppe H. Efficient implementation of progressive meshes.
Computers & Graphics, Vol. 2 No. 1,1998, 27-36
The user can visualize the spatial saturation planes around [9] Schroeder W., Zarge J. and Lorensen w. Decimation of triangular
the tumor using the plane manipulation tools provided in the meshes. Computer Graphics (SIGGRAPH ’92 Proc.), 26(2): 65-
70, July 1992.
slider bar interface. This allows the user to fine tune the [10] Garland M., Heckbert P. Surface simplification Using Quadric
solution by moving the planes in X, Y and Z directions, or Error Metrics. Proceedings of SIGGRAPH 97, 1997
by collectively moving the planes inwards or outwards by [11] W. Lorensen and H. Cline. Marching Cubes: A High resolution 3
D Surface Reconstruction Algorithm
the same amount. [12] Barber, C.B., Dobkin, D.P., and Huhdanpaa, H.T.The Quickhull
algorithm for convex hulls, ACM Trans. on Mathematical
Software, 22(4): 469-483, Dec 1996
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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