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Background:
Accurate and highly specific diagnosis of intra-cranial tumors is
critical for minimizing the morbidity and mortality for brain
cancer. The early
detection of tumor response to therapy is critically important in
the management and staging of cancer treatment. However, there are
currently no widely accepted or effective noninvasive approaches for
assessing brain tumor response to therapeutic intervention. Though
biopsy is still considered the gold-standard for diagnosis and
grading, this approach is too invasive to be used routinely in
long-term assessment of tumor response and it comes with significant
risks including a 1.7% mortality rate and 8% chance of other serious
complications. Therefore, the availability of significantly less
invasive monitoring approaches would be of great value in the
management of cancer patients. Imaging
plays an important role in such diagnosis since there is a trade-off
between over-aggressive resection of tumor and maintenance of
critical brain function. Structural
magnetic resonance imaging (MRI) helps in non-invasive tumor assess,
but the specificity is highly variable on different tumor types.
Research Project:
Chemical shift imaging (CSI) is an imaging modality whereby high
resolution MR spectra are acquired across a volume of tissue. In
vivo CSI allows for non-invasive characterization and quantification
of molecular markers with clinical utility for improving detection,
identification and treatment for brain cancers.
Together with MRI, CSI offers the potential for an integrated
biochemical and morphological view of biological tissue and disease
processes (an example of 31P CSI spectra is shown in
figure 1).
Figure 1. Example of 31P CSI data. Shown is an 8-by-8 voxel axial slice of spectra taken of the brain from a healthy subject.
Each
tissue type can be viewed as having a characteristic spectral
profile corresponding to the chemical composition of the tissue.
In tumors, for example, metabolites are heterogeneously distributed
and in a given voxel multiple metabolites and tissue types may be
present. The observed spectra X are therefore a combination of
different constituent spectra and can be explained as
X = AS + N (1)
where
the columns of A represent the abundance
of the constituent materials and the rows in S their
corresponding spectra. Nrepresents additive noise. The abundance matrix A has M columns ( one for
each constituent) and N rows (one for each voxel). X
and S
have L columns (one for
each resonance band). Physically, in absorption spectra, abundances A
and resonance
amplitudes S are both non-negative,
but in reality due to noise some components of X
can be negative.
Given
this view, the challenge in MRSI is to estimate
A
and the spectra in
S
given
observations X (see
Figure 2-1 and Figure 2-2).
Figure 2 (a) Illustration plot of unmixing problem. (b) Illustration plot of MRS unmixing problem. X is the observed spectra, A is mixing matrix and S is constituent spectra matrix. The square with question mark is a specific approach taken in the source separation to estimate A and S.
There
are conventional methods to estimate A and
S by constrained
least-square problem. More recently, there have been efforts to
simultaneously exploit the statistical structure of multi-voxel
spectra to solve Equation 1 as a blind source separation (BSS)
problem, e.g., Nuzillard’s
second order blind identification
(SOBI),
Ochs formulates Equation 1 within a Bayesian framework to
simultaneously solve for A and
S using a Markov chain
Monte Carlo (MCMC) procedure, etc. But SOBI requires constituent
spectra to be orthogonal, which cannot be fulfilled since they seem
to be highly correlated in many cases. Bayesian MCMC works well on
MRS data, but due to its MCMC procedure, the method is computational
expensive.
We
present constrained non-negative matrix factorization (cNMF) , which
is developed based on Lee and Seung’s NMF, to estimate A and
S in
a maximum-likelihood framework. In cNMF we only constrained A and
S to be non-negative. The
method can be viewed as a subspace reduction whereby negative values
of the constituent sources are forced to zero–i.e. forced to the
edges of a polygonal conic subspace spanned by the constituent
spectra. Although no sparsity constraint is applied, the algorithm
can recover sparse spectra quite well. The cNMF algorithm is four
orders of magnitude faster than the BSD MCMC approach and converges
to the same solution for real-mixture, multi-voxel CSI data.
In
figure 3, cNMF 31P spectral separation results are shown.
And the cNMF algorithm only requires 0.48 seconds (2.3 GHz Intel
Processor) to converge while BSD takes 12,000 seconds (1.2 GHz Intel
Processor).
cNMF can be further developed and applied in a hierarchical framework, with the spectral recovery and subspace reduction constraining which observations are used in the next hierarchical level of recovery. A flow diagram, shown in Figure 4, illustrates the basic idea. The approach enables a “drilling down” of the source space, ultimately increasing the specificity of the recovered spectra based on a natural, and physically meaningful, hierarchy (e.g. head, brain, tumor, proliferation/grade, etc). Two types of hierarchical decomposition can be considered, single resolution and multi-resolution. In figure 5, the single resolution hierarchical decomposition results are shown.
Figure
3. 31P spectral separation results using cNMF. (a) cNMF
recovered spectra, (top) source 1 and (bottom) source 2. Source 1
shows sharp ´
ATP peaks centered at
-18.62 ppm with PCr at -2.52 ppm, indicative of muscle tissue.
Source 2 shows ATP
centered at -18.92 ppm with PCr at -2.52 ppm, which is indicative of
brain tissue. (b) shows the spatial distribution of the recovered
spectra for the 5th axial relative concentrations of muscle and
brain spectra. Note that these images are interpolated from 8-by-8
to 4-by-64 for visualization purposes.
Figure
4. Hierarchical blind
recovery. At the first level an image (data volume) having a given
resolution is used as observations for recovering source spectra
(e.g., 1, 2 and 3 shown in the figure). The source spectra are
analyzed given prior information about the spectral signatures (e.g.
spectra for brain versus muscle) and their spatial distributions
(e.g. location of the head/brain in the image and its approximate
shape). The result is the construction of a spatial mask, used for
selecting those voxels which will be processed at the next level. At
the second level, only those voxels passing through the mask are
used as observations and cNMF is reapplied to recover new source
spectra (e.g., 1’,2’ and 3’). This process is iterative and
continues until the desired specificity is reached.
Figure
5. (a) First level of single-resolution hierarchy separation
results. First row is the recovered spectra, second row is the
spatial distribution of each recovered spectra and the third row is
an upsampled version of the spatial distribution. Clear in the
spectra are biochemical markers for brain (2nd column, top image)
with peaks for CHO, CR and NAA. Note that the corresponding spatial
distribution of these markers maps to the region of the brain. Also
recovered are the sinuses (column 3) indicated by a water peak as
well as a physically realistic spatial distribution for lipids
(columns 1). (b) Second level of hierarchy separation results.
Column 1, 2, 4 spectra are indicative of normal brain tissue: low
CHO, high CR and high NAA. Note the small amount of residual water
(4th column at 4.7 ppm); Column 3 spectrum (top image) indicates
low-grade cancerous tissue: moderately elevated CHO, significantly
decreased NAA and appearance of LAC peak.
Besides
MRS, cNMF can also be applied on other spectral data, such as
hyperspectral data and Raman spectra, etc.
Publication
P.
Sajda, S. Du, T. R. Brown, L. C. Parra, and R. Stoyanova,
"Recovery of constituent spectra in 3D chemical shift imaging
using non-negative matrix factorization", in 4th International
Symposium on Independent Component Analysis an Blind Source
Separation (ICA2003), 2003, Tokyo, Japan, pp.71-76.
P.
Sajda, S. Du, and Lucas Parra," Recovery of constituent spectra
using non-negative matrix factorization", SPIE 2003.
S. Du, X. Mao, D. Shungu, and P. Sajda, "Blind recovery of biochemical markers of brain cancer in MRSI", SPIE's 2004 Medical Imaging Symposium, San Diego, USA (This paper won the 1st Place Best Student Paper Award: The Michael B. Merickel Award).
S.
Du, P. Sajda, X. Mao, and D. Shungu, "Multi-resolution
hierarchical blind recovery of biochemical markers of brain cancer
in MRSI", International Symposium of Biomedical Imaging 2004,
Arlington, VA, USA.
S.
Du, X. Mao, D. Shungu, and P. Sajda, "Blind removal of lipids
in 1H MRSI using constrained non-negative matrix
factorization", International Society of Magnetic Resonance in
Medicine 2004, Kyoto, Japan.
P. Sajda, S. Du, T. Brown, R. Stoyanova, D. Shungu, X. Mao, and L. Parra, "Non-negative matrix factorization for rapid recovery of constituent spectra in magnetic resonance chemical shift imaging of the brain", IEEE transaction on Medical Imaging, Vol. 23(12), 1453-1465, 2004.
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