High-dimensional MRI data analysis using a large-scale manifold learning approach

Abstract

A novel manifold learning approach is presented to efficiently identify low-dimensional structures embedded in high-dimensional MRI data sets. These low-dimensional structures, known as manifolds, are used in this study for predicting brain tumor progression. The data sets consist of a series of high-dimensional MRI scans for four patients with tumor and progressed regions identified. We attempt to classify tumor, progressed and normal tissues in low-dimensional space. We also attempt to verify if a progression manifold exists—the bridge between tumor and normal manifolds. By identifying and mapping the bridge manifold back to MRI image space, this method has the potential to predict tumor progression. This could be greatly beneficial for patient management. Preliminary results have supported our hypothesis: normal and tumor manifolds are well separated in a low-dimensional space. Also, the progressed manifold is found to lie roughly between the normal and tumor manifolds.

Appendix

GMM Classifier:

Gaussian mixture models (GMM) are commonly used for classification. A benefit of using GMM is that a posterior probability mapping can be generated. Here, the landmarks chosen in the landmark selection step were used for training the parameters of the GMM using the Expectation Maximization (EM) algorithm [31]:

• Step 1: Initialize the means $\mu _{k}$, covariance $\sigma _{k}$, and mixing coefficients $\pi _{k}$ for the $k$-th component in the GMM.

• Step 2: Expectation step. Evaluate the responsibilities $\gamma \left( z_{nk}\right)$ using the current parameters

  $$\begin{aligned} \gamma \left( z_{nk}\right) =\frac{\pi _{k}\mathcal N \left( x_{n}|\mu _{k},\sigma _{k}\right) }{\sum _{j=1}^{K} \pi _{j}\mathcal N \left( x_{n}|\mu _{j},\sigma _{j}\right) } \end{aligned}$$

  where $K$ is the total number of components in the GMM. The responsibilities, $\gamma (z_{nk})$, at each point can be viewed as the posterior probability of each Gaussian function.

• Step 3: Maximization step. Compute new parameter values for the means $\mu _{k}^{new}$, standard deviations $\sigma _{k}^{new}$, and mixing coefficients $\pi _{k}^{new}$ using the current responsibilities to maximize the responsibilities of Step 2.

  $$\begin{aligned} \mu _{k}^{new}&= \frac{1}{N_{k}}\sum _{n=1}^{N}\gamma \left( z_{nk} \right) z_{n}\\ \sigma _{k}^{new}&= \frac{1}{N_{k}}\sum _{n=1}^{N}\gamma \left( z_{nk} \right) \left( x_{n}-\mu _{k}^{new}\right) \left( x_{n}-\mu _{k}^{new} \right) ^{T}\\ \pi _{k}^{new}&= \frac{N_{k}}{N} \end{aligned}$$

  where

  $$\begin{aligned} N_{k}=\sum _{n=1}^{N}\gamma \left( z_{nk}\right) \end{aligned}$$

  and $N$ is the total number of data samples for training.

• Step 4: Evaluate the log likelihood

  $$\begin{aligned}&\ln p\left( X|\mu ,\sigma ,\pi \right) \nonumber \\&\quad =\sum _{n=1}^{N}\ln \left\{ \sum _{k=1}^{K}\pi _{k}\mathcal N \left( x_{n}|\mu _{k},\sigma _{k} \right) \right\} \end{aligned}$$

  (4)

  If the values do not converge, then the process is reiterated at step 2. With each iteration of the Expectation and Maximization step, the log likelihood function in Eq. (4) will always increase and is guaranteed to converge to a local optimum.