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IEEE Trans Biomed Eng. Author manuscript; available in PMC 2010 July 16.

Published in final edited form as:

Published online 2008 October 3. doi: 10.1109/TBME.2008.2006017

PMCID: PMC2904987

NIHMSID: NIHMS218867

Lejian Huang, Elizabeth A. Thompson, Member, IEEE, Vincent Schmithorst, Scott K. Holland, Member, IEEE, and Thomas M. Talavage, Member, IEEE

Lejian Huang, School of Electrical and Computer Engineering, Purdue University, West Lafayette, IN 47907;

(Email: ude.eudrup@naijel; Email: ude.eudrup@tmt). (Email: gro.cmhcc@tsrohtimhcS.ecniV; Email: gro.cmhcc@dnalloH.ttocS).

The publisher's final edited version of this article is available at IEEE Trans Biomed Eng

In this work, the architectures of three partially adaptive STAP algorithms are introduced, one of which is explored in detail, that reduce dimensionality and improve tractability over fully adaptive STAP when used in construction of brain activation maps in fMRI. Computer simulations incorporating actual MRI noise and human data analysis indicate that element space partially adaptive STAP can attain close to the performance of fully adaptive STAP while significantly decreasing processing time and maximum memory requirements, and thus demonstrates potential in fMRI analysis.

SPACE-TIME adaptive processing (STAP), originally developed for radar signal processing [1], has recently been shown to exhibit potential for detecting cortical activations in functional magnetic resonance imaging (fMRI) [2]-[5]. Unlike the methods in [6][7], it is not a subspace approach. Rather, it is a Fourier-based joint beamformer-frequency filter. Previous research in applying STAP to fMRI has focused on a fully adaptive version of the algorithm, which is computationally intensive because it applies a separate adaptive weight to every data point in the 2D or 3D data set. Here we introduce a partially adaptive STAP scheme to reduce dimensionality of the algorithm and make it more tractable.

Partially adaptive STAP was first introduced by Brennan and Reed in the form of element/beamspace algorithms [1][8]. For the partially adaptive implementation, a large set of input signals is first transformed into a smaller number of signals prior to applying STAP. The goal is to reduce processing time relative to fully adaptive STAP while maintaining comparable ability to detect true positives without introducing false positives. The method of element space is described in this paper, adapted for application to fMRI, and demonstrated to perform comparably with standard fMRI analysis techniques.

The architecture of partially adaptive STAP consists of three parts: partially-selective processor, partially-adaptive processor and post-processor. The partially-selective processor is a non-adaptive preprocessor comprising two components—a temporal filter **J*** _{p}* and a spatial filter

Combinations of temporal and/or spatial filtering using **J*** _{p}* and

STAP is a two-dimensional filter which simultaneously determines spatial location and frequency of cortical activations. STAP requires both a spatial steering matrix, which controls the ability of the algorithm to correctly locate the fMRI activation in space, and also a temporal steering matrix, which dictates its ability to properly identify the frequency/frequencies of fMRI activation. In [2], Thompson, et al. defined a spatial steering matrix, **A**(x, y), to be applied to k-space, representing the activation patterns and spatial locations of all possible activations. In [9], the model was simplified for application to image space fMRI data, reducing the Hermitian transpose in many of the STAP equations to a simple transpose, given that data in image space are real-valued (magnitude images) rather than complex. However, in the discussion below, the more general formula for complex valued data is provided with the understanding that real data is a subset of this general case.

The temporal steering matrix for activation frequency *ω*, similar to that in [2], is

$$\mathbf{B}\left(\omega \right)={\left[\begin{array}{ccc}\hfill 1\phantom{\rule{thickmathspace}{0ex}}{e}^{j2\pi {\omega}_{1}}\hfill & \hfill {e}^{j2\pi {\omega}_{1}\cdot 2}\hfill & \hfill \dots {e}^{j2\pi {\omega}_{1}\cdot (N-1)}\hfill \\ \hfill \dots \hfill & \hfill \dots \hfill & \hfill \dots \hfill \\ \hfill 1\phantom{\rule{thickmathspace}{0ex}}{e}^{j2\pi {\omega}_{k}}\hfill & \hfill {e}^{j2\pi {\omega}_{k}\cdot 2}\hfill & \hfill \dots {e}^{j2\pi {\omega}_{k}\cdot (N-1)}\hfill \end{array}\right]}^{T},$$

(1)

where *N* and *k* represent the number of time points and the number of known frequencies of activation, respectively, associated with periodic task or stimulus presentations. The reduction of the temporal steering matrix to include only the frequencies in which we are interested represents a revision from that presented in our original work of [2], [3], and [5].

As in [2], the spatiotemporal steering matrix is defined as

$$\mathbf{V}=\mathbf{B}\left(\omega \right)\otimes \mathbf{A}(x,y)$$

(2)

and represents all combinations in space and frequency where activations may occur. The symbol denotes Kronecker product.

Element space partially adaptive STAP retains the spatial dimensionality of fully adaptive STAP but reduces the number of temporal degrees of freedom prior to adaptation. It does so by combining the time course data of the 3D data sets into several subsets prior to computing reduced weights. Thus, a subset is defined comprising *K _{t}* successive frames from a full

$${\overrightarrow{\chi}}_{p}={({\mathbf{J}}_{p}\otimes {\mathbf{I}}_{M})}^{H}{\overrightarrow{\chi}}_{\mathrm{u}},$$

(3)

where *p* represents the *p*th subset (*p* = 0, 1,... *N'*–1), H denotes hermitian transpose, ${\overrightarrow{\chi}}_{\mathrm{u}}$ is the original full dimensional column-wise reshaped resting state (baseline) data set as defined in [2], **I**_{M} is an identity matrix of size *M* = *M _{x}* ×

$${\mathbf{J}}_{p}=[{0}_{p{K}_{t}\times {K}_{t}};{\mathbf{I}}_{{K}_{t}};{0}_{(N-p{K}_{t}-{K}_{t})\times {K}_{t}}]$$

(4)

which chooses a particular subset of frames. The notation ${0}_{p{K}_{t}\times {K}_{t}}$ and ${0}_{(N-p{K}_{t}-{K}_{t})\times {K}_{t}}$ refer to $p{K}_{t}\times {K}_{t}$ and $(N-p{K}_{t}-{K}_{t})\times {K}_{t}$ matrix of zeros, respectively. For each subset *p*, a reduced weight matrix, **W*** _{p}* is calculated as

$${\mathbf{W}}_{p}={\mathbf{R}}_{\mathbf{u}p}^{-1}{\mathbf{V}}_{p},$$

(5)

where

$${\mathbf{R}}_{\mathbf{u}p}=E\left\{{\overrightarrow{\chi}}_{p}{\overrightarrow{\chi}}_{p}^{H}\right\}$$

(6)

and

$${\mathbf{V}}_{p}={({\mathbf{J}}_{p}\otimes {\mathbf{I}}_{M})}^{H}\mathbf{V}$$

(7)

where **R**_{up} is the reduced noise correlation matrix with *E*{ } denoting the expected value, and **V*** _{p}* is the reduced spatiotemporal steering matrix. The individual weights (

$$\mathbf{W}=\sum _{p=0}^{N-1}({\mathbf{J}}_{p}\otimes {\mathbf{I}}_{M}){\mathbf{W}}_{p}\left(diag\left({{\mathbf{W}}_{p}}^{H}{\mathbf{V}}_{p}\right)\right)$$

(8)

where *diag* returns a diagonal matrix. Equation (8) has been adapted from Seliktar et al. [10] to accommodate a spatiotemporal steering matrix and a corresponding weight matrix rather than vectors. This weight matrix **W** is subsequently used to obtain the filter output,

$$\overrightarrow{\mathrm{z}}={\mathbf{W}}^{H}\overrightarrow{\chi}.$$

(9)

Here, $\overrightarrow{\chi}$ is the original column-wise reshaped activated data signal of full dimensionality as defined in [2]. This filter output consists of one frame for each of the stimulus frequencies present rather than output frames for all possible frequencies as represented by the DFT. Note that since $\overrightarrow{\mathrm{z}}$ is complex, it can potentially provide phase information about the signal, in addition to the frequency and location information inherent to STAP.

Validation of the proposed algorithm was effected using synthetic activation data superimposed on human baseline data. For purposes of comparison to the previously incorporated fully adaptive STAP techniques presented in [2][3], subsets of the same 64× 64× 1490 human data set were used here. For the simulations, reduced data sets of size 10 × 10× 60 were extracted so as to maintain the spatial and temporal continuity of the original data set. Two sets of *in vivo* images were used for the simulation. A time series of fMRI data, **χ**_{u}, was first acquired from a subject in the resting state, with no stimulation, in order to establish the baseline signal characteristics of the echo planar image data and to estimate the reduced noise correlation matrix, **R**_{up}. In the same subject we also obtained a second resting state fMRI data set and then superimposed simulated activation in 10 voxels from a subset of 100 voxels in the image time series by adding a simulated activation signal (square wave) with amplitude of 4% relative to the mean intensity value of the time course data.

The first set of simulations was performed to establish suitable values for the parameter *K _{t}* of (4). Simulations were conducted on seven distinct subsets. The number of true positives (TP) and false positives (FP) were recorded as a function of threshold, and averaged over each of the seven data set simulations, providing a measure of algorithm performance. In addition, measures of CPU time and maximum memory requirements as a function of the parameter

Having identified a preferred value for the parameter *K _{t}*, another set of simulations was performed to compare the existing and widely-used fMRI analysis method of cross-correlation [11] to fully and partially adaptive STAP. Each of the three algorithms was tested using the same seven data subsets noted above, and results were averaged to obtain composite receiver-operator-characteristic (ROC) curves [12]. Furthermore, to illustrate the consistency of STAP results at lower signal intensity, additional simulations were run using a response amplitude of 2% relative to the mean intensity value of the time course data.

To demonstrate that the proposed algorithm can be applied to actual human activation data, human data were collected from an fMRI experiment in which two blocked paradigms were conducted in an overlapping manner: (1) a finger-tapping activity with period 60 s, and (2) visual stimulation (flickering checkerboard) with period 40 s. Echo-planar images were acquired using the following parameters: TR/TE = 3000/38 ms; FOV = 25.6 cm × 25.6 cm; BW = 125 kHz; 24 axial slices; 5 mm slice thickness; 120 time points; acquisition matrix = 64 × 64. Given that it is impossible to know the true activation region in a human experiment, it is necessary to choose a traditional analysis scheme as a gold-standard for subsequent comparison. For this work, the STAP algorithms were compared with the simple and widely-used cross-correlation and general linear model (GLM) approaches [13].

Table I displays the average result for each tested *K _{t}* as a function of threshold. Instances in which all 10 activated voxels were successfully identified with no false positives are highlighted in bold. From Table I, we observe that for these simulations, element space partially adaptive STAP attains its highest degree of detection accuracy at thresholds between 0.4 and 0.6, but that performance is relatively independent of

Average True/False Positive performance for element space partially adaptive STAP as function of *K*_{t}

Thus, given a desired detection accuracy, *K _{t}* may be selected to achieve a balance between processing time and memory requirements. Fig. 1 displays the CPU time and memory requirements as a function of

Fig. 2 depicts the ROC curves for the three tested algorithms as computed based on simulations with each of 4% and 2% activation. The ROC curves indicate that both element space and fully adaptive STAP exhibited better detection accuracy than cross-correlation in the simulations. Further, the performance of element space partially adaptive STAP was nearly equal to that of fully adaptive STAP.

Results from single slices within (a) the motor and somatosensory cortices and (b) the primary visual cortex in a single subject are presented in Fig. 3. Note that the activation maps obtained for both forms of STAP are, for both the motor and visual tasks, consistent with those obtained using cross-correlation and GLM analysis. The selected slices in sensory motor cortex and visual cortex were chosen as examples from the full 24-slice data set to demonstrate the sensitivity of the partially adaptive STAP algorithm for detection of brain activation in expected locations. As seen in Fig. 3, element space partially adaptive STAP produced results nearly identical to those of fully adaptive STAP while the latter required triple the computation time. Therefore, element space partially adaptive STAP produces appropriate results and can significantly reduce the processing time and memory requirement, making it practical to use this spatio-temporal analysis procedure on human data.

We introduced a partially adaptive version to reduce dimensionality and improve tractability over that of fully adaptive STAP. The use of known activations, in the form of superimposed simulated activations enabled an objective and accurate comparison to the widely accepted method of cross-correlation.

Element space partially adaptive STAP exhibits potential in detecting cortical activations in fMRI and in reducing processing time and/or memory requirements of fully adaptive STAP. Within a wide range of parameters that specify the size of the reduced dimension data sets, element space partially adaptive STAP can exhibit better accuracy in detecting true activations than that of cross-correlation while greatly reducing running time and memory requirements of fully adaptive STAP. Human data analysis showed that the functional map of element space partially adaptive STAP was almost identical to that of fully adaptive STAP, and both were consistent with cross-correlation and the GLM approach.

This work was supported by NIH grant 1R21MH68267-01A1 and NIH grant 1R01EB003990.

Lejian Huang, School of Electrical and Computer Engineering, Purdue University, West Lafayette, IN 47907.

Elizabeth A. Thompson, Department of Engineering, Purdue University, Ft. Wayne, IN 46805 (Email: ude.wfpi.rgne@nospmoht).

Vincent Schmithorst, Imaging Research Center, Children's Hospital Medical Center, Cincinnati, OH 45229.

Scott K. Holland, Imaging Research Center, Children's Hospital Medical Center, Cincinnati, OH 45229.

Thomas M. Talavage, School of Electrical and Computer Engineering, Purdue University, West Lafayette, IN 47907.

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