Optimal and efficient designs for fMRI experiments via two-level circulant almost orthogonal arrays☆
Introduction
Functional magnetic resonance imaging (fMRI) is a way to study neural correlates of consciousness involving perception, memory, learning, thinking, and affection by measuring hemodynamic response to mental stimuli. In an fMRI experiment, the experimental subject is asked to participate in mental tasks in response to the stimuli, while the subject’s brain is scanned by a magnetic resonance (MR) scanner at regular time intervals to collect observation data. For event-related (ER) fMRI experiments, mental stimuli as brief as several milliseconds can be detected by high-speed MR scanners so that transient brain activity can be explored. Each observation at a constant time interval is supposed to be affected by not only the current stimulus but also the preceding stimuli. In that sense, observations from an ER-fMRI experiment may be viewed as time series data and an experiment is designed based on a multiple linear regression model to estimate hemodynamic response functions (HRFs) for plural types of stimuli. In an ER-fMRI experiment, the number of stimuli in a brief duration can be huge and this gives rise to challenging research problems for statistical designs. In this paper, we focus on designs of ER-fMRI experiments (fMRI experiments for short). We refer the reader to Lazar (2008) for an overview of statistical methods in fMRI experiments, Kao and Stufken (2015) for a review of some previous works on designs for fMRI experiments, and Cheng and Kao, 2015, Cheng et al., 2017 for a general theory of guiding the selection of fMRI designs to estimate the HRF. A detailed description of experimental settings and the statistical model for an fMRI experiment with a single type of stimulus is given in the next section.
M-sequences can provide fMRI designs (see Buračas and Boynton, 2002). In the binary case, such designs are shown to be optimal under certain optimality criterion in estimating the HRF (see Kao, 2014). However, for such fMRI designs, only few stimulus effects can be estimated. Lin et al. (2017b) tried to find fMRI designs overcoming this problem by introducing the concept of a circulant almost orthogonal array (CAOA) as a comprehensive framework of fMRI designs.
Definition 1.1 A circulant array with alphabet of size is said to be a circulant almost orthogonal array (CAOA) with levels, strength and bandwidth , denoted by CAOA, if, in any subarray of , holds for any pair of ordered -tuples and of , where is a subarray-dependent frequency of as column vectors. The first row of is called the generating vector of . Especially when , is simply called a circulant orthogonal array (COA).
Example 1.2 The following is an example of CAOA. Its generating vector is . It can be observed that, in any pair of distinct rows of , or
In an fMRI experiment, the parameter of a CAOA is considered as the number of time points taken into account to estimate HRFs for the respective types of stimuli. In this sense, is ideal to be at least the duration of an HRF from the onset of a stimulus to the HRF’s complete return to baseline. However, even for , the maximum values of for ‘optimal’ CAOAs with small have not been determined for all , and the values of for previously known CAOA with are bounded above by . That is why our purpose is to find ‘efficient’ CAOAs with larger as well as ‘optimal’ CAOAs.
As a general theory of selecting ‘optimal’ CAOA, we can refer to Cheng and Kao (2015) for , and Lin et al. (2017b) for . In Lin et al. (2017b), we can also find, for and , the generating vectors of optimal CAOA and the values of searched by computers as well as the previously known values of . For , Lin et al. (2017b) classified CAOA into three classes called , , and , where -CAOAs are ‘optimal’. Through computer search, Lin et al. (2017b) also found examples of ‘optimal’ CAOA other than -CAOAs, but they did not go deep into it. In this paper, restricting to the case , we characterize the class of CAOA of which Lin et al. (2017b) did not make a deeper investigation.
In the next section, we review the statistical model for estimating HRFs in an fMRI experiment with a single type of stimulus, and a class of optimality criteria, called “type 1”, for selection of designs. In Section 3, we propose the notion -CAOA as a subclass of -CAOAs and show their optimality under type 1 criteria. In Section 4, we discuss the asymptotic optimality of CAOA. In Section 5, we demonstrate that algebraic constructions for perfect binary sequences can be applied to get efficient CAOA with large . Concluding remarks are given in the last section together with observations by computer search and some further problems.
Section snippets
Preliminaries
An fMRI experiment we suppose is as follows: A mental stimulus such as a 1.5-second flickering checkerboard image is presented to a subject at some of the time points in the experiment (see Boynton et al., 1996, Miezin et al., 2000). Let () be the measurement of a brain voxel collected by an fMRI scanner at the th time point. We consider the following linear model: where is a nuisance parameter, is the unknown height (magnitude)
Other optimal CAOAs
In this section, we investigate a class of optimal circulant arrays other than the class given by Theorem 2.5. For and , let be a circulant -array and be the information matrix of defined as in (2.5). As stated in Section 2, for the parameter to be estimable without confounding with , it is necessary that, in each row of , the numbers of and are equally . If this is the case, holds, which is an integer matrix such that the diagonal entries
Asymptotic optimality and D-efficiency
As stated in Proposition 3.3, when the value of for optimal CAOA is bounded above by . However, from the practical point of view, it is desirable that is as large as possible to estimate a hemodynamic response function (HRF) with a long tail.
In this section, we show the asymptotic optimality of CAOA for as becomes large. To do that, we need the following lemma (see Appendix B for its proof).
Lemma 4.1 Let be a positive integer. Given , let
CAOAs derived from perfect binary sequences
In this section, we establish the relationship between a -CAOA and a perfect balanced binary sequence. As a consequence, by applying two constructions of such binary sequences involving finite fields, the following two infinite families of -CAOA for are proposed:
- (i)
(Sidelnikov-type) -CAOA with , where is a power of a prime;
- (ii)
(Ding–Helleseth–Martinsen-type) -CAOA with , where is a prime such that at least one of
Concluding remarks
As fMRI experimental designs with a single type of stimulus, we discussed circulant -arrays with from the perspective of type 1 optimality and D-efficiency to theorize empirical results obtained by Lin et al. (2017b).
In Section 3, we recharacterized CAOA with by their information matrices, and gave a class of optimal CAOA other than the previously known class with respect to any type 1 criterion for estimating in model (2.2).
For
Acknowledgments
The authors would like to express their gratitude to the Associate Editor and reviewers for their valuable comments and suggestions in improving the quality of this work.
References (25)
- et al.
Binary sequences with optimal autocorrelation
Theoret. Comput. Sci.
(2009) - et al.
Optimal and efficient designs for functional brain imaging experiments
J. Statist. Plann. Inference
(2017) On the optimality of extended maximal length linear feedback shift register sequences
Statist. Probab. Lett.
(2013)A new type of experimental designs for event-related fMRI via Hadamard matrices
Statist. Probab. Lett.
(2014)Sequences and arrays with desirable correlation properties
- et al.
Almost difference sets and their sequences with optimal autocorrelation
IEEE Trans. Inform. Theory
(2001) - et al.
Balanced perfect sequences of period 38 and 50
J. Comb. Inf. Syst. Sci.
(2010) - et al.
Perfect binary sequences of even period
J. Stat. Appl.
(2009) - et al.
Linear systems analysis of functional magnetic resonance imaging in human V1
J. Neurosci.
(1996) - et al.
Efficient design of event-related fMRI experiments using M-sequences
NeuroImage
(2002)
Optimality of certain asymmetrical experimental designs
Ann. Statist.
Optimal experimental designs for fMRI via circulant biased weighing designs
Ann. Statist.
Cited by (0)
- ☆
This work was supported in part by JSPS KAKENHI Nos. 15H03636, 18H01133, 19K11866, 19K14585, 20K03715.
- 1
X.-N. Lu was supported by Leading Initiative for Excellent Young Researchers, MEXT, Japan.