Optimal and efficient designs for fMRI experiments via two-level circulant almost orthogonal arrays

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Highlights

  • A new class of optimal CAOA(n,k,2,2,1) for n2(mod4) with larger k is provided.

  • Asymptotic optimality of CAOA(n,k,2,2,1) for n2(mod4) and k<n is shown.

  • The relationship between a class of CAOAs and perfect sequences is established.

  • Efficient CAOAs are obtained from perfect sequences via algebraic constructions.

Abstract

In this paper, we investigate a class of optimal circulant {0,1}-arrays other than the previously known class of optimal designs for fMRI experiments with a single type of stimulus. We suppose throughout the paper that n2(mod4) and discuss the asymptotic optimality and the D-efficiency of k×n circulant almost orthogonal arrays (CAOAs) with 2 levels (presence/absence of the stimulus), strength 2 and bandwidth 1, denoted by CAOA(n,k,2,2,1). We show that for n2(mod4) the largest possible value of k for statistically optimal CAOA(n,k,2,2,1) cannot exceed n2. We also clarify that CAOA(n,k,2,2,1) with high D-efficiency and k greater than n2 can be obtained via perfect binary sequences. By applying algebraic constructions for perfect binary sequences and by computer search, lists of such efficient CAOAs and the new class of optimal CAOAs are provided.

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 k×n array A with alphabet S of size s is said to be a circulant almost orthogonal array (CAOA) with s levels, strength t and bandwidth b, denoted by CAOA(n,k,s,t,b), if, in any t×n subarray of A, |λ(a1)λ(a2)|b holds for any pair of ordered t-tuples a1 and a2 of S, where λ(a) is a subarray-dependent frequency of aSt as column vectors. The first row of A is called the generating vector of A. Especially when b=0, A is simply called a circulant orthogonal array (COA).

Example 1.2

The following is an example of CAOA(10,5,2,2,1). A=10010010111100100101111001001001110010011011100100.Its generating vector is (1,0,0,1,0,0,1,0,1,1). It can be observed that, in any pair of distinct rows of A, λ((0,0))=λ((1,1))=2andλ((0,1))=λ((1,0))=3,or λ((0,0))=λ((1,1))=3andλ((0,1))=λ((1,0))=2.

In an fMRI experiment, the parameter k 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, k 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 s=t=2, the maximum values of k for ‘optimal’ CAOAs with small b have not been determined for all n, and the values of k for previously known CAOA(n,k,2,2,1) with n2(mod4) are bounded above by n2. That is why our purpose is to find ‘efficient’ CAOAs with larger k as well as ‘optimal’ CAOAs.

As a general theory of selecting ‘optimal’ CAOA(n,k,2,2,b), we can refer to Cheng and Kao (2015) for n0,1,3(mod4), and Lin et al. (2017b) for n2(mod4). In Lin et al. (2017b), we can also find, for n50 and b1, the generating vectors of optimal CAOA(n,k,2,2,b) and the values of k searched by computers as well as the previously known values of k. For n2(mod4), Lin et al. (2017b) classified CAOA(n,k,2,2,1) into three classes called T1, T2, and T3, where T1-CAOAs are ‘optimal’. Through computer search, Lin et al. (2017b) also found examples of ‘optimal’ CAOA(n,k,2,2,1) other than T1-CAOAs, but they did not go deep into it. In this paper, restricting to the case n2(mod4), we characterize the class of CAOA(n,k,2,2,1) 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 T3-CAOA(n,k,2,2,1) as a subclass of T3-CAOAs and show their optimality under type 1 criteria. In Section 4, we discuss the asymptotic optimality of CAOA(n,k,2,2,1). In Section 5, we demonstrate that algebraic constructions for perfect binary sequences can be applied to get efficient CAOA(n,k,2,2,1) with large k. 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 n time points in the experiment (see Boynton et al., 1996, Miezin et al., 2000). Let yi (i=1,2,,n) be the measurement of a brain voxel collected by an fMRI scanner at the ith time point. We consider the following linear model: yi=γ+xih1+xi1h2++xik+1hk+εi,for i=1,2,,n,where γ is a nuisance parameter, hj 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 n2(mod4) and k2, let A be a k×n circulant {0,1}-array and M(A) be the information matrix of A defined as in (2.5). As stated in Section 2, for the parameter h to be estimable without confounding with γ, it is necessary that, in each row of A, the numbers of 0 and 1 are equally n2. If this is the case, M(A)=X̃X̃ holds, which is an integer matrix such that the diagonal entries

Asymptotic optimality and D-efficiency

As stated in Proposition 3.3, when n2(mod4) the value of k for optimal CAOA(n,k,2,2,1) is bounded above by n2. However, from the practical point of view, it is desirable that k 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(n,k,2,2,1) for 2k<n as n becomes large. To do that, we need the following lemma (see Appendix B for its proof).

Lemma 4.1

Let α be a positive integer. Given 2kα, let Φf(λ1,λ2,,λk

CAOAs derived from perfect binary sequences

In this section, we establish the relationship between a T3-CAOA(n,k,2,2,1) 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 T3-CAOA(n,k,2,2,1) for kn1 are proposed:

  • (i)

    (Sidelnikov-type) T3-CAOA(n,k,2,2,1) with n=q1, where q3(mod4) is a power of a prime;

  • (ii)

    (Ding–Helleseth–Martinsen-type) T3-CAOA(n,k,2,2,1) with n=2p, where p5(mod8) is a prime such that at least one of p1

Concluding remarks

As fMRI experimental designs with a single type of stimulus, we discussed k×n circulant {0,1}-arrays with n2(mod4) 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(n,k,2,2,1) with n2(mod4) by their information matrices, and gave a class of optimal CAOA(n,k,2,2,1) other than the previously known class with respect to any type 1 criterion for estimating h=(h1,,hk) 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)

  • ChengC.-S.

    Optimality of certain asymmetrical experimental designs

    Ann. Statist.

    (1978)
  • ChengC.-S. et al.

    Optimal experimental designs for fMRI via circulant biased weighing designs

    Ann. Statist.

    (2015)
  • 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.

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