Universally optimal fMRI designs for comparing hemodynamic response functions

Abstract

We consider an experimental design problem in functional magnetic resonance imaging (fMRI), a dominant technology for studying brain activity in response to mental stimuli presented to the experimental subject. In contrast to previous studies, we develop analytical results on optimal designs for comparing hemodynamic response functions, each describing the effect of the corresponding type of stimulus. In particular, for studies with two stimulus types, we derive a sufficient condition for an fMRI design to be universally optimal, and show that designs constructed via m-sequences or Paley difference sets satisfy this sufficient condition.

Circulant Partial Hadamard Matrices: Construction via General Difference Sets and Its Application to fMRI Experiments

“…, v. When (v, k, λ) = (4t − 1, 2t − 1, t − 1) and v is a prime power, we have the well-known Paley difference set (Paley (1933)) that consists of all the quadratic residues in GF (v) (The Galois field with v elements). Following (Kao (2015)), the Paley difference sets can also be adopted to obtain some 0-H(m × n), although this was not pointed out in that paper. While this method gives an infinite number of 0-H(m × n), the value of m for each n may be relatively small compared to the maximum possible m. For example, the Paley difference set can achieve an 0-H(5 × 20), but the maximum m for this n = 20 is m = 7 (Craigen et al (2013); Low et al (2005)).…”

“…Another useful combinatorial construction in design of experiments is the 'r-row-regular circulant partial Hadamard matrix' (Craigen et al (2013)). An application of these matrices in constructing good fMRI designs is also discussed in (Kao (2015)). With given integers m, n, and r, an m-by-n, r-row-regular CPHM, abbreviated r-H(m × n), is a ±1 circulant matrix H with Hj = rj and HH T = nI m ; here, j is an all-ones vector, I m is an identity matrix of order m, and H T is the transpose of H. When m = n, an r-H(m × n) is a circulant Hadamard matrix that as conjectured by Ryser (1963), may not exist when n > 4.…”

Optimal experimental designs for fMRI via circulant biased weighing designs

“…, N . For simplicity, we adopt the following assumptions from previous studies [Kao (2013) and references therein]; see also Kao (2014Kao ( , 2015 for discussions on these assumptions. First, the last K − 1 elements of d are also presented in the pre-scanning period, that is, before the collection of y 1 .…”

“…However, designs attaining the optimal stimulus frequency can still be sub-optimal since the onset times and presentation order of the stimuli play a vital role. Working on this research line, Kao (2015) provided a sufficient condition for fMRI designs to be universally optimal in the sense of Kiefer (1975), and proposed to construct optimal designs for comparing two HRFs via an extended m-sequence (or de Bruijn sequence), a Paley difference set or a circulant partial Hadamard matrix. A major limitation of this recent contribution is that the proposed designs exist only when the design length N is a multiple of 4.…”

Pseudococyclic Partial Hadamard Matrices over Latin Rectangles

“…We remind the reader in this regard that a partial Hadamard matrix is an r × n (binary) matrix H with r ≤ n such that HH t = nI r . The recent implementation of these types of matrices in cryptography [19], experimental design [20], and quantum information [21] has awakened the interest in describing different ways of constructing them [22][23][24][25]. In addition, Latin rectangles may be implemented in Internet of Things (IoT) studies [26], coding theory [27,28], and modern 5G wireless networks [29].…”