Optimal design for nonlinear estimation of the hemodynamic response function

INTRODUCTION

The hemodynamic response function (HRF) describes the functional magnetic resonance imaging (fMRI) signal change to one short stimulus based on the blood oxygen level-dependent contrast [Logothetis and Wandell, 2004]. Most statistical analyses of fMRI experiments are based on a general linear model where one specific shape for the HRF for all subjects and voxels is assumed, e.g., the canonical double gamma function with fixed parameter values [Friston et al., 1998]. However, it is well established that the HRF varies per subject [Aguirre et al., 1998; Boynton et al., 1996; Zarahn et al., 1997] and brain region [Handwerker et al., 2004; Miezin et al., 2000; Zarahn et al., 1997]. For this reason, Aguirre et al. (1998) suggested to handle the variability of the HRF by applying subject-specific HRFs. Region-specific functions might also be justified but in general the variability has been found to be more expressed between subjects than between regions in one subject [Handwerker et al., 2004]. Using a more accurate model for the HRF by allowing variability between subjects increases sensitivity [Aguirre et al., 1998]. In the following we will focus on subject-specific HRFs, while our methods and results can equally be applied to determine optimal designs for nonlinear estimation of region-specific HRF parameters.

Handwerker et al. ( 2004) compared the use of a subject-specific HRF to a canonical HRF with regard to false negatives, t-values and misestimation of the amplitude of the HRF. They showed that the subject-specific HRF performed better than the canonical HRF. Although differences between subject-specific and canonical HRF were small, they were statistically significant and will especially be relevant for cross-subject analyses. Acknowledging the relevance and importance of using subject-specific HRFs, the optimal design for estimation of such a HRF has to be found. The purpose of this article is to find the optimal design for nonlinear estimation of the parameters of the double gamma function which is a common model for the HRF and provides a good fit to the general form of the HRF.

A first experiment with one stimulus type can be performed to estimate the subject-specific HRF independently of subsequent experiments. From this first experiment, activated voxels or brain regions can be identified by fitting a general linear model based on a Fourier basis set or finite impulse response (FIR) basis set and corrected for multiple comparisons and dependence across voxels [Handwerker et al., 2004]. Furthermore, the HRF can be estimated nonlinearly from the identified voxels or brain regions. The subject-specific HRF can then be employed in fMRI data analyses for subsequent experiments with the same subject and more stimulus types, as variability across scan sessions has been proven to be minor [Aguirre et al., 1998]. Despite the fact that additional scanning time is necessary to firstly derive a subject-specific HRF, Handwerker et al. (2004) showed that if the fit of the canonical HRF is poor a subject-specific HRF will be beneficial.

Besides using subject-specific HRF, several other methods to capture variability of the HRF have been applied. Fourier basis sets [Josephs et al., 1997], splines [Crellin et al., 1998] and FIR sets [Ollinger et al., 2001a] are flexible in modeling the shape of the hemodynamic response, but are less parsimonious and provide less powerful tests than a double gamma function [Henson and Friston, 2007]. The interpretation of the parameters of these sets is also less direct than with a derived subject-specific double gamma function where only one parameter reflecting the amplitude of the HRF has to be estimated. Goutte et al. (2000) extend the approach of HRF estimation based on FIR sets by assuming a Gaussian prior on the FIR coefficients. This forces the HRF to be smooth and reduces the risk of ill determined parameters and overfitted models [Goutte et al., 2000].

Another approach to handle variability of the hemodynamic response from the standard form is the use of the canonical HRF plus partial derivatives [Friston et al., 1998]. The partial derivatives with respect to time and dispersion can capture small derivations from the canonical form. However, the amount of deviations that can be captured are small, i.e., up to 1 s in peak latency [Lindquist et al., 2009]. Subject-specific HRFs could capture more variability. Lindquist and Wager (2007) proposed a HRF model based on the superposition of three inverse logit functions and showed that by using their model parameter estimates of response height, delay and width are more independent of each other than with other common HRF models. Furthermore, deconvolution analysis, e.g., via a Wiener filter [Glover, 1999], can be used to estimate the HRF but deconvolution analysis is a noisy process and the deconvolution parameters must be chosen carefully [Glover, 1999]. Apart from the HRFs models mentioned so far, several other models [Buxton et al, 2004; Ciuciu et al., 2003; Rajapakse et al., 1998; Woolrich et al., 2004] have been proposed with their own advantages and disadvantages. Further details of other HRFs models, comparisons between models and the relevance of precise HRF estimation can be found in Lindquist and Wager (2007). The focus in this article is on using subject-specific parameters for the double gamma HRF which can be estimated from a nonlinear model by a nonlinear iterative least squares fitting algorithm, e.g., the Levenberg-Marquardt algorithm which uses a first order Taylor approximation [Handwerker et al., 2004; Lindquist and Wager, 2007; Miezin et al., 2000].

Optimal design for estimation of the HRF by means of a nonlinear model has not been considered yet. Previous research on optimal design for fMRI experiments has focused on optimizing detection power and estimation efficiency [Birn et al., 2002; Buračas and Boynton, 2002; Dale, 1999; Liu et al., 2001]. Detection power assumes a fixed shape of the HRF with known HRF parameters and refers to the power of a design to detect task-specific activation of brain regions whereas estimation efficiency refers to the efficiency of a design to estimate the HRF based on a linear model with a binary design matrix indicating stimulus presentations at certain time points [Dale, 1999; Henson and Friston, 2007; Liu et al., 2001]. Several designs, e.g., m-sequences [Buračas and Boynton, 2002], random event-related designs [Friston et al., 1999; Hagberg et al., 2001], blocked designs, and optimal designs generated by genetic algorithms (GAs) [Kao et al., 2009; Wager and Nichols, 2003] have been studied and applied to increase detection power and estimation efficiency. In this article, these commonly used designs are applied to find an optimal design for nonlinear estimation of the double gamma function parameters, and a thorough and balanced comparison of the efficiency of these various designs is provided. By means of the GA, optimal designs can be obtained without previous specialization of the design properties. Because the design space of all possible fMRI designs is enormous [Buračas and Boynton, 2002], this algorithm is useful to provide an efficient search within the design space [Kao et al., 2009]. It is known that blocked designs are optimal for detection power, whereas rapid event-related designs with randomized interstimulus intervals are optimal for estimation efficiency based on linear models [Birn et al., 2002; Liu and Frank, 2004; Liu, 2004]. However, not any randomly generated rapid event-related design achieves high estimation efficiency [Buračas and Boynton, 2002], and the GA provides more efficient designs than a random search in the design space [Wager and Nichols, 2003]. Our results show that a mixture of a blocked design and a rapid event-related design will be most efficient for the nonlinear model but that rapid event-related designs can also be very efficient for the nonlinear model.

To calculate the efficiency of a given design, e.g., an m-sequence, a Taylor approximation of first order is applied to linearize the model with respect to the unknown parameters of the HRF [Atkinson et al., 2007]. Based on the linearized model, the covariance matrix of the generalized least squares (GLS) estimator can be calculated for a given expansion point of the Taylor approximation. The D-optimality criterion is then considered to determine the optimal design which minimizes the determinant of the covariance matrix of the GLS estimators for the linearized model. The determinant of the GLS estimator is proportional to the area of the confidence ellipsoid for the estimator. Thus, minimizing the determinant leads to a minimized confidence ellipsoid. However, for the Taylor approximation an expansion point is needed. The optimal design may then depend on the expansion point and is thus locally optimal. To handle local optimality, a range for HRF parameters describing commonly observed shapes of the HRF was chosen from literature [Aguirre et al., 1998; de Zwaart et al., 2005; Glover, 1999; Handwerker et al., 2004; Le et al., 2001] and a maximin approach [Atkinson et al., 2007] was applied. The D-optimality criterion and the maximin criterion will be explained in the following theory section section.

THEORY

The fMRI signal Y to one stimulus type can be described by the following model

(1)

where Z is an T × 1 vector, β is the amplitude of the hemodynamic response, S is a nuisance matrix modeling, e.g., scanner drift, and γ is the vector with unknown nuisance parameters. We assume the error ε to be normally distributed with expectation 0 and covariance matrix Σ. T is the number of time points for the fMRI experiment. The design vector Z is obtained by multiplication of the zero-one T × k matrix X with the assumed hemodynamic response vector h(θ) = (h(1, θ),…,h(k, θ))^T, where k is the number of time points for the HRF and θ is the assumed parameter vector for the HRF. The matrix X is the stimulus convolution matrix, and the product Xh(θ) equals the discrete convolution of the stimulus sequence, which indicates an occurrence of a stimulus at a certain time point, with the HRF h for parameter θ [Dale, 1999]. To estimate the HRF, Y will here be the spatially averaged time course from a set of activated voxels, i.e., a region of interest.

The vector h(θ) depends on the parameters for the HRF. A commonly employed function is the double gamma function with parameter vector θ = (a₁, b₁, c, a₂, b₂):

(2)

where Γ is the Gamma function and t is time in seconds since stimulus onset. The first gamma distribution in Eq. (2) models the peak of the HRF and the second gamma distribution in Eq. (2) models the undershoot of the HRF. For detection of activation by stimuli presentation, the main interest is normally in the peak of the HRF and its amplitude. However, more variation in the undershoot than in the peak has been reported and incorrect modeling of the undershoot might decrease statistical sensitivity [Handwerker et al., 2004]. Thus, all parameters were included in the Taylor approximation to study optimal designs for simultaneous estimation of all double gamma parameters. First order Taylor expansion of h(t, θ) with respect to θ and expansion point θ₀ gives

(3)

where θ_0,j is the jth entry in the expansion point θ₀ and θ_j is the jth entry in the point θ. Equation (3) is a Taylor expansion of h(t, θ) with respect to the several variables in θ = (a₁, b₁, c, a₂, b₂). The partial derivatives of these variables are denoted by ∂h(t, θ)/∂θ_j, e.g., ∂h(t, θ)/∂θ₁ is the partial derivative with respect to a₁ etc. These partial derivatives with respect to θ_j are evaluated in Eq. (3) at the expansion point θ₀ to give the slope of the tangent at h(t, θ) in point θ₀ into the direction of θ_j. These slopes are multiplied with the distance (θ_j – θ_0,j) between the jth entries of θ and θ₀, and the sum of these products plus h(t, θ₀) gives an estimate of h(t, θ).

Substituting Eq. (3) in (1) we obtain

(4)

where the matrix H(θ₀) contains the HRF (double gamma function) and its derivatives sampled at time points 1, …, k and evaluated at expansion point θ₀:

(5)

and τ = (β, β(θ₁ − θ_0,1),…,β(θ₅ − θ_0,5))^T. The first column of the matrix H(θ₀) contains the double gamma function h sampled at time points 1, …, k, and evaluated at the HRF parameter vector θ₀. The second column contains the partial derivative of h with respect to θ₁, thus a₁, evaluated at expansion point θ₀ and sampled at time points 1,…, k. Analogously, the other columns consists of the partial derivatives with respect to θ₂ (b₁), θ₃ (c), θ₄ (a₂), and θ₅ (b₂) sampled at time points 1, …, k and evaluated at expansion θ₀. Equation (4) can be rewritten as

(6)

with Z = XH (θ₀). This illustrates that the model in Eq. (6) is similar to a model for detection of task-related activation with a HRF basis set consisting of the canonical double gamma function for parameter θ₀ and the derivatives of the double gamma function with respect to a₁, b₁, c, a₂, and b₂ evaluated at θ₀ [Kao et al, 2009; Henson and Friston, 2007; Liu and Frank, 2001]. However, the model in Eq. (6) is also related to a model for linear estimation of the HRF as the parameters of the double gamma function will be estimated from Eq. (6). The GLS estimator for (τ^T, γ^T)^T in Eq. (6) is given by

(7)

From Eq. (7) using Theorem 8.5.11 in Harville (1997) for the inversion of partitioned matrices, it is seen that the GLS estimator of τ is given by

(8)

with covariance matrix equal to

(9)

where P_VS = VS[(VS)^T VS]⁻ (VS)^T is the projection matrix onto the vector space spanned by the columns of the matrix VS, [(VS)^T VS]⁻ is the generalized inverse of (VS)^T VS and V = Σ^−1/2 is the whitening matrix so that Vε is white noise (uncorrelated errors). Denoting (I-P_VS)VZ by Z_⟂, Eq. (8) and Eq. (9) can be rewritten as

(10)

Based on the covariance τ matrix, the D-optimality criterion can be used to evaluate the efficiency of a given design [Atkinson et al., 2007]. The D-optimality criterion is based on the determinant or generalized variance of the covariance matrix of the estimators. An alternative to the D-optimality criterion is the A-optimality criterion which uses the trace of the covariance matrix of the estimators [Atkinson et al., 2007; Liu et al., 2001; Wager and Nichols, 2003]. A general advantage of the D-optimality criterion is that the optimal design will remain the same after linear transformation of the predictors in the design matrix. It is possible to rewrite Eq. (4) with τ^* = (β, βθ₁, …, βθ₅)^T. However, the determinant of the covariance matrix of is equal to the determinant of the covariance matrix of , while this equality is not guaranteed for the A-optimality criterion. Therefore, we focused on the D-optimality criterion. The following criterion based on D-optimality was applied

(11)

where r is the rank of Cov( ). The higher F is, the better the design is.

Because the matrix H(θ₀) depends on the expansion point θ₀, the obtained optimal design for a given value of θ₀ is locally optimal. To handle the problem of local optimality the maximin criterion was applied for all designs ξ₁ in the design space Ξ and all parameters θ₀ in the parameter space θ. Here, the parameter space was the set of all considered values for the expansion point θ₀ and is further described in the section on the parameter space for the expansion point. The maximin criterion was applied for each design type (m-sequences, blocked designs etc.) separately. Thus, each design type had its own design space which was the set consisting of all considered designs for this design type. The parameter space however remained the same for all design types. The design space for the GA consisted of all locally optimal designs obtained by the GA as described in the section on design types. Furthermore, the section on design types presents the design spaces for the other design types studied in this article, such as m-sequences, blocked designs and rapid event-related designs.

All locally optimal designs of a design type, e.g., m-sequences, were compared to each other by the relative efficiency to finally select the maximin design from the locally optimal designs. We denote the locally optimal design obtained for a given expansion point θ₀ by . The relative efficiency of a locally optimal design ξ₁ versus the locally optimal design is given by

(12)

with and being the GLS estimators obtained by designs ξ₁ and at the expansion point θ₀. The covariance matrices of the GLS estimators Cov( ) and Cov( ) given by Eq. (9) depend on their respective designs since the matrix X in Z = XH(θ₀) differs per design. The relative efficiency of the design ξ₁ versus the design in Eq. (12) depends on the expansion point θ₀ because firstly the design ξ₁ is compared to the locally optimal design for the expansion point θ₀ and secondly the GLS estimators obtained by the designs ξ₁ and are both evaluated at the expansion point θ₀. The relative efficiency in Eq. (12) is a measure between 0 and 1 with a relative efficiency RE(ξ₁| ) of almost 0 for the design ξ₁ versus the design meaning that the design ξ₁ is very inefficient compared to the locally optimal design at the expansion point θ₀. A relative efficiency close to 1 indicates that the design ξ₁ is almost as efficient as the locally optimal design .

A locally optimal design is maximin D-optimal if it maximizes the smallest relative efficiency [Atkinson et al., 2007]. Given a design ξ₁, the smallest relative efficiency of this design ξ₁ over all expansion points θ₀ in the parameter space θ, i.e., against all locally optimal designs , is its worst relative efficiency. The maximin design MMD thus achieves the best-worst case for the relative efficiencies as it has the maximum minimum relative efficiency over all locally optimal designs ξ₁ in the design space Ξ:

(13)

When the maximin value MMV = is high, the maximin design is very efficient versus the locally optimal designs at all parameter values in the parameter space.

The relative efficiency was further employed to compare the different maximin designs. The relative efficiency of a design ξ₁ versus a design ξ₂ is then a positive measure and is defined by

(14)

The estimator (i = 1,2) is the GLS estimator for the design ξ_i and depends on the design matrix belonging to design ξ_i. Using the model in Eq. (6), the relative efficiency in Eq. (14) depends on the expansion point θ₀. The inverse of the relative efficiency gives the number of times that design ξ₁ has to be repeated to be as efficient as the design ξ₂.

METHODS

The GA was used to find locally optimal designs for the simultaneous nonlinear estimation of the parameters a₁, b₁, c, a₂ and b₂ for the double gamma function. From the locally optimal designs obtained by the GA (with one locally optimal design per considered expansion point of the Taylor approximation) the maximin design MMD_GA was chosen. Furthermore, we obtained maximin designs from different specific design types which are commonly applied or known to be efficient, i.e., m-sequences, event-related designs with fixed ISI, blocked designs and rapid event-related designs with a geometric or uniform distribution for the ISI.

Parameter Space for Expansion Point

The expansion point for the Taylor approximation was varied in a₁, b₁, c, a₂ and b₂. Note that while the estimation of the amplitude parameter β is also optimized by maximizing Eq. (11), the locally optimal designs are not dependent on the value for β, but only on the expansion point θ₀. Thus, only the HRF parameters a₁, b₁, c, a₂ and b₂ are varied for determination of the locally optimal designs. Parameter a₁ had values 4.8, 5, 5.2, b₁ was varied from 0.9 to 1.3 in steps of 0.1, c was varied between 4 and 8 in steps of 1 and a₂ had values 14.5, 15 and 15.5. The parameter b₂ was equal to 0.9, 1 or 1.1 for a₂ = 14.5, equal to 0.9, 1, 1.1 or 1.3 for a₂ = 15 and equal to 0.9, 1, 1.1 or 1.3 for a₂ = 15.5. For each expansion point, the GA by Kao et al. ( 2009) was applied to find the locally optimal design which maximizes Eq. (11). The MATLAB code of Kao et al. (2009) was adapted to use the linearized model in Eq. (4). This resulted in 825 locally optimal designs from which the maximin design (MMD_GA) was ascertained. The applied MATLAB code is available on request from the first author.

The chosen range for parameter values was obtained from the literature showing that the time to peak (TTP = a₁/b₁) of the HRF can lie between 3 s and 7 s [Aguirre et al., 1998; de Zwaart et al., 2005; Handwerker et al., 2004; Le et al., 2001] while the canonical values for the double gamma function are a₁ = 5 and b₁ = 1. The full width at half maximum (FWHM) of the first gamma function, given by , was found to lie in the interval [3 s, 6 s] [Bandettini and Cox, 2000; de Zwaart et al., 2005; Glover, 1999; Le et al., 2001]. The time to undershoot (TTU = a₂/b₂) was found to vary between 9 s and 18 s for real data [Aguirre et al., 1998; Friston et al., 2002; Handwerker et al., 2004] and simulations [Woolrich et al., 2004; Penny et al., 2004]. The FWHM of the undershoot was assumed to be in the interval 7 s to 11 s because the canonical HRF with a₁ = 5, b₁ = 1, c = 6, a₂ = 15 and b₂ = 1 has a FWHM of 9.10 s for the second gamma function. To reduce computation time we restricted the parameters to the values as given above which satisfy the conditions for TTP, TTU and FWHM of both gamma functions. Furthermore, the parameter steps were based on the results of a regression analysis which was performed to study the influence of the parameters on the accuracy of the Taylor approximation. Concluding from the results of the regression analysis, we chose smaller steps for a₁ and b₁ than for a₂ and b₂. As a result, the modeled range of the time to peak is [3.69, 5.78], the modeled FWHM of the peak is [3.96, 5.95], the modeled range of the time to undershoot is [11.92, 17.22] and the modeled FWHM of the undershoot is [7.12, 10.28]. In Figure 1 the range of the HRFs given by the parameter space is presented. The values for a₁ and b₁ in Figure 1a and the values for a₂ and b₂ in Figure 1b were chosen so that the minima and maxima of the TTP, TTU and the FWHMs for the first and second gamma function were obtained.

RESULTS

The maximin design MMD_GA obtained from the GA, proved to be most efficient in comparison to the other maximin designs (MMD_m, MMD_CI, MMD_B, MMD_geo and MMD_uni) obtained under the assumption of linearity of the HRF. Likewise, the maximin design MMD_GANL was most efficient under the assumption of nonlinearity of the HRF. Therefore, we restrict our design comparison in the following sections to the pairwise comparison of the maximin designs obtained by the GA to the maximin designs obtained by m-sequences, constant ISI designs, blocked designs, designs with geometric or uniform distribution for ISI under the assumption of linearity or nonlinearity.

The minimum relative efficiency of the maximin designs within their own design type and the range of relative efficiencies against the MMD_GA or MMD_GANL over the expansion points are given in Tables III and IV. Furthermore, the relative efficiencies of each maximin design from the specific design types against the maximin design MMD_GA are illustrated in Figure 2. Note that the calculation of the relative efficiencies against MMD_GA and MMD_GANL in Figure 2, Tables III and IV is based on Eq. (14) and not on the by Eq. (12) calculated relative efficiencies, which were used to determine the maximin design from the locally optimal designs of each design type.

After the presentation of the results for the different maximin designs, the maximin design MMD_GA is compared in the section on optimal designs to the optimal design for HRF estimation, using a linear model with the stimulus convolution matrix as design matrix, and to the optimal design for detection of task-related activation, using a linear model where the regressors are obtained by multiplication of the stimulus convolution matrix with the sampled HRF. The latter design will be called the optimal design for detection power whereas the former design will be called the optimal design for estimation efficiency. The corresponding models will be named detection power model or estimation efficiency model. These comparisons are interesting for evaluation of the efficiency loss when the model and optimality criterion do not match with the ultimate data analysis of an fMRI design. Linearity of the HRF was assumed for the calculation of both designs which were obtained by applying the original code for the GA of Kao et al. ( 2009). More details are given in the section on optimal designs.

Linearity of HRF

The maximin design MMD_GA presented in Figure 3a is a mixture of a blocked and a rapid event-related design, as it contains task and null blocks while other parts of the design sequence have an event-related form. The maximin design MMD_m in Figure 3b is more similar to a rapid event-related design than the maximin design MMD_GA. The maximin design MMD_CI for the constant ISI designs is the slow event-related design with 17.5 s ISI (see Fig. 3c) and the maximin design MMD_B for the blocked designs has block length 25 s and one stimulus block at the end (see Fig. 3d). The maximin geometric ISI design MMD_geo in Figure 3e is the design with ISI_min = ISI_step = 2.5 s, p = 0.5 and the maximin design MMD_uni shown in Figure 3f is the design with ISIs 2.5 s, 7.5 s and 12.5 s.

The properties of the maximin designs are presented in Table III. Furthermore, the minimum relative efficiency of the maximin designs within their own design class and the range of relative efficiencies against the MMD_GA over the expansion points are given. The range of relative efficiencies against the maximin design MMD_GA is also illustrated in Figure 2. Further details will now be discussed per design type.

Genetic algorithm

The maximin design MMD_GA has a high frequency of ISI 2.5 s and some longer ISIs of 5 s, 7.5 s, 10 s, 12.5 s, 15 s and 27.5 s as can be seen in Figure 4. The longer ISIs indicate null blocks. Null events had a slightly lower frequency than stimulus events in the maximin design MMD_GA (Table III). This was also the case for all the locally optimal designs, except two which had equal frequencies. The minimum relative efficiencies of the 825 locally optimal designs were on average 0.87, and the range was from 0.80 to 0.93. The maximin design MMD_GA was obtained for a₁ = 4.8, b₁ = 1.1, c=4, a₂ = 15.5 and b₂ = 1.1, and its minimum relative efficiency was equal to 0.93. The minimum relative efficiency of the locally optimal design for the canonical expansion point (a₁ = 5, b₁ = 1, c = 6.0, a₂ = 15 and b₂ = 1) was equal to 0.88. In Figure 5 the convergence of the GA for all expansion points is shown. It can be seen that for most expansion points the GA converged very quickly to 99% of its maximum and final value F_max obtained for 10,000 generations. For some expansion points however reaching 99% of F_max needed more generations. The worst convergence was for expansion point number 668 with 3507 generations to reach 99% of F_max.

M-sequence, constant ISI designs, blocked designs

The maximin design MMD_GA was compared to the maximin design MMD_m of the m-sequences by calculating the relative efficiency as in Eq. (14) of the maximin design MMD_m versus MMD_GA at each expansion point. Likewise, the relative efficiency of the maximin design MMD_B and MMD_CI versus MMD_GA was determined. Figure 2a shows that the maximin design MMD_m was slightly less efficient than the maximin design MMD_GA at all expansion points. It can be seen in Figure 2a that the maximin design MMD_CI had a low relative efficiency and the maximin design MMD_B a medium relative efficiency.

Geometric distribution

In Figure 6 the minimum relative efficiencies of the 5400 random sequences versus the locally optimal designs are presented. On the x-axis the 27 designs with geometric distribution for ISI are given (see Table I) and on the y-axis the range of minimum relative efficiencies is indicated. For each design a boxplot of the 200 minimum relative efficiencies for this design is shown. The vertical lines between the boxplots indicate the three sets of nine designs which have the same ISI_min and ISI_step but differ in probability p. Figure 6 shows that for the geometric distribution, designs with middle probability around p = 0.5 perform better with respect to minimum relative efficiency than designs with high and low p as long as the other factors such as ISI_min are kept constant. Furthermore, designs with high or low p have more variability in the minimum relative efficiencies. A small ISI_min is more efficient than a higher ISI_min. The maximin design for the geometric distribution which is shown in Figure 3e had an ISI_min of 2.5 s with possible ISIs in steps of 2.5 s and p was equal to 0.5 (design G5 in Fig. 6). This results in a theoretical mean ISI of 5 s. Figure 2b illustrates that the maximin design MMD_geo performs well against the maximin design MMD_GA.

Uniform distribution

In Figure 7 the boxplots of minimum relative efficiencies for each of the 13 uniform designs are shown. The minimum relative efficiencies are calculated by comparison of the 5200 random sequences versus the locally optimal sequences. The maximin design in Figure 3f for the uniform distribution was the design with ISIs 2.5 s, 7.5 s and 12.5 s (design U6 in Fig. 7). Figure 7 illustrates that designs with minimum ISI 2.5 s have higher minimum relative efficiencies. The maximin design MMD_uni performs well versus MMD_GA but not as well as MMD_geo, see Figure 2c.

DISCUSSION

The general advantage of using a maximin design for nonlinear models is that the maximin criterion chooses a design which performs well within the range of possible values for the unknown parameters. The unknown parameters in our study were the parameters of the double gamma function. The main result was that the maximin design based on the GA performed best. However, due to less computation time maximin designs based on m-sequences, random event-related designs with a uniform or geometric distribution for ISI or even an optimal design for estimation efficiency can be recommended as an alternative to a maximin design obtained by the GA. It was furthermore found that the maximin design MMD_GA, the most efficient design, is between a blocked design and a rapid event-related design. In the following we will discuss the results per design types and finalize this discussion with the properties of an optimal design for the nonlinear model.

Buračas and Boynton ( 2002) showed that m-sequences performed in general better for estimation efficiency than randomly generated sequences, and Kao et al. (2009) found that for estimation efficiency the design obtained by the GA was more efficient than m-sequences. The results of both papers are based on a linear model to estimate the HRF time points and are thus not directly comparable to our results from a model to estimate the amplitude of the HRF and the nonlinear parameters of the double gamma function. However, some similarities can be found between our results and their results. Under the assumption of empirically estimated correlated noise and for one stimulus type, Buračas and Boynton (2002) found randomly generated sequences which were more efficient than m-sequences. Similarly in our study, the best random sequence with a geometric distribution for ISI was more efficient than the best m-sequence when compared to the maximin design MMD_GA (Fig. 2) but of course several realizations of the random sequences are necessary to find an efficient sequence. Furthermore, it is noticeable that in this article and in Kao et al. (2009) m-sequences performed well with relative efficiencies versus designs obtained from the GA from 0.8 to almost 1.

The major advantage of m-sequences is that the computation time for calculation of MMD_m is much less than the computation time for calculation of the maximin design MMD_GA based on the GA (Table III). This advantage in computation time of the MMD_m comes along with a slightly lower efficiency of the MMD_m in comparison to the MMD_GA for one stimulus type. Generally, m-sequences are only available for certain sequence lengths N and number of stimulus types Q, i.e., N+1 and Q+1 have to be a prime or power of a prime. The problem of the sequence length can be handled by concatenating several m-sequences or truncating m-sequences so that the obtained sequence still maintains to a high degree desirable properties of m-sequences. These properties, which lead to a high efficiency of m-sequences especially for estimation of the HRF, are almost perfect counterbalancing, equal number of trials for different stimulus types and orthogonality of the m-sequence to cyclically shifted versions of itself (Buračas and Boynton, 2002). The estimation efficiency of m-sequences seems to be robust to truncation [Liu, 2004]. Liu (2004) also generated clustered m-sequences which are constructed from m-sequences by clustering events of the same stimulus type and provide a trade off between estimation efficiency and detection power. The possibility of constructing efficient sequences from m-sequences widens the applicability of m-sequences. To our knowledge not much research has been done to construct alternative efficient sequences from m-sequences when m-sequences are not available for a given number of stimulus types.

Our results for the geometric distribution showed that an event occurrence probability of about p = 0.5 was optimal. This probability is similar to previous results for estimation efficiency and detection power by Friston et al. ( 1999), Birn et al. (2002) and Zarahn and Friston (2002). For the uniform distribution, a smallest ISI of 2.5 s together with ISIs which were 5 s apart from each other, i.e., 2.5 s, 7.5 s, 12.5 s, 17.5 s, proved to be efficient, and the maximin design had ISIs 2.5 s, 7.5 s and 12.5 s for linearity as well as nonlinearity. The maximin designs MMD_geo and MMD_uni are useful because they have a high relative efficiency, needed less computation time than the maximin design MMD_GA in our situation (Table III) and they are above all available for any number of stimulus types in contrast to m-sequences. However, it is necessary to simulate several random sequences for the maximin design MMD_geo or MMD_uni and choose the best one among these sequences in terms of its minimum relative efficiency as not any randomly generated sequence might be efficient. The least efficient designs were the maximin blocked design and the maximin constant ISI design. The maximin blocked design had a block length of 25 s for linearity and a block length of 22.5 s for nonlinearity. To avoid nonlinearity completely and if one wants to use a constant ISI design, the maximin constant ISI design with 17.5 s can be recommended.

Optimal designs for estimation and detection power were studied to show how much efficiency is lost by choosing the incorrect model and optimization criterion for the final data analysis. It was seen that optimal designs for one purpose might not be efficient for another purpose. For example, the optimal design for detection power Opt_det had a medium relative efficiency compared to the maximin design MMD_GA based on the nonlinear model. In addition, the maximin design MMD_GA did not perform well for estimation efficiency or detection power. It is thus relevant to specify the model and the optimization criterion in consistency with the final analysis.

The properties of the optimal design for nonlinear estimation can be determined from the most efficient and most flexible design MMD_GA. The maximin design MMD_GA had a stimulus frequency close to 0.5 which was also the optimal stimulus frequency for the geometric distribution under linearity. It can thus be concluded that for one stimulus type and under linearity of the HRF the optimal stimulus frequency for the design for nonlinear estimation is 0.5 or a close value. This frequency 0.5 equals the D-optimal frequency for detection power and estimation efficiency p_D = 1/(Q+1) for Q=1 and uncorrelated errors [Maus et al., 2010]. Under linearity a mean ISI of 5 s can be recommended while under nonlinearity the mean ISI should be slightly higher.

Furthermore, a mixture of a blocked and a rapid event-related design seems to be the optimal design for nonlinear estimation. The explanation for this might be that the nonlinear parameters of the double gamma function push the optimal design into another direction than the amplitude parameter β. The optimal estimation for the nonlinear double gamma parameters and the optimal estimation of the second till sixth entry in τ, which are products of the nonlinear parameters with the amplitude parameter β, may demand a rapid event-related design. In contrast, the first parameter of τ in Eq. (6), the amplitude parameter β, may be efficiently estimated by a blocked design.

One restriction of our study could be that a limited range for the HRF parameters was considered. The high minimum relative efficiencies above 0.8 of the locally optimal designs obtained by the GA could be due to the limited range for the HRF parameters. Additional numerical calculations were performed with a broader range of the HRF parameters a₁ and b₁, while the parameters for the undershoot were set to the canonical value of a₂ = 15, b₂ = 1 and c = 6. Moreover, results were obtained for a broader range of the parameters for the undershoot with the parameters of the peak a₁ and b₁ fixed to a standard value of a₁ = 5 and b₁ = 1. For these two parameter spaces, the minimum relative efficiencies of the locally optimal designs obtained by the GA were all above 0.78 which is similar to the results in the section on the genetic algorithm for the limited simultaneous parameter space of all HRF parameters.

Another restriction might be that we considered only one stimulus type, and that the results may not extend to multiple trial types. However, to estimate the subject-specific parameters of a nonlinear HRF model like the double gamma function, it would be useful to perform a run of a simple design with only one stimulus type. From this run, the active voxels or regions can be localized and the HRF parameters can be estimated such that they can be used for further data analyses from experiments with more stimulus types. Our code for the GA can also be used for multiple trial types assuming that the HRFs of these different trial types have the same double gamma function parameters and differ only in amplitude. In the following, we will discuss which results are expected to be valid for more than one stimulus type. Further work is nevertheless needed to draw firm conclusions on optimal designs with multiple trial types for nonlinear estimation of the HRF.

The best maximin design seems to be a mixture of a blocked and a rapid event-related design, and the optimal trade off between “blockiness” and “event-relatedness” can best be achieved by an open search in the design space as performed by the GA. As a consequence, we expect that the following order of efficiency from most efficient to least efficient will extend to multiple trial types: MMD_GA, MMD_m/MMD_geo/MMD_uni/Opt_est, MMD_B/Opt_det, MMD_CI. The computation time for the maximin designs is expected to be from highest to lowest: MMD_GA, MMD_geo/MMD_uni, MMD_m/MMD_B/MMD_CI.

The optimal stimulus frequency in this article was close to 1/(Q+1) and may extend to higher number of stimulus types Q. The reason for this extension is that the optimality criterion for nonlinear estimation favors a mixture of a blocked design and a rapid event-related design and seems thus to behave like a multiobjective criterion combining detection power and estimation efficiency [Wager and Nichols, 2003; Kao et al., 2009]. The D-optimal stimulus frequency for detection power and for estimation efficiency is 1/(Q+1) and a combined criterion should have the same D-optimal frequency [Maus et al., 2010]. Furthermore, as the MMD_GA is presumed to be between a blocked design and a rapid event-related design, the mean block length of the maximin design MMD_GA is expected to be between the mean block length of the optimal design for estimation efficiency and detection power.

We presented a common method in optimal design theory to determine optimal designs for nonlinear models. A Taylor approximation was used to linearize the nonlinear model and the maximin design was chosen from the locally D-optimal designs. This method can be used to perform further research on optimal designs for estimation of the nonlinear parameters of the double gamma function or parameters of other nonlinear models in fMRI data analysis.