fMRI analysis with the general linear model: removal of latency-induced amplitude bias by incorporation of hemodynamic derivative terms
Introduction
The traditional approach for analyzing functional magnetic resonance imaging (fMRI) data utilizes the general linear model using a hypothesized neural model convolved with a canonical hemodynamic response function. Mismatches of the data to the specified hemodynamic model can be induced by, for example, small hemodynamic delays or slice-timing differences. The use of a hemodynamic model and its temporal derivative for fMRI analysis was proposed in Friston et al. (1998) as a parsimonious model with additional flexibility to address delay-induced modeling mismatches. In Friston et al. (1998), it was suggested to calculate the amplitude parameter estimates from the nonderivative terms only. The effects modeled by the derivative terms were interpreted as a shift of the hemodynamic model in time. In subsequent work, it was concluded that the hemodynamic response function plus temporal derivative produced the most sensitive analyses for event-related fMRI analyses (Hopfinger et al., 2000). It has since become common practice to fit the full model (nonderivative and derivative together) but to use only the nonderivative terms as estimates of hemodynamic amplitude and to test for amplitude differences using a t test (e.g., Bunge et al., 2002, Cabeza et al., 2003, Kiehl et al., 2001, McGonigle et al., 2002).
Such an approach for estimating the amplitude ignores the potential for an amplitude bias induced by a delay difference between the hemodynamic model and the data. This amplitude bias is due to the use of only the nonderivative portion of the model in the test for significant amplitudes. This effect has been observed previously, but was used to justify not using the temporal derivative (Della-Maggiore et al., 2002). We propose instead testing an amplitude estimate that is a function of both the nonderivative and the derivative terms of the model. Using simulations, we show that the proposed amplitude test does not suffer from delay-induced bias, is a more natural test for amplitude differences when using a model incorporating temporal derivatives, and improves the fit of the model to the data (when compared to a model not using the temporal derivative term). We apply the proposed test in a random-effects analysis of 100 subjects and reveal increased amplitudes in areas consistent with the task, with the largest increases in regions consistent with greater hemodynamic delays.
Section snippets
Theory
In the simplest case, the data (assumed to be zero mean) at a given voxel are modeled as:where y is the data, x is the signal activation model, and ε is the residual error. The error is often modeled as zero-mean Gaussian, independent and identically distributed, with variance σ2, written . The activation amplitude, β̂1, is then typically estimated using least-squares and tested using a t test as:where ε , the residual error, is N × 1 for time
Simulations
We generated a synthetic fMRI model waveform consisting of five events modeled as delta functions spaced 15-s apart and convolved with the default hemodynamic response function in the Statistical Parametric Mapping software package, SPM99 (Worsley and Friston 1995). Simulated MRI data with different delays were created by adding Gaussian noise and shifting this model by (1) 0 s, (2) 1 s, (3) 2 s, or (4) 3 s relative to the model waveform. Each of these four artificial fMRI data sets was
Results
The simulations show that, as expected, the amplitude estimates for either the nonderivative term or for both terms is the same as the underlying “true” response when no delay is present. As the delay increases, the amplitude estimate for the nonderivative terms decreases, with significant decreases occurring even for delays as small as 1 s. For delays of 3 s, the amplitude estimate is reduced by a factor of 3 from the correct value. When utilizing both terms, the amplitude estimate is much
Discussion
The current work has two main purposes. First, we have demonstrated that the practice of incorporating temporal derivatives into an fMRI analysis and contrasting only the nonderivative terms can introduce significant latency-induced amplitude bias, even for delays as small as 1–2 s. Secondly, we propose a method for incorporation of both the nonderivative and derivative terms into the analysis and show that it mitigates the latency-induced amplitude bias even for larger delays. This is
Acknowledgements
We would like to thank the research staff and MR technicians at the University of British Columbia and Hartford Hospital for their invaluable assistance in this project. This research was supported in part by grants from the Dr. Norma Calder Foundation (Liddle), Medical Research Council of Canada (Liddle) and the Institute of Living at Hartford Hospital Open Competition Grant (Kiehl), and a NARSAD Young Investigator Grant (Kiehl).
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