Effect of voluntary repetitive long-lasting muscle contraction activity on the BOLD signal as assessed by optimal hemodynamic response function

Abstract

Objective

Among other neuroimaging techniques, functional magnetic resonance imaging (fMRI) can be useful for studying the development of motor fatigue. The aim of this study was to identify differences in cortical neuronal activation in nine subjects on three motor tasks: right-hand movement with minimum, maximum, and post-fatigue maximum finger flexion.

Materials and methods

fMRI activation maps for each subject and during each condition were obtained by estimating the optimal model of the hemodynamic response function (HRF) out of four standard HRF models and an individual-based HRF model (ibHRF).

Results

ibHRF was selected as the optimal model in six out of nine subjects for minimum movement, in five out of nine for maximum movement, and in eight out of nine for post-fatigue maximum movement. As compared to maximum movement, a large reduction in the total number of active voxels (primary sensorimotor area, supplementary motor area and cerebellum) was observed in post-fatigue maximum movement.

Conclusion

This is the first approach to the evaluation of long-lasting contraction effort in healthy subjects by means of the fMRI paradigm with the use of an individual-based hemodynamic response. The results may be relevant for defining a baseline in future studies on central fatigue in patients with neuropathological disorders.

Appendix: HRF models

In the fMRI data analysis, for each voxel, the observed response vector (${\mathbf{y}} \in \Re^{N \times 1}$ with N equal to the number of the acquired volumes), is expressed as a linear combination of explanatory variables, contained in the known matrix (${\mathbf{X}} \in \Re^{N \times M}$ with M equal to the number of the unknown regression coefficients), commonly called the design matrix, via the unknown parameter vector $({\varvec{\beta}} \in \Re^{M \times 1} )$, plus an error term $({\varvec{\varepsilon}} \in \Re^{N \times 1} )$:

$$ y = {\mathbf{X\beta }} + {\varvec{\varepsilon}}. $$

(2)

The theoretical basis of GLM in fMRI analysis can be found in [13].

The design matrix X is crucial for the results of this study. Its first column is a vector of ones and the second column contains values, at acquisition time points n = 1,2, … ,N, equal to the convolution of the stimulus function, u, with the impulse response, HRF, representing the hemodynamic response function:

$$ {\mathbf{X}}(n) = \left[ {1(n)\;{\mathbf{u}} \otimes {\mathbf{HRF}}(n)} \right] $$

(3)

where $\otimes$ denotes the convolution operation. Additional columns are eventually present when the time and dispersion derivatives of the HRF are introduced. The external stimulus function u is known since it depends on the experimental set-up, with values equal to 1 or 0 denoting the presence or absence of a stimulus, respectively.

Instead of postulating a model for HRF, we selected an optimal model out of five candidates. The general template for all models, except for the FIR, is derived from [25]:

$$ {\mathbf{HRF}}(k) = \left\{ {\begin{array}{*{20}c} 0 & {0 < k < \delta_{1} } \\ {a_{1} \; \cdot \;[(k - \delta_{1} )/\tau_{1} ]^{2} \; \cdot \;\exp [1 - ((k - \delta_{1} )/\tau_{1} )^{2} ]} & {\delta_{1} < k < \delta_{2} } \\ {a_{1} \; \cdot \;[(k - \delta_{1} )/\tau_{1} ]^{2} \; \cdot \;\exp [1 - ((k - \delta_{1} )/\tau_{1} )^{2} ] - a_{2} \; \cdot \;[(k - \delta_{2} )/\tau_{2} ]^{2} \; \cdot \;\exp [1 - ((k - \delta_{2} )/\tau_{2} )^{2} ]} & {\delta_{2} < k} \\ \end{array} } \right. $$

(4)

where HRF is sampled at time points k = 1, 2…K; a ₁ and a ₂ are the magnitude of the peak and undershoot terms, respectively; τ ₁ and τ ₂ are related to the width, peak height and time to peak; δ ₁ and δ ₂ to the time-to-onset. The time points of occurrence of the positive (k _max) and negative (k _min) peaks are:

$$ k_{ \hbox{max} } = \delta_{1} + \tau_{1} $$

(5)

$$ k_{ \hbox{min} } = \delta_{2} + \tau_{2} . $$

The vector of model parameters is: (a ₁, a ₂, τ ₁, τ ₂, δ ₁)^T.

Differently, the most flexible FIR model makes minimal assumptions about the shape of the response [36]. FIR functions increase the degrees of freedom used in the design matrix and allow less powerful statistical tests. The FIR set consists of contiguous stick functions, each lasting T/K _FIR s, where T is the duration of HRF and K _FIR the number of functions of the basis set [61]. In our case, we assumed K _FIR equal to the ratio of window length and TR (K _FIR = 12). Although we made this assumption, the number of basis function can be independent of the sampling period [62].