Jeff J. Orchard, M. Stella AtkinsSchool of Computing Science, Simon Fraser University

Freire et al. (1) pointed out that least squares based registration methods are sensitive to the BOLD signal present in fMRI experiments, resulting in stimulus-correlated motion errors.

References

1. Freire L, Mangin J-F. Motion correction algorithms of the brain mapping community create spurious functional activations”, IPMI 2001:246-258.

Fig 6 plots the motion parameter estimates calculated by the least squares registration algorithm and the theoretical estimate of the error, based on Eqn [5].

The shape and distribution of the region of activation affects the motion errors. The theory outlined here is a tool that can be used to study how the region of activation impacts the motion estimates.

A least squares algorithm that incorporates both the motion and activation simultaneously might avoid the interference between the two.

Volume RegistrationLeast squares rigid-body registration involves finding a transformation T that optimally aligns volumes U and V. Thus, we seek a solution (in the least squares sense) to the equation,

),( mT

VU

m

mTTmT

GV

VVV

)0,()0,(),(

where G holds the gradients of V with respect to the motion parameters. Thus, Eqn [1] can be re-written,

The volume U can also be approximated as a copy of V with added activation,

BOLD Signal

AVU

where A is the known activation map, and is the level of activation in U.

Linear Error EstimateThus, for small motions, we can combine Eqns [2] and [3] to get,

AGAVU

GVU

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A simulated dataset of 40 volumes was created by duplicating an EPI volume with dimensions 64x64x30 (Fig 1), adding activation (Fig 2) and 2.5% Gaussian noise. No motion was added to the dataset.

Using a linear stimulus function (Fig 4), a linear trend was also observed in the motion estimates (Fig 5).

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Fig 4. Linear Stimulus Function

Fig 5. Motion estimates for a motionless dataset containing activation governed by the linear stimulus function shown in Fig 4. Notice the linear trends in the motion estimates.

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Fig 3. Motion estimates for a motionless dataset containing activation governed by the stimulus function shown in Fig 2. Notice the high correlation with the stimulus function.

The lease squares solution of Eqn [4] can be found for (the motion parameters). If we assume that U and V are properly aligned, this gives the first-order error in the motion estimate due to activation.

m

m

GVU

Simulations

AGm

[5]

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Fig 6. Motion estimates of the least squares registration algorithm (blue) and the theoretical model (red).

The theoretical model tracks the actual registration error from the least squares registration algorithm. In this case, the model does particularly well at predicting the translation in the anterior/posterior direction, and the pitch angle.

Conclusions

Future Directions

We have shown that the BOLD signal from activation does, in fact, influence the motion estimates calculated by least squares registration algorithms.

This work offers a theoretical justification for the errors observed in least squares registration algorithms due to the presence of the BOLD signal.

[2]

[3]

[1]

More detail...Depending on where linear approximations are applied, different approximation formulas can be derived for the same quantity. For example, from Eqn [2] we can solve for the motion parameters to get,

)( VUG m

Compare the right-hand sides of Eqns [5] and [6]. They are the same, except Eqn [6] has a large, additional matrix. For small motions, that matrix is very close to the identity matrix, and numerical simulations yield almost identical results. Similar arguments can be made for other approximating formulas.

At convergence of the registration algorithm, we have 0G+(U-V). Since we can approximate U as T (V+A), which can in turn be approximated linearly, we get,

AGGG

VAGAVG

VAVG

1

)()(0

)(0

p

p

T

[6]

where is the gradient of A with respect to the motion parameters, and holds the motion estimates (negated because the motion approximation is applied after the addition of the activation, instead of before, as above).

Ap

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Fig 2. Stimulus Function

Fig 1. Slice from the original EPI volume. The overlay shows the region of activation.

+

Fig 3 shows the least squares motion estimates for the simulated dataset. The dataset is motionless, so any detected motion is erroneous. Notice the high correlation between the motion plots and the stimulus function in Fig 2.

[4]

where holds the 6 motion parameters. For small motions, the transformation can be linearly approximated by,

m

Our goal is to offer a theoretical justification for these BOLD-induced registration errors.

Acknowledgments

The authors thank Dr. Bruce Bjornson of the British Columbia’s Children’s Hospital for helpful discussions. Also, this work was supported in part by the Natural Sciences and Engineering Research Council of Canada.