TE (In)Dependence Models

Functional MRI signal can be described in terms of fluctuations in S_0 and T_2^*. In the below equation, S(t, TE_k) is signal at a given time t and for a given echo time TE_k. \bar{S}(TE_k) is the mean signal across time for the echo time TE_k. {\Delta}{S_0}(t) is the difference in S_0 at time t from the average S_0. {\Delta}{R_2^*}(t) is the difference in R_2^* at time t from the average R_2^*.

S(t, TE_k) = \bar{S}(TE_k) * (1 + \frac{{\Delta}{S_0}(t)}{\bar{S}_0} - {\Delta}{R_2^*}(t)*TE_k)

If we ignore time, this can be simplified to

\frac{{\Delta}S(TE_k)}{\bar{S}(TE_k)} = \frac{{\Delta}S_0}{S_0}-{\Delta}{R_2^*}*TE_k

In order to evaluate whether signal change is being driven by fluctuations in either S_0 or T_2^*, one can break this overall model into submodels by zeroing out certain terms.


Remember- R_2^* is just \frac{1}{T_2^*}

For a TE-independence model, if there were no fluctuations in T_2^*:

\frac{{\Delta}S(TE_k)}{\bar{S(TE_k)}} = \frac{{\Delta}S_0}{S_0}

{\Delta}S(TE_k) = {\bar{S}(TE_k)} * \frac{{\Delta}S_0}{S_0}

Note that TE is not a parameter in this model. Hence, this model is TE-independent.

Also, \frac{{\Delta}S_0}{S_0} is a scalar (i.e., doesn’t change with TE), so we can just ignore that, which means we only use {\bar{S}(TE_k)} (mean echo-wise signal).

Thus, the model becomes {\Delta}S(TE_k) = {\bar{S}(TE_k)} * X, where we fit X to the data using regression and evaluate model fit.

For TEDPCA/TEDICA, we use regression to get parameter estimates (raw PEs; not standardized beta values) for component time-series against echo-specific data, and substitute those PEs for {\bar{S}(TE_k)}. Thus, to assess the TE-independence of a component, we use the model PE(TE_k) = {\bar{S}(TE_k)} * X, fit X to the data, and evaluate model fit.

For a TE-dependence model, if there were no fluctuations in S_0:

\frac{{\Delta}S(TE_k)}{\bar{S}(TE_k)} = -{\Delta}{R_2^*}*TE_k

{\Delta}S(TE_k) = {\bar{S}(TE_k)} * -{\Delta}{R_2^*}*TE_k

{\Delta}S(TE_k) = {\bar{S}(TE_k)} * TE_k * X

PE(TE_k) = {\bar{S}(TE_k)} * TE_k * X

Note that TE is a parameter in this model. Hence, it is TE-dependent.

Applying our models to signal decay curves

As an example, let us simulate some data. We will simulate signal decay for two time points, as well as the signal decay for the hypothetical overall average over time. In one time point, only S_0 will fluctuate. In the other, only T_2^* will fluctuate.


To make things easier, we’re simulating these data with echo times of 0 to 200 milliseconds, at 1ms intervals. In real life, you’ll generally only have 3-5 echoes to work with. Real signal from each echo will also be contaminated with random noise and will have influences from both S_0 and T_2^*.


We can see that {\Delta}S(TE_k) has very different curves for the two simulated datasets. Moreover, as expected, \frac{{\Delta}S(TE_k)}{\bar{S}(TE_k)} is flat across echoes for the S_0-fluctuating data and scales roughly linearly with TE for the T_2^*-fluctuating data.

We then fit our TE-dependence and TE-independence models to the {\Delta}S(TE_k) data, which gives us predicted data for each model for each dataset.


As expected, the S_0 model fits perfectly to the S_0-fluctuating dataset, while the T_2^* model fits quite well to the T_2^*-fluctuating dataset.

The actual model fits can be calculated as F-statistics. Then, the F-statistics per voxel are averaged across voxels into the Kappa and Rho pseudo-F-statistics.

Applying our models to spatiotemporal decompositions

Now let us see how this extends to time series, components, and component parameter estimates.

We have the means to simulate T_2^*- and S_0-based fluctuations, so here we have generated two time series- one T_2^*-based and one S_0-based. Both time series share the same level of percent signal change (a standard deviation equivalent to 5% of the mean), although the mean S_0 (16000) is very different from the mean T_2^* (30).

We can then average those two time series with different weights to create components that are T_2^*- or S_0-based to various degrees. In this case, both T_2^* and S_0 contribute equally to the simulated time series. This simulated time series will act as our ICA component for this example.


We also simulate multi-echo data for a single voxel with the same levels of T_2^* and S_0 fluctuations as in the pure T_2^* and S_0 time series above. Here we show time series for a subset of echo times.


And here we can see how those signals decay with echo time (again for only a subset of echo times):


We then run a regression for each echo’s data against the component time series, producing one parameter estimate for each echo time. The parameter estimates match the signal decay curve for {\Delta}S(TE_k), as seen above. We can thus apply the same TE-dependence and -independence models as above, in order to calculate single-voxel \rho and \kappa values. Note that the metric values are extremely high, due to the inflated degrees of freedom resulting from using so many echoes in the simulations.


You may also notice that, despite the fact that T_2^* and S_0 fluctuate the same amount and that both contributed equally to the component, \rho is much higher than \kappa.