dependence_metrics.rst

TE (In)Dependence Models

Functional MRI signal can be described in terms of fluctuations in S₀ and T₂^*. In the below equation, S(t, TE_k) is signal at a given time t and for a given echo time TE_k. S̄(TE_k) is the mean signal across time for the echo time TE_k. ΔS₀(t) is the difference in S₀ at time t from the average S₀. ΔR₂^*(t) is the difference in R₂^* at time t from the average R₂^*.

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

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 S₀ or T₂^*, one can break this overall model into submodels by zeroing out certain terms.

Important

Remember- R₂^* is just $\frac{1}{T_2^*}$

For a TE-independence model, if there were no fluctuations in T₂^*:

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

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 S̄(TE_k) (mean echo-wise signal).

Thus, the model becomes ΔS(TE_k) = S̄(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 S̄(TE_k). Thus, to assess the TE-independence of a component, we use the model PE(TE_k) = S̄(TE_k) * X, fit X to the data, and evaluate model fit.

For a TE-dependence model, if there were no fluctuations in S₀:

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

ΔS(TE_k) = S̄(TE_k) *  − ΔR₂^* * TE_k

ΔS(TE_k) = S̄(TE_k) * TE_k * X

PE(TE_k) = S̄(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 S₀ will fluctuate. In the other, only T₂^* will fluctuate.

Caution

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 S₀ and T₂^*.

We can see that ΔS(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 S₀-fluctuating data and scales roughly linearly with TE for the T₂^*-fluctuating data.

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

As expected, the S₀ model fits perfectly to the S₀-fluctuating dataset, while the T₂^* model fits quite well to the T₂^*-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 T₂^*- and S₀-based fluctuations, so here we have generated two time series- one T₂^*-based and one S₀-based. Both time series share the same level of percent signal change (a standard deviation equivalent to 5% of the mean), although the mean S₀ (16000) is very different from the mean T₂^* (30).

We can then average those two time series with different weights to create components that are T₂^*- or S₀-based to various degrees. In this case, both T₂^* and S₀ 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 T₂^* and S₀ fluctuations as in the pure T₂^* and S₀ 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 ΔS(TE_k), as seen above. We can thus apply the same TE-dependence and -independence models as above, in order to calculate single-voxel ρ and κ 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.

Attention

You may also notice that, despite the fact that T₂^* and S₀ fluctuate the same amount and that both contributed equally to the component, ρ is much higher than κ.