**3. Simulation results**

It is noted that the spatial pattern correlations are applied to the multivoxel matrices (in notation of [**r**]) of the χ source, the fieldmap, and the images, which are all discretized at a

It is straightforward to implement 4D BOLD fMRI simulation based on the repetition of volumetric BOLD fMRI simulations for each snapshot capture over a BOLD activity. First, we need to define a task-evoked 4D BOLD χ change, as illustrated in **Figure 6**. Specifically, we configure a 3D vasculature-laden VOI and provide a 3D χ distribution for a brain VOI state. A local χ change is simulated with a spatiotemporal modulation by a NAB and a task paradigm (in Eq. (1)). For the weak BOLD response detection, the task paradigm is usually designed as a boxcar waveform for repetition measurement of BOLD signals. We may define a positive NAB for an excitatory BOLD response and a negative NAB for an inhibitory response. The static background χ0 may be assigned to a water pool (χ0 = χwater) or be empty (χ0 = 0) with an

**Figure 6.** Illustration of 4D BOLD χ response simulations. A VOI is filled local Δχ change with positive and negative Δχ responses superimposed on a static background in the presence of noise. A BOLD event is represented by a times‐ eries of the 3D Δχ snapshot distributions in Eq. (1) through a spatiotemporal modulation by NAB(**r**) and task(*t*).

The 4D BOLD fMRI simulation involves a predefined 4D source χ[**r**,*t*] and two output 4D images (*A*[**r**,*t*] for magnitude and *P*[**r**,*t*] for phase, as defined in Eq. (8)). Conventional BOLD fMRI exploits the 4D magnitude dataset *A*[**r**,*t*] for functional analysis. For a task-evoked BOLD fMRI simulation, the functional activity map can be extracted from a 4D dataspace, Λ[**r**,*t*] =

*x y z t task t x y z tcorr x y z t task t std x y z t std task t*

cov( [ , , , ], [ ]) [ , , ] [ , , , ], [ ] ( [ , , , ]) ( [ ])

L

(14)

t

{χ[**r**,*t*], *A*[**r**,*t*], *P*[**r**,*t*]} by a temporal correlation (*tcorr)* map that is defined by

( ) <sup>L</sup> L =L º

*true recon*

L cc

with 'A' 'P' ' ', ' '}

 *={ , ,*

*tcorr*

spatial resolution of the same voxel size.

14 Numerical Simulation - From Brain Imaging to Turbulent Flows

**2.8. 4D BOLD fMRI simulation**

additive Gaussian noise.
