**2. Models and methods**

An overview of a brain BOLD fMRI model is diagramed in **Figure 1**, which consists of a cascade of three modules (stages). Specifically, the "**Source Magnetism**" module provides the pheno‐ typical χ expression of a brain functional biophysiological state, which serves as the source of the "**MRI technology**" module that produces a complex-valued MR image. Upon data acquisition of a 4D fMRI, a postprocessing stage of "**Statistic image analysis**" is performed to extract the brain functional map (fmap). A complete BOLD fMRI simulation implements the three cascaded stages in **Figure 1** by numerical representations and computations.

**Keywords:** magnetic resonance imaging (MRI), blood oxygenation level dependence (BOLD), magnetic susceptibility source, dipole effect, voxelization, complex-valued magnetic resonance signal (image), intravoxel dephasing signal, multivoxel image,

Magnetic resonance imaging (MRI) is a versatile non-invasive imaging technology that has been widely accepted for brain imaging (probing a magnetic state of brain interior). When applied to brain functional imaging, MRI produces a timeseries of images that are construed as an image representation of a brain functional activity. It is believed that any brain activity incurs a cerebral blood oxygenation level dependent (BOLD) magnetic state change that can be detected by MRI [1–4]. Brain functional imaging based on MRI and the endogenous BOLD

In principle, the MRI output is a complex-valued image consisting of a pair of magnitude and phase [5]. Nevertheless, only the MR magnitude image has been exploited for brain imaging (structural or functional). Recent research shows that neither the MR magnitude nor the phase could faithfully represent the brain magnetic state. This is due to a cascade of MRI transfor‐ mations (including linear dipole-convolved magnetization and nonlinear complex modulo/ argument operations [6]). Consequently, conventional BOLD fMRI that is based on MR magnitude imaging may deviate from the underlying brain magnetic source change due to nonlinear data transformations associated with MR magnitude image formation. Since there is a lack of analytic formulation for describing the imaging aspects of BOLD fMRI, we conduct

In the past decades, there have been reports on single-voxel BOLD signal simulations [7–9] and multivoxel 3D BOLD imaging simulations [6, 10, 11]. In this chapter, we first provide a tutorial on the numeric simulations of single-voxel signals and multivoxel images and move

An overview of a brain BOLD fMRI model is diagramed in **Figure 1**, which consists of a cascade of three modules (stages). Specifically, the "**Source Magnetism**" module provides the pheno‐ typical χ expression of a brain functional biophysiological state, which serves as the source of the "**MRI technology**" module that produces a complex-valued MR image. Upon data acquisition of a 4D fMRI, a postprocessing stage of "**Statistic image analysis**" is performed to extract the brain functional map (fmap). A complete BOLD fMRI simulation implements the

three cascaded stages in **Figure 1** by numerical representations and computations.

BOLD fMRI simulation, task correlation

4 Numerical Simulation - From Brain Imaging to Turbulent Flows

numeric simulations to understand the BOLD fMRI model.

forward to address implementing 4D BOLD fMRI simulations.

**1. Introduction**

contrast is termed BOLD fMRI.

**2. Models and methods**

**Figure 1.** A BOLD fMRI model consists of three stages. The stage of "Source Magnetism" provides a dynamic magnetic susceptibility source for the stage of "MRI Technology". The MRI detection produces a 4D complex-valued fMRI data‐ set, which are used for functional imaging and mapping by "Statistical Image Analysis".
