**2.1. Definition of 3D vasculature and magnetic susceptibility source (χ)**

The initial step of BOLD fMRI simulation is to configure a χ-expressed BOLD activity, thereby providing a BOLD χ source for fMRI. We define a brain cortex volume of interest (VOI) with a tissue background and fill it with a cortex vasculature, thus simulating a brain cortex region. Let χ0(**r**) denote a static 3D χ distribution of parenchymal tissue in VOI and Δχ(**r**,*t*) the vascular blood χ change associated with a BOLD activity, with **r** = (*x,y,z*) denoting the spatial coordinates in VOI, then the dynamic 4D χ source is given by

$$\begin{aligned} \boldsymbol{\chi}(\mathbf{r},t) &= \boldsymbol{\chi}\_{\text{o}}(\mathbf{r}) + \Delta \boldsymbol{\chi}(\mathbf{r},t) + \text{noise} \\ \text{with} \\ \boldsymbol{\chi}\_{\text{o}}(\mathbf{r}) &= \boldsymbol{\chi}\_{\text{min}}(\mathbf{r}) & \text{(parencchynami tissue)} \\ \Delta \boldsymbol{\chi}(\mathbf{r},t) &= \text{Hct} \cdot \boldsymbol{\chi}\_{\text{do}} \cdot (\mathbf{l} - Y(t)) \cdot \text{NAB}(\mathbf{r}) \cdot V(\mathbf{r},t) & \text{(BOLD activity)} \end{aligned} \tag{1}$$

where *Hct* denotes the blood hemocrit (*Hct* = 0.4 for normal blood), χdo the magnetic suscept‐ ibility difference between deoxygenated and oxygenated blood tissues (χdo = 0.27 × 4π ppm (in SI unit)), *Y*(*t*) the blood oxygenation level (*Y* ∈ [0,1]), NAB(**r**) the local neuroactive blob distribution, and *V*(**r**,*t*) the vasculature geometry in VOI. The explicit *t* variable indicates a possible change during a BOLD activity, such as cerebral blood volume change in *V*(**r**,*t*) and oxygenation level change in *Y*(*t*). For the sake of simulating fMRI signals, a pure BOLD activity is expressed by a dynamic blood magnetic susceptibility change, Δχ(**r**,*t*), which serves as the magnetic source for BOLD MRI simulation. In practice, the BOLD activity provides an additive term, Δχ(**r**,*t*) (a perturbation term), on a background distribution χ0(**r**) in Eq. (1).

A local functional activity is defined by a 3D spatial distribution of NAB(**r**) (a neuroactive blob centered at **r** in VOI). For the sake of numerical representation, we assume a NAB by a Gaussian-shaped blob (with soft boundary) or a ball-shaped blob (with hard boundary). A NAB defines a local activity distribution in VOI, which presents with an ON state (active state) and vanishes with an OFF state (resting state) by a temporal modulation of a designed task paradigm. We may define an excitatory activity by a positive distribution (NAB(**r**) > 0) or an inhibitory activity by a negative distribution (NAB(**r**) < 0), in relation to the static background distribution. For the numerical simulation of a BOLD activity, we define a BOLD χ response by a spatiotemporal modulation model in Eq. (1). A brain active state gives rise to Δχ(**r**,*t*) ≠ 0 in NAB and at a task "ON" epoch, and a brain resting state is numerically characterized by Δχ(**r**,*t*) = 0 over the VOI in Eq. (1).

It is mentioned that the BOLD χ expression in a brain activity is simply simulated by a spatial modulation model in Eq. (1), where a neuronal activity is defined by a local blob that shapes a local blood Δχ map by a spatial multiplication. We also simplify the BOLD χ source simu‐ lation by ignoring the hemodynamic response function (hrf), which otherwise could be accounted for by convoluting Δχ(**r**,*t*) with a kernel of hrf (usually adopting a canonical hrf that is characteristic of a high upshoot followed by a small undershoot).

A BOLD χ change happens inside the vascular blood stream. We need to configure the vasculature geometry *V*(**r**,*t*) by filling the VOI with cluttered vessels with a blood volume fraction (*bfrac*), as expressed by

#### BOLD fMRI Simulation http://dx.doi.org/10.5772/63313 7

$$V(\mathbf{r},t) = \begin{cases} 1, & \mathbf{r} \in \text{vessel} \\ 0, & \text{otherwise} \end{cases} \text{ s.t. } \text{ } b\text{frac}(t) = \frac{\sum\_{(\mathbf{r},\mathbf{y},z)} V(\mathbf{r},t)}{\text{size}(V)} \tag{2}$$

where the *t* variable is reserved to incorporate the change in cerebral blood volume as a result of vasodilation/vasoconstriction in a BOLD activity. A static vasculature is included as a binary volume *V*(**r**) that remains stationary during a BOLD activity. The random vascular geometry is generated under a control of *bfrac* = [0.02, 0.04] for cortex vasculature simulation [1, 8, 11–13].

Let χ0(**r**) denote a static 3D χ distribution of parenchymal tissue in VOI and Δχ(**r**,*t*) the vascular blood χ change associated with a BOLD activity, with **r** = (*x,y,z*) denoting the spatial coordinates

> ( ) ( ) (parenchymal tissue) ( , ) (1 ( )) ( ) ( , ) (BOLD activity)

where *Hct* denotes the blood hemocrit (*Hct* = 0.4 for normal blood), χdo the magnetic suscept‐ ibility difference between deoxygenated and oxygenated blood tissues (χdo = 0.27 × 4π ppm (in SI unit)), *Y*(*t*) the blood oxygenation level (*Y* ∈ [0,1]), NAB(**r**) the local neuroactive blob distribution, and *V*(**r**,*t*) the vasculature geometry in VOI. The explicit *t* variable indicates a possible change during a BOLD activity, such as cerebral blood volume change in *V*(**r**,*t*) and oxygenation level change in *Y*(*t*). For the sake of simulating fMRI signals, a pure BOLD activity is expressed by a dynamic blood magnetic susceptibility change, Δχ(**r**,*t*), which serves as the magnetic source for BOLD MRI simulation. In practice, the BOLD activity provides an additive

A local functional activity is defined by a 3D spatial distribution of NAB(**r**) (a neuroactive blob centered at **r** in VOI). For the sake of numerical representation, we assume a NAB by a Gaussian-shaped blob (with soft boundary) or a ball-shaped blob (with hard boundary). A NAB defines a local activity distribution in VOI, which presents with an ON state (active state) and vanishes with an OFF state (resting state) by a temporal modulation of a designed task paradigm. We may define an excitatory activity by a positive distribution (NAB(**r**) > 0) or an inhibitory activity by a negative distribution (NAB(**r**) < 0), in relation to the static background distribution. For the numerical simulation of a BOLD activity, we define a BOLD χ response by a spatiotemporal modulation model in Eq. (1). A brain active state gives rise to Δχ(**r**,*t*) ≠ 0 in NAB and at a task "ON" epoch, and a brain resting state is numerically characterized by

It is mentioned that the BOLD χ expression in a brain activity is simply simulated by a spatial modulation model in Eq. (1), where a neuronal activity is defined by a local blob that shapes a local blood Δχ map by a spatial multiplication. We also simplify the BOLD χ source simu‐ lation by ignoring the hemodynamic response function (hrf), which otherwise could be accounted for by convoluting Δχ(**r**,*t*) with a kernel of hrf (usually adopting a canonical hrf that

A BOLD χ change happens inside the vascular blood stream. We need to configure the vasculature geometry *V*(**r**,*t*) by filling the VOI with cluttered vessels with a blood volume

is characteristic of a high upshoot followed by a small undershoot).

(1)

in VOI, then the dynamic 4D χ source is given by

6 Numerical Simulation - From Brain Imaging to Turbulent Flows

*t t*

do

D = × ×- × ×

 c

*t Hct Y t NAB V t*

term, Δχ(**r**,*t*) (a perturbation term), on a background distribution χ0(**r**) in Eq. (1).

**r r r**

( , ) ( ) ( , ) noise

= +D +

 c

0

**r rr**

0 static

=

 c

**r r**

with

c

Δχ(**r**,*t*) = 0 over the VOI in Eq. (1).

fraction (*bfrac*), as expressed by

c

cc

**Figure 2.** Illustrations of VOI vasculature and BOLD Δχ source. The VOI is filled with (a1) random vessels (cylindrical segments) and (b1) spheric beads. The NAB-modulated Δχ distributions are shown in (a2) and (b2), respectively, with a *y*0-slice. It is noted Δχ may assume positive and negative values in local NAB regions.

In order to maintain a control of constant *bfrac* for cortex vasculature over different regions or across multiresolution subregions, we may fill a VOI with random beads instead of cluttered vessels. In **Figure 2** (**a1**,**b1**) are illustrated two brain local vasculature geometries with cluttered cylinders and random beads, under local stimuli by an excitatory blob (in red) and an inhibitory blob (in green). The NAB-modulated BOLD χ response distributions (in an active ON state) are shown in **Figure 2 (a2, b2)** with a *y*0-slice in which the inactive regions (far from NAB) have little or no BOLD responses (Δχ ≈ 0).

In order to numerically depict the vasculature geometry, we need to define the VOI with a large finely gridded 3D matrix with a tiny grid element (gridel) at a scale of micronmeter [14]. For example, a matrix of 2048 × 2048 × 2048 gridels, where a gridel = 2 × 2 × 2 μm3 , is used to represent a small VOI of 4.1 × 4.1 × 4.1 mm3 . The large matrix resulting from VOI gridel sampling offers a quasi-continuous representation of a continuous distribution over VOI. A gridel represents a spin packet (or isochromat) that contains numerous identical proton spins, serving as a mesoscopic representation (at micronmeter scale) between microscopic structure (at atomic and molecular angstrom scale) and macroscopic structure (at millimeter scale of MRI voxels) [15, 16].
