The Mind Research Network and LBERI, Albuquerque 87106, NM, USA

Department of Electrical and Computer Engineering, Albuquerque 87131, NM, USA

Abstract

Background

The blood oxygenation-level dependent (BOLD) functional magnetic resonance imaging (fMRI) modality has been numerically simulated by calculating single voxel signals. However, the observation on single voxel signals cannot provide information regarding the spatial distribution of the signals. Specifically, a single BOLD voxel signal simulation cannot answer the fundamental question: is the magnetic resonance (MR) image a replica of its underling magnetic susceptibility source? In this paper, we address this problem by proposing a multivoxel volumetric BOLD fMRI simulation model and a susceptibility expression formula for linear neurovascular coupling process, that allow us to examine the BOLD fMRI procedure from neurovascular coupling to MR image formation.

Methods

Since MRI technology only senses the magnetism property, we represent a linear neurovascular-coupled BOLD state by a magnetic susceptibility expression formula, which accounts for the parameters of cortical vasculature, intravascular blood oxygenation level, and local neuroactivity. Upon the susceptibility expression of a BOLD state, we carry out volumetric BOLD fMRI simulation by calculating the fieldmap (established by susceptibility magnetization) and the complex multivoxel MR image (by intravoxel dephasing). Given the predefined susceptibility source and the calculated complex MR image, we compare the MR magnitude (phase, respectively) image with the predefined susceptibility source (the calculated fieldmap) by spatial correlation.

Results

The spatial correlation between the MR magnitude image and the magnetic susceptibility source is about 0.90 for the settings of T_{E }= 30 ms, B_{0 }= 3 T, voxel size = 100 micron, vessel radius = 3 micron, and blood volume fraction = 2%. Using these parameters value, the spatial correlation between the MR phase image and the susceptibility-induced fieldmap is close to 1.00.

Conclusion

Our simulation results show that the MR magnitude image is not an exact replica of the magnetic susceptibility source (spatial correlation ≈ 0.90), and that the MR phase image conforms closely with the susceptibility-induced fieldmap (spatial correlation ≈ 1.00).

Background

Blood oxygenation-level dependent (BOLD) functional magnetic resonance imaging (fMRI) has been widely accepted for brain functional mapping and neuroimaging _{2}). The neuroactivity-induced biomagnetic susceptibility perturbation can be detected by T2*-weighted MRI (T2*MRI)

In past decades, there have been many published reports on numerical simulations of BOLD mechanism

Since T2*MRI is designed to sense an inhomogeneous fieldmap that is established via magnetization of an inhomogeneous susceptibility distribution, the underlying source of fMRI is the susceptibility-expressed distribution of a neurovascular coupling state. Therefore, for numerical BOLD fMRI simulation, we need numerically characterize the neurovascular coupling process in terms of biomagnetism susceptibility perturbation for the purpose of T2*MRI detectibility. Given a susceptibility map (representing a snapshot of dynamic BOLD susceptibility perturbation), we can carry out T2*MRI simulation in a way similar to BOLD voxel signal simulations

In this paper, we propose a volumetric BOLD fMRI model that deals with a cortical field of view (FOV), as a whole matrix (without dividing into submatrices), in which the magnetic influence among the intra-FOV vasculatures are accounted for during the fieldmap calculation using a Fourier technique

The neurovascular coupling process is a complicated neurophysiological process. Experiments have shown that the BOLD activity in response to a neuroactivity may be described as a linear response model _{2}:cerebral metabolism rate of oxygen).

A BOLD fMRI study produces a 4D MR dataset, of which each 3D volume at a time point is interpreted as a snapshot of a dynamic BOLD state. The 4D BOLD fMRI data acquisition can be numerically simulated by a train of 3D T2*MRI snapshots provided the dynamic susceptibility perturbation is numerically specified for each snapshot time. In this report, we will only focus on the snapshot capture of a susceptibility-expressed BOLD physiological state at a specific time point. Our approach is general and we can implement a dynamic 4D BOLD fMRI simulation by repeating the snapshot imaging at a series of time points at more computation cost.

Methods

We motivate our approach by looking at a typical neuroimage as shown in Figure

Illustration of a local neuroactive blob (NAB) observed at cerebral cortex by fMRI experiment

**Illustration of a local neuroactive blob (NAB) observed at cerebral cortex by fMRI experiment**. Conventionally, a NAB is inferred from an fMRI data and is considered as a local neuronal activity. However this convention has not been numerically verified from fMRI physics. This paper proposes a volumetric BOLD fMRI computation model to quantitatively examine the multivoxel fMRI formation of a magnetic susceptibility expressed BOLD state under a linear NAB-modulated neurovascular coupling assumption. Simulations are rendered over a 3D cortical FOV that contains a NAB with adequate margins in an array of vasculature-laden voxels. (The image shown in Figure 1 was taken with the written consent of the subject for a separate study in our group. The image acquisition protocol was approved by the Mind Research Network review board).

The overall diagram for the computational model of neurovascular coupling and BOLD fMRI is shown in Figure

Overall diagram of the computational model for neurovascular coupling and BOLD fMRI

**Overall diagram of the computational model for neurovascular coupling and BOLD fMRI**. It is decomposed into a neurophysiology module (upper dashed box) and a MRI technology module (down dashed box). The linear neurovascular coupling model accounts for the local neuroactivity, blood oxygenation, and cortical vasculature by a spatiotemporal modulation. The complex fMRI dataset is resulted from susceptibility magnetization and intravoxel dephasing (T2* imaging). The complex fMRI dataset can be used for backward mappings as diagrammed by double-directed arrows. It is noted the conventional neuroimaging is diagrammed by an overall backward mapping as diagrammed by a grey double-directed arrow of A ~ NAB on the left-hand side. In this paper, we focus on the mapping from the MR magnitude image to the susceptibility source (as diagrammed by the black double-directed arrow of A~Δχ), and that from the MR phase image to the fieldmap (P~ΔB).

Neurovascular coupling formulation

It is known in neurobiology and neurophysiology that a neuroactivity is accompanied by a complicated process of cellular, metabolic, and vascular processes. For simplification of computational implementation, we express the spatial distribution for the functional parcellation of a neuroactivity by a 3D Gaussian-shaped NAB embedded in a cortical FOV (with size D_{0 }× D_{0 }× D_{0}) by

where (_{x}, σ_{y}, σ_{z}
_{x }
_{y }
_{z}
_{x }= _{y }<< _{z }for long ellipsoid or cylinder, _{x }≠ _{y }≠ _{z }for general ellipsoid), (x_{0}, y_{0}, z_{0}) denotes the location of the NAB in the FOV, and c = max(NAB) represents the maximum activity at the NAB center (the activity strength is scaled by 0 < c < 1). We should mention that a non-spherical blob like the one in Figure _{0}, y_{0}, z_{0}) and (_{x}, _{y}, _{z}). The graded local neuroactivity strength over the FOV is reflected in the spatially distributed multivalues in the range of [0, c] (maximum activity strength at the NAB center, moderate strength at the NAB boundary, and zero strength outside the NAB).

In response to the neuroactivity in a NAB, the neurovascular unit regulates the cerebral microcirculation by vasodilation (increase in CBV) and a burst of blood inflow (increase in CBF). The intravascular blood oxygenation level is subject to change due to neuromodulation in the NAB, primarily occurring in the capillaries and venules. Figure

A 2D illustration of a localized neurovascular coupling model

**A 2D illustration of a localized neurovascular coupling model**. The local neuroactivity is represented by a ball-shaped neuroactive blob (NAB), which confines the locality and the graded strength in a cortical FOV. The red and blue lines illustrate the oxygen-rich arterioles and oxygen-poor venules. Due to spatial weighting by NAB, the cortical regions outside the NAB has no contribution to the BOLD state. (In numerical BOLD fMRI simulation, the neuron-glial clusters are considered as the extravascular parenchyma, thus being omitted).

At a snapshot time, the cortical vasculature geometry determines the intravascular blood volume, that only takes up a fractional space of the cortical FOV, as described by blood volume fraction (

where

It is also known that only the red blood cells in blood stream convey oxygen that contribute to intravascular blood susceptibility perturbation. The total volume of red blood cells in normal blood is about 40%, as described in terms of hematocrit (Hct≈0.4). Blood physiology also shows that a red blood cell can carry up to 4 oxygen molecules (via attachment to 4 heme groups in a hemoglobin). Due to the oxygen detachment during microcirculation, the deoxygenated blood reveals more paramagnetism than the oxygenated blood, that is _{deoxy }> _{oxy}. Let Y(t) represent the dynamic blood oxygenation level, then (1-Y(t)) represents the dynamic deoxygenation level, that is a parameter to reflect the cerebral metabolism of oxygen (CMRO_{2}). It is noted that Y(t) = [0,1], with Y = 1 for the fully oxygenated blood in artery and Y = 0 for the fully deoxygenated blood in vein.

Based on the neurovascular-coupled blood biomagnetic perturbation mechanism, we propose a biomagnetic susceptibility expression formula for a linear neurovascular coupling model by

where _{do }= _{deoxy}-_{oxy }= 0.27 × 4_{total }includes contributions from both intravascular blood and extravascular tissue parenchyma. The susceptibility distribution of a selected baseline state is denoted by χ_{base}. In reference to χ_{base}, we can characterize a BOLD susceptibility perturbation state by Δχ in Eq. (3). Suppose the BOLD fMRI system is a linear digital imaging system, then the behavior of MR image change in reference to its baseline state can represent the intravascular susceptibility perturbation Δ

This is a computational model for linear neurovascular coupling, which provides a mathematical formula for numerically expressing the NAB-modulated BOLD response process in a vasculature-filled FOV. Specifically, CMRO_{2 }is accounted for by Y(t), CBF and CBV by _{do }and the blood physiology parameter Hct assume for normal blood, which are experimentally determined constants (see Table

Parameters and settings for numerical simulations

**Parameters**

**Settings**

**Remarks**

B_{0}

3 Tesla

Main static magnetic field

Hct

0.4 (dimensionless ratio)

Blood hematocrit

Y

0.6 (dimensionless ratio)

Oxygenation level = [0,1]

χ_{do}

0.27 × 4π ppm (SI metrics)

= χ_{dexoy }- χ_{oxy}

NAB

NAB(x, y, z) Gaussian ellipsoid

Neuroactive blob (Eq(1))

V(x, y, z)

Binary volume (radii =

Cortical vasculature (Eq(2))

FOV: D_{0 }× D_{0 }× D_{0}

2 × 2 × 2 mm^{3}

Cortical field of view

FOV support matrix

2048 × 2048 × 2048

Digital geometry (1 μm grid)

voxel: d_{0}× d_{0}× d_{0}

32 × 32 × 32, 64 × 64 × 64, 128 × 128 × 128

Three voxel sizes (1 μm grid)

C[x_{n}, y_{n}, z_{n};T_{E}]

64 × 64 × 64, 32 × 32 × 32, 16 × 16 × 16

Complex multivoxel image

A[x_{n}, y_{n}, z_{n};T_{E}]

Assuming non-negative

MR magnitude image(Eq(9))

P[x_{n}, y_{n}, z_{n};T_{E}]

Assuming positive/zero/negative

MR phase image (Eq(9))

corrA

In a range of [0, 1]

Magnitude mapping (Eq(10)

corrP

In a range of [0, 1]

Phase mapping (Eq(11))

T_{E}

In a range of [0, 60] ms

Gradient echo time

For the neurovascular coupling formula in Eq. (4), we need to point out following aspects:

1. It is a linear neurovascular coupling formula that accounts for different neurovascular parameters by a spatiotemporal modulation. For simplicity, we do not consider the hemodynamic time lag, spatial displacement, or spatial response spread in this work, though our model will support it.

2. The BOLD susceptibility perturbation is due to the temporal modulation by the blood deoxygenation level (1-Y(t)), which is an embodiment of CMRO_{2}. The interplay among CBF, CBV and vasculature are numerically characterized by a single parameter

3. Only intravascular blood deoxyhemoglobin contributes to the BOLD susceptibility perturbation; no contribution from extravascular tissue (due to V(x, y, z) = 0 for extravascular region), nor from oxyhemoglobin (due to 1-Y = 0 for Y = 1), nor from neuronal inactive regions (due to NAB = 0).

4. The volumetric computational model can be considered as a generalization of the single voxel neurovascular coupling model (Δ_{do}⋅Hct⋅(1-Y)) that has been accepted for single voxel BOLD signal simulation

Overall, a multivoxel BOLD fMRI simulation requires a predefined magnetic susceptibility distribution as the input source of T2*MRI. The susceptibility expression for a neurovascular coupling process plays a bridge between the neuroscience and MRI technology. Although the neurovascular coupling process is not fully understood so far, we propose a linear spatiotemporal modulation model (in Eq. (4)) that allows us to look into the effects of CBF, CBV, CMRO_{2}, and NAB on the BOLD susceptibility perturbation (which is detected by T2*MRI). In Figure _{0 }through t_{5 }in Figure

Illustration of dynamic magnetic susceptibility perturbation in a NAB-weighted cortical region

**Illustration of dynamic magnetic susceptibility perturbation in a NAB-weighted cortical region**. It is shown that the voxel at the central NAB region produces the maximum susceptibility perturbation (in comparison with the voxels outside the central NAB), and the dynamic susceptibility perturbation may assume a pres-stimulus transient initial dip (at t_{1}) and small post-stimulus undershoot (at t_{5}) in addition to the prevailing response mode (between t_{2 }and t_{4}). For dynamic BOLD fMRI simulation, the neurovascular coupling process should be numerically characterized by a dynamic magnetic susceptibility perturbation in a cortical region. It is noted that the voxel susceptibility timecourses are not necessarily mathematically tractable for numerical representations.

Forward BOLD fMRI simulation

From magnetism perturbation to fieldmap establishment

Given a snapshot of neurovascular-coupled BOLD state, Δχ(x, y, z), we can calculate its magnetization field distribution (resulting from the blood magnetization in a main field B_{0}), called the fieldmap henceforth, by

where _{x}, _{y}, _{z}) coordinates in the Fourier domain, conv the convolution, and

Multivoxel partition of FOV

We simulate the cortical FOV by filling it with vasculature. Figure _{0 }× D_{0 }× D_{0}, which is filled with random vascular networks (the cortical vasculature generation technique has been reported previously _{0 }× d_{0 }× d_{0}), thereby we can represent the FOV in a small array of voxels, in which each voxel can be assigned a value by intravoxel average. The process of spatial partition into voxels and intravoxel average is called voxelization. For the vasculature V(x, y, z) in the FOV, the voxelization is expressed by

A typical geometry of cortical FOV that is filled with vasculature and encloses a local neuroactivity blob (NAB in red)

**A typical geometry of cortical FOV that is filled with vasculature and encloses a local neuroactivity blob (NAB in red)**. The fMRI-detectable neurovascular coupling state is expressed as a NAB-weighted intravascular blood magnetic susceptibility perturbation distribution. The vasculature-laden FOV in size of D_{0}xD_{0}xD_{0 }is voxelized by a voxel size of d_{0}xd_{0}xd_{0}, thus producing a reduced matrix in size of [D_{0}/d_{0}, D_{0}/d_{0}, D_{0}/d_{0}]. The vasculature is randomly generated under a control of blood volume fraction (

where * denotes the convolution, and V [x_{n}, y_{n}, z_{n}] connotes the voxelization of V(x, y, z). It is shown that the voxelization operation (denoted by squared brackets "[ ]") suppresses the intravoxel details and produces a digital image representation of the vasculature configuration in the FOV. By applying voxelization to the 3D distributions of NAB(x, y, z), Δχ(x, y, z), ΔB(x, y, z), we obtain the corresponding 3D matrices of NAB[x_{n}, y_{n}, z_{n}], Δχ[x_{n}, y_{n}, z_{n}], and ΔB[x_{n}, y_{n}, z_{n}], respectively. For example, in our simulation (see later), the fieldmap is originally represented as a large matrix in size of 2048 × 2048 × 2048 (with a fine grid resolution as used for the digital FOV representation), which can be reduced a smaller matrix in size of 32 × 32 × 32 by voxelization with voxel size of 64 × 64 × 64.

Voxel signal calculation by intravoxel dephasing integration

Exposed to an inhomogeneous fieldmap, a proton precesses with a phase angle Δ_{E}
_{E}
_{E }(Larmor law). Due to the finite dimension of a voxel (for example, d_{0 }= 128 micron in our simulation), its voxel signal is formed by a vector sum of all spin packets (or isochromats) inside the voxel, called intravoxel dephasing integration _{0 }× d_{0 }× d_{0 }voxel at discrete position [_{n}, y_{n}, z_{n}

where the echo time T_{E }is explicitly retained to remind of the T_{E }dependence of voxel signal (as will be demonstrated in our simulation later).

From voxel signal values to a multivoxel image

After calculating the voxel signals for all voxels in the FOV (with a specific T_{E}), we assemble the voxel signal values into a 3D MR matrix according to the voxelization scheme in Eq. (6) by

which is complex-valued and explicitly T_{E}-dependent. Since the MR magnitude image contrast is due to a spatial distribution of voxel signal decay, we are concerned with the magnitude loss. For the phase image, we are concerned with the phase angle accumulation during the period of T_{E}. The magnitude loss map and phase accumulation map for a given T_{E }are calculated by

Where |C[x_{n}, y_{n}, z_{n}; T_{E }= 0]|denotes the non-decay initial magnitude image and ∠C[x_{n}, y_{n}, z_{n}; T_{E }= 0] the initial phase image. Henceforth, we will denote the MR magnitude and phase images as A[x_{n}, y_{n}, z_{n}; T_{E}] and P[x_{n}, y_{n}, z_{n}; T_{E}], respectively.

Backward mappings

In the forward BOLD fMRI calculation, we obtain a pair of MR magnitude image and phase image by Eq. (9). Considering the NAB[x_{n}, y_{n}, z_{n}] (the voxelized version of the NAB(x, y, z) in Eq. (6)) as the neuronal origin, the conventional neuroimaging effort consists in establishing a backward mapping from the MR magnitude images to the neuronal origin, as designated by "A ~ NAB" in Figure

It is known that the underlying source of BOLD fMRI is the susceptibility-expressed BOLD state, which is highly dependent upon the vasculature configuration in the FOV. To reduce the effect of vessel randomness on our numerical simulation, we propose to measure the similarity of the MR magnitude image A[x_{n}, y_{n}, z_{n}] and the predefined BOLD susceptibility map Δχ[x_{n}, y_{n}, z_{n}] by a spatial correlation coefficient as defined by

Where cov(x, y) denotes the covariance between vector x and y, std(x) the standard deviation, and ":" a nD-to-1D operator (as used by Matlab language) that reorders a high-dimensional array entries into one dimensional vector. The correlation coefficient defined in Eq. (10) gives rise to corrA ∈ [0, 1] with corrA = 1 for a perfect match (linear mapping) and corrA ≠1 for mismatch (nonlinear mapping). Since there is little decay for a short relaxation time (A[x_{n}, y_{n}, z_{n}; T_{E}]→0 for T_{E }→ 0), a meaningful corrA(T_{E}) should be evaluated at a relative long T_{E }(T_{E }> 0). However, a long T_{E }will also introduce a diffusion smearing effect, which is not addressed herein. It is noted that corrA defined in Eq. (10) is a numerical measure of the backward mapping "A~Δχ" in Figure

Likewise, we can measure the spatial correlation between the MR phase image P[x_{n}, y_{n}, z_{n}; T_{E}] and the fieldmap ΔB[x_{n}, y_{n}, z_{n}] by

Likely, corrP is a numerical measure of the phase-vs-fieldmap correlation, a backward mapping designated by "P~ΔB" in Figure

In summary, our computational BOLD fMRI model can be used for backward mappings: magnitude-vs-susceptibility correlation (corrA) and phase-vs-fieldmap correlation (corrP) by Eqs. (10) and (11), respectively. The goal of conventional neuroimaging and brain mapping is to render a backward mapping from MR magnitude image to neuronal origin, which consist of two steps: from the MR magnitude image to its vascular origin (as designated by "A~Δχ" and numerically measured by corrA) and then from the vascular response to its neuronal origin. In this paper, we are concerned with the mapping of "A~Δχ", with which we show the effect of MRI technology on the imaging performance of BOLD fMRI.

Results

The main parameters and their settings for the simulations we performed are listed in table

The overall numerical simulation scheme is described by a flowchart in Figure _{0 }= 3 T, we calculate the fieldmap from a 3D susceptibility perturbation in size of 2048 × 2048 × 2048 (assuming the same support matrix as FOV) by Eq. (5). By spatially partitioning the FOV into voxels, we calculate the voxel signals for a T_{E }by using the intravoxel dephasing integration in Eq. (7). At last, we assemble the voxel signal values into a 3D image matrix by Eq. (8). By repeating the multivoxel BOLD fMRI simulations for a range of T_{E }setting (T_{E }= 0:2:60 ms with an increment of 2 ms), for the different voxel sizes (128 × 128 × 128, 64 × 64 × 64, 32 × 32 × 32 micron^{3}), and for different vessel sizes (radii = 2 and 4 micron), we obtain a collection of MR magnitude images and phase images; from which we may observe the effect of MRI technology on the imaging performance of BOLD fMRI with respect to the echo time T_{E}, the image resolution, and the vessel size. In particular, we are concerned with the magnitude-vs-susceptibility correlation (corrA in Eq. (10)) and the phase-vs-fieldmap correlation (corrP in Eq. (11)). In what follows, we present the simulation results via figures.

Figure

With 1-micron digital grid resolution, we originally represent a FOV with large support matrix in size of 2048 × 2048 × 2048. After the fieldmap calculation, we partition the fieldmap into multivoxel image arrays for three voxel sizes: 16 × 16 × 16 matrix (voxel size: 128 × 128 × 128 micron^{3}), 32 × 32 × 32 matrix (voxel size: 64 × 64 × 64 micron^{3}), and 64 × 64 × 64 matrix (voxel size 32 × 32 × 32 micron^{3}). With these three voxel sizes, we demonstrate the effect of image resolution on BOLD fMRI.

Figure _{E }= 30 ms for voxel size = 64 × 64 × 64 micron^{3}. It is noted that the magnitude image resembles the susceptibility map to a great extent, and the phase image replicates the fieldmap very well. Figure _{E }dependence of the MR magnitude and phase images (with the central z-slice).

Montage displays of (a) a 3D NAB-induced multivoxel susceptibility map Δχ (in size of 32 × 32 × 32 matrix) and (b) the corresponding multivoxel fieldmap ΔB (calculated for B_{0 }= 3 T) for z-slices of z = [1,2,...,16] (z = 1 at FOV surface and z = 16 through FOV center)

**Montage displays of (a) a 3D NAB-induced multivoxel susceptibility map Δχ (in size of 32 × 32 × 32 matrix) and (b) the corresponding multivoxel fieldmap ΔB (calculated for B _{0 }= 3 T) for z-slices of z = [1,2,...,16] (z = 1 at FOV surface and z = 16 through FOV center)**. The 32 × 32 × 32 multivoxel matrix is calculated from a 2048 × 2048 × 2048 fine grid matrix (grid resolution = 1 micron), by voxelization with the voxel size of 64 × 64 × 64 micron

Montage displays of (a) the 3D multivoxel magnitude image A[x, y, z] for z = 1:1:16 and (b) the multivoxel phase image P[x, y, z] (calculated with T_{E }= 30 ms voxel size = 64 × 64 × 64 micron^{3 }and B_{0 }= 3 T)

**Montage displays of (a) the 3D multivoxel magnitude image A[x, y, z] for z = 1:1:16 and (b) the multivoxel phase image P[x, y, z] (calculated with T _{E }= 30 ms voxel size = 64 × 64 × 64 micron^{3 }and B_{0 }= 3 T)**. Note that the central z-slice of the multivoxel image is at z = 16.

Montage displays of the T_{E }dependence of MR magnitude image at the central z-slice A[x, y, z_{0}; T_{E}] (in(a)) and the corresponding MR phase image P[x, y, z_{0}; T_{E}] (in (b)) for T_{E }=

**Montage displays of the T _{E }dependence of MR magnitude image at the central z-slice A[x, y, z_{0}; T_{E}] (in(a)) and the corresponding MR phase image P[x, y, z_{0}; T_{E}] (in (b)) for T_{E }= **

With the MR magnitude and phase images generated for a variety of parameter settings, we calculate the magnitude-vs-susceptibility correlation and the phase-vs-fieldmap correlation by Eqs. (10) and (11), respectively. The results are shown in Figure _{E }in a range of 0 to 60 ms and three different image resolutions (see the legend). From Figure _{E }and voxel size; on contrary, the phase-vs-fieldmap correlation decreases with respect to T_{E }and voxel size.

(a1) and (a2) corrA for the magnitude-vs-susceptibility correlations with respect to T_{E }= [0, 60] ms, two vasculatures of radii = 2 and 4 micron, and three voxel sizes (see legend); (b1) and (b2) corrP for the corresponding phase-vs-fieldmap correlations

**(a1) and (a2) corrA for the magnitude-vs-susceptibility correlations with respect to T _{E }= [0, 60] ms, two vasculatures of radii = 2 and 4 micron, and three voxel sizes (see legend); (b1) and (b2) corrP for the corresponding phase-vs-fieldmap correlations**. It is shown that a long T

In Table ^{3}), and two selected echo times (T_{E }= 1 ms and 30 ms).

Spatial matching results in terms of spatial correlation coefficients corrA and corrP calculated by Eqs.(10) and (11) with the setting of B_{0 }= 3 T, _{E }= 1 and 30 ms for three image resolutions (voxel sizes)

**2- μm-radius vessels**

**4- μm-radius vessels**

**T _{E }= 1 ms**

**T _{E }= 30 ms**

**T _{E }= 1 ms**

**T _{E }= 30 ms**

**Voxel size(μm ^{3})**

**corrA**

**corrP**

**corrA**

**corrP**

**corrA**

**corrP**

**corrA**

**corrP**

32 × 32 × 32

0.867

1.000

0.885

0.998

0.834

1.000

0.854

0.998

64 × 64 × 64

0.883

1.000

0.899

0.999

0.871

1.000

0.887

0.999

128 × 128 × 128

0.892

1.000

0.907

0.999

0.898

1.000

0.910

0.999

Discussion

In this paper, we propose a magnetic susceptibility expression formula for a linear neurovascular coupling model (accounting for the neurovascular aspects: CBF, CBV, CMRO_{2}, FOV, and NAB) and a computational model for volumetric BOLD fMRI simulation. Overall, our goal is to numerically examine the principles of MRI-based neuroimaging: implementing the forward imaging from a neuronal origin (a numerical NAB), to vascular response (a numerical susceptibility distribution Δχ, which also serves as the vascular origin of T2*MRI), to complex multivoxel MR image formation (MR magnitude images), and then rendering the backward mappings (as designated by "A ~ NAB", "A~Δχ" and "P~ΔB" in Figure

We decompose the overall BOLD fMRI model into two modules in Figure _{2}, and NAB. It should be mentioned that the linear model neurovascular coupling model has the experimental supports

The cortical vasculature mainly consists of capillaries with radii in a range from 2 to 10 micron. Voxelization may suppress the intravoxel vasculature details (due to voxel size > > vessel size), consequently, we cannot discern the vascular structures in Figures _{E }(< 30 ms). However the corrP tends to drop for long T_{E }and high resolution (see Figure

In our simulation implementation, the cortex FOV is represented by a large 3D support matrix in size of 2048 × 2048 × 2048 (with 1-micron grid resolution) in order to depict small capillaries (digital geometry requirement). The output MR images are represented in much smaller multivoxel matrices (via voxelization): 16 × 16 × 16, 32 × 32 × 32, and 64 × 64 × 64 for three image resolutions. It is reminded that the fieldmap should be calculated from the finely-gridded susceptibility map (in the large support matrix) as a whole by a FT technique (in Eq. (5)), rather than trying to divide the large matrix into smaller submatrices. The 3D FFT on a large matrix (that is too large to be processed as a whole in computer memory) is implemented by a scheme of 2D + 1D decomposition

The simulation results for magnitude-based and phase-based backward mappings are graphically shown in Figure _{E }(shown in a range from 0 to 60 ms), whereas the phase-vs-fieldmap correlation (corrP) decreases for T_{E }> 30 ms; 2) Large voxel size (or low image resolution) cause both corrA and corrP decrease (noticeable corrP decrease for T_{E }> 30 ms); 3) Large vessels cause both corrA and corrP drop.

In addition to the study on the backward mapping between the MR magnitude image and its vascular origin, we also provide the backward mapping between the MR phase image and the fieldmap in a measure of corrP. Our simulation results show that corrP ≈ 1.00, which indicates that the MR phase image conforms very well with the fieldmap. This suggests the possibility of susceptibility reconstruction by computed inverse MRI

Numerical simulation provides a powerful tool to look into a digital imaging system as long as all the involved parameters can be numerically characterized. In light of numerical method, a parameter must be digitized for computation. The parameter digitization does not necessarily require an analytic formula, it may be fulfilled in any way. For example, the BOLD susceptibility perturbation over a FOV in Figure

Conclusions

In this report we propose a computational model for numerically simulating volumetric BOLD MRI with a magnetic susceptibility expression for a linear neurovascular coupling state. The forward procedure for this model includes: 1) defining a neuronal origin (a numerical NAB) in a cortex region(FOV); 2) expressing the NAB-modulated vascular response by a spatial distribution of intravascular blood biomagnetic susceptibility perturbation; and 3) calculating the susceptibility-induced fieldmap by accounting for the magnetization of all vascular blood in the FOV; 4) calculating the multivoxel MR image by intravoxel dephasing integration. Upon the completion of the forward simulation, we compare the output MR magnitude image with the predefined susceptibility map in a numerical measure of magnitude-vs-susceptibility correlation, thereby quantitatively examining the reproducibility of the vascular origin by the MR magnitude image. Meanwhile, we can also compare the output MR phase image with the precomputed fieldmap in terms of phase-vs-fieldmap correlation, showing that the MR phase image conforms with the fieldmap very well.

Based on our simulation results, we conclude that, 1) the magnitude image of BOLD fMRI can approximately, but not exactly, represent the vascular origin; and 2) the phase image conforms very well with the fieldmap. Considering that the vascular response to the neuronal origin is subject to a neurovascular coupling process, we can infer that the mapping between the MR magnitude image and neuronal origin suffers more mismatch than that between the MR magnitude image and the vascular origin, even if under a linear neurovascular coupling model (because the vasculature exerts a spatial modulation that imposes additional mismatch). The volumetric computational model provides a general framework for simulating many neurovascular, physiological, and biophysical aspects. It can be extended to accommodate a nonlinear neurovascular coupling process over a broad physiological range if the magnetic susceptibility expression is numerically available (not necessarily formulable). Our simulation results pose a caveat to the MRI-based neuroimaging and brain mapping study: the MR magnitude image is not an exact reproduction which may in part explain the nonlinearity of BOLD fMRI. In future research we plan to to look into the overall nonlinearity of BOLD fMRI by incorporating the intrinsic nonlinear neurovascular coupling process.

Abbreviations

BOLD: Blood oxygenation level dependent; MR: Magnetic resonance; MRI: Magnetic resonance imaging; fMRI: Functional magnetic resonance imaging; FOV: Field of view; NAB: Neuroactive blob; CBV: Cerebral blood volume; CBF: Cerebral blood flow; CMRO_{2}: Cerebral metabolic rate of oxygen;

Competing interests

The authors declare that they have no competing interests.

Authors' contributions

Zikuan Chen conceived the computational model, implemented the algorithm by program, and drafted the manuscript. Vince Calhoun analyzed the model, algorithm, data, and edited the manuscript. All authors have approved the content of the manuscript.

Acknowledgements

This research was supported by the NIH (1R01EB006841, 1R01EB005846), NSF (0612076), and the MRN internal grant 6003-154.

Pre-publication history

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