Voxel Spread Function (VSF) Method for Correction of Magnetic Field Inhomogeneity Effects in Quantitative Gradient-Echo-Based MRI

Abstract

Purpose

Macroscopic magnetic field inhomogeneities adversely affect different aspects of MRI images. In quantitative MRI when the goal is to quantify biological tissue parameters, they bias and often corrupt such measurements. The goal of this paper is to develop a method for correction of macroscopic field inhomogeneities that can be applied to a variety of quantitative gradient-echo-based MRI techniques.

Methods

We have re-analyzed a basic theory of gradient echo (GE) MRI signal formation in the presence of background field inhomogeneities and derived equations that allow for correction of magnetic field inhomogeneity effects based on the phase and magnitude of GE data. We verified our theory by mapping R2* relaxation rate in computer simulated, phantom, and in vivo human data collected with multi-GE sequences.

Results

The proposed technique takes into account voxel spread function (VSF) effects and allowed obtaining virtually free from artifacts R2* maps for all simulated, phantom and in vivo data except of the edge areas with very steep field gradients.

Conclusion

The VSF method, allowing quantification of tissue specific R2*-related tissue properties, has a potential to breed new MRI biomarkers serving as surrogates for tissue biological properties similar to R1 and R2 relaxation rate constants widely used in clinical and research MRI.

INTRODUCTION

Macroscopic (on the scale bigger than the imaging voxel) B₀ magnetic field inhomogeneities that are always present in MR experiments adversely affect different aspects of MRI images. Specifically, in quantitative MRI when the goal is to quantitatively measure biological tissue or any other system parameters, macroscopic magnetic field inhomogeneities bias and often corrupt such measurements. Numerous methods have been proposed in the past to deal with this problem both at the level of data collection (1–10) and post processing approaches (11–18).

In the presence of macroscopic magnetic field inhomogeneities in MR experiment, the signal from an imaging voxel can be represented in a general form as follows (11)

$$S(T,\{p\})=S_0(T,\{p\})·F(T)$$ [1]

where S₀(T, {p}) is a signal that would exist in the absence of macroscopic magnetic field inhomogeneities, {p} is a set of parameters that characterize system under consideration, F(T) is a function that describes contribution to the signal from macroscopic magnetic field inhomogeneities and T is a time after initial RF excitation pulse. For SE-based experiments it is often convenient to define time origin (T = 0) at a spin echo time.

A contribution into F-function in Eq. [1] from through-plane magnetic field B₀ inhomogeneities in 2D MRI can usually be achieved using the sinc-function approach (11,12) that assumes that the through-plane magnetic field inhomogeneity can be described in terms of a constant magnetic field gradient. More sophisticated methods take into account quadratic cross-slice B₀ inhomogeneity (16) and effects of spatial response function (17,18) introduced in (19) and also used for correction of EPI data (e.g. (20)). Evaluation of contribution of field inhomogeneities to the F-function in the readout and phase encoding directions is different due to the well-known Gibbs ringing artifacts that lead to inter-voxel signal leakage. These effects are dramatically enhanced in the presence of B₀ inhomogeneities (8). A semi-phenomenological method proposed in (11) accounts for the presence of the F-function in Eq. [1] but requires additional fitting parameters.

Herein we reexamine influence of B₀ magnetic field inhomogeneities on MRI signal formation. Based on this new analysis, we developed a mathematical algorithm that utilizes both magnitude and phase of MR images, allowing quantitative correction of magnetic field inhomogeneity effects in phase encoding and frequency encoding directions. This new method can be applied to a variety of MRI pulse sequences. Examples include but are not restricted to traditional gradient echo (GE) sequences, multi-gradient echo sequences such as GESFID (gradient echo sampling of Free Induction Decay (FID)) (11), GESSE (gradient echo sampling of Spin Echo (SE)) (11), GESFIDE (gradient echo sampling of FID and spin Echo) (21) and also for spectroscopy CSI (chemical shift imaging).

THEORY

Voxel Spread Function (VSF) Method - General Consideration

Consider a 3D GE experiment with X and Y phase encoding directions and Z being a readout. In the presence of inhomogeneous magnetic field b(r) MR signal during the readout period t can be represented as follows:

$$\mathrm{S̃}(\mathbf{k};\mathit{\text{TE}})={\displaystyle \int }d\mathbf{r}·\rho (\mathbf{r};\mathit{\text{TE}})\text{exp}[-2\pi i\mathbf{\text{kr}}+i\gamma b(\mathbf{r})(\mathit{\text{TE}}+t)+i{\phi }_0(\mathbf{r})]$$ [2]

Here we presented time T in Eq. [1] as T = TE + t, so that TE is the gradient echo time, t is time during gradient echo acquisition (t is zero at the center of the gradient echo), γ is the gyromagnetic ratio, φ₀(r) is the signal phase shift at TE=0 (mainly resulted from RF field inhomogeneities), and k-space is defined in a standard manner as

$$2\pi k_x=\gamma G_x{t}_x;2\pi k_y=\gamma G_y{t}_y;2\pi k_z=\gamma G_{z}t$$ [3]

where G_x, G_y, and G_z are phase encoding (x and y) and read-out (z) gradients, t_x and t_y are durations of x and y gradients, ρ(r; TE) is the “ideal” signal evolution that would exist in the absence of magnetic field inhomogeneities. The integral in Eq. [2] is over the object volume. It should be kept in mind that the “ideal” signal ρ (r;TE) in Eq. [2], in general, depends on pulse sequence parameters, tissue magnetic properties and sensitivity of RF coil used for MR signal excitation and receiving. Note that we adopted a very specific signs in the exponential in Eq. [2]. If data are acquired with different signs, the theory should be modified accordingly.

Our general goal is to solve Eq. [2] and to find spatial distribution of an “ideal” signal ρ(r; TE). One of the approaches to this problem, NL-CSI, was proposed in our previous publication (8) which extended SLIM reconstruction technique (22) by accounting for the presence of magnetic field inhomogeneities. The method (8), however, required acquisition of additional high resolution MR images. Herein we propose another method that does not require any additional data but is based on certain “linear” approximations that are appropriate for high resolution MRI. These approximations are outlined below.

Consider first MR image that is IFT (Inverse Fourier Transform) of the signal:

$$S_n(\mathit{\text{TE}})=\frac{1}{N}·{\displaystyle \sum _{\mathbf{k}}}\mathrm{S̃}(\mathbf{k};\mathit{\text{TE}})·\text{exp}(2\pi i{\mathbf{\text{kr}}}_n)$$ [4]

where r_n is a position of the nth voxel. The sum in Eq. [4] is taken over k-space defined in Eq. [3] with

$$k_j=(-1/2+1/N_j,-1/2+2/N_j,\dots ,1/2)/a_j$$ [5]

where j = x, y, z, N_j are the imaging matrix dimensions in x, y and z directions, a_j are the corresponding voxel dimensions, N = N_xN_yN_z.

In a linear approximation (reasonable for high resolution images), the distribution of inhomogeneous macroscopic magnetic field B₀, b(r), can be represented in the nth imaging voxel as

$$b_n(\mathbf{r})=b_n+g_{\mathit{\text{nx}}}x+g_{\mathit{\text{ny}}}y+g_{\mathit{\text{nz}}}z$$ [6]

As it is well known, the term b(r)·t in Eq. [2] leads to the voxel displacement and distortions (e.g. (23–26)). However, if the readout gradient is much bigger than the background gradients in Eq. [6], these effects are small. While they can be included in our theoretical consideration, we will focus our attention on another term - (b(r)·TE) - which is responsible for the signal losses. Even if imaging gradients are bigger than the background gradients, the influence of this term grows with gradient echo time TE. Hence it cannot be ignored in MR imaging with long times TE.

Assuming that the “ideal” signal ρ(r; TE) only slightly varies across the voxels, we substitute it by the averaged values across a given voxel:

$$\rho (\mathbf{r};\mathit{\text{TE}})\to {\rho }_n(\mathit{\text{TE}})={\langle \rho (\mathbf{r};\mathit{\text{TE}})\rangle }_n$$ [7]

where 〈…〉 _n means averaging across the nth voxel. At the same time for the phase φ₀(r) we make a linear approximation similar to Eq. [6]:

$${\phi }_0(\mathbf{r})={\phi }_{0,n}+{\phi }_{\mathit{\text{nx}}}x+{\phi }_{\mathit{\text{ny}}}y+{\phi }_{\mathit{\text{nz}}}z;{\phi }_{0,n}={\langle {\phi }_0(\mathbf{r})\rangle }_n$$ [8]

Thus, the signal S̃(k; TE) in Eq. [2] can be written as

$$\mathrm{S̃}(\mathbf{k};\mathit{\text{TE}})={\displaystyle \sum _n}{\sigma }_n(\mathit{\text{TE}})·\text{exp}(+i\gamma b_n·\mathit{\text{TE}}+i{\phi }_{0,n})·\text{exp}(-2\pi i{\mathbf{\text{kr}}}_n)·\text{sinc}[(k_x-k_{\mathit{\text{nx}}})a_x]·\text{sinc}[(k_y-k_{\mathit{\text{ny}}})a_y]·\text{sinc}[(k_z-k_{\mathit{\text{nz}}})a_z]$$ [9]

The sum in Eq. [9] is over all imaging voxels n, each with the volume V = a_xa_ya_z; σ_n(TE) = V · ρ_n(TE) is the “ideal” signal from the nth voxel, and sinc(u) = sin(πu) / (πu). Parameters k_nj are defined as

$$2\pi k_{\mathit{\text{nj}}}=\gamma g_{\mathit{\text{nj}}}\mathit{\text{TE}}+{\phi }_{\mathit{\text{nj}}},j=x,y,z$$ [10]

and describe the signal shifts in the k-space due to the presence of background field gradients, Eq. [6] and RF field inhomogeneities, Eq. [8].

Equations [9] represent a set of equations with respect to σ_n (TE), where signal S̃(k; TE) is a measured MR signal. To solve this set of equations, we need to define all other parameters in Eqs. [9]. The distribution of magnetic field b(r) can be evaluated from the same data set in Eqs. [4]. Indeed, the complex image in Eqs. [4] can be represented as

$$S_n(\mathit{\text{TE}})=|S_n(\mathit{\text{TE}})|·\text{exp}[i{\phi }_n(\mathit{\text{TE}})]$$ [11]

where φ_n (TE) is a signal phase. For small phase dispersion across the voxel, the phase φ_n (TE) is

$${\phi }_n(\mathit{\text{TE}})={\phi }_{0,n}+\gamma b_n·\mathit{\text{TE}}$$ [12]

Hence, the phase φ_{0, n}, and the average magnetic field b_n in each voxel can be found by fitting this equation to experimental data in the form of Eq. [11] with multiple TE. However, numerical simulations show that the field inhomogeneities can distort linear behavior of the signal phase in Eq. [12] and only short TE can be used for evaluation of φ_{0, n} and b_n (see Results and Discussion sections).

If magnetic field and initial signal phase change slowly on the scale of a voxel dimension, we can use linear approximation and calculate field gradients g_nj and phase gradients φ_nj as follows:

$$g_{\mathit{\text{nj}}}=\frac{1}{2a_j}(b_{\mathit{\text{nj}}+1}-b_{\mathit{\text{nj}}-1});{\phi }_{\mathit{\text{nj}}}=\frac{1}{2a_j}({\phi }_{0,\mathit{\text{nj}}+1}-{\phi }_{0,\mathit{\text{nj}}-1}),$$ [13]

and then find parameters k_nj from Eq.[10]. After that Eqs. [9] become a set of linear equations that could be solved using different mathematical algorithms (e.g. similar to those described in (8)). This solution defines a set of values σ_n (TE) that represents an MR image free of artifacts resulting from MR signal decay due to B₀ field inhomogeneities. Note also that this solution is free from Gibbs ringing artifacts.

Herein we propose another method in which the MRI signal is analyzed in the imaging domain rather than the k-space signal in Eq. [9]. Since Eq. [9] has a similar structure in all three directions, we can analyze first its 1D analog when data are acquired in the x direction. In the 1D case, the image in Eq. [4] is:

$$S_n(\mathit{\text{TE}})=\frac{1}{N}{\displaystyle \sum _m}{\sigma }_m(\mathit{\text{TE}})·\text{exp}(+i\gamma b_m\mathit{\text{TE}}+i{\phi }_{0,m})·{\displaystyle \sum _q}\text{sinc}(q-q_m)·\text{exp}[2\pi \mathit{\text{iq}}(n-m)]$$ [14]

where q = k_xa_x. The parameters q_m, representing the phase dispersion across the 1D voxel, are defined as

$$2\pi q_m(\mathit{\text{TE}})=(\gamma g_{\mathit{\text{mx}}}\mathit{\text{TE}}+{\phi }_{\mathit{\text{mx}}})a_x$$ [15]

It is convenient to rewrite Eq. [14] in the form:

$$S_n(\mathit{\text{TE}})={\displaystyle \sum _m}{\mathrm{\Psi }}_{\mathit{\text{nm}}}(\mathit{\text{TE}})·{\sigma }_m(\mathit{\text{TE}})$$ [16]

where

$${\mathrm{\Psi }}_{\mathit{\text{nm}}}(\mathit{\text{TE}})=\eta (n,m;q_m(\mathit{\text{TE}}))·\text{exp}(i{\phi }_{0,m}+i\gamma b_m\mathit{\text{TE}}),$$ [17]

and

$$\eta (n,m;q_m(\mathit{\text{TE}}))={\displaystyle \sum _q}\text{sinc}(q-q_m(\mathit{\text{TE}}))\text{ exp }(2\pi \mathit{\text{iq}}(n-m))$$ [18]

will be referred to as a voxel spread function or VSF.

Equations [14]–[18] can be readily generalized for 2D and 3D cases. In the 2D case, the indices n, m and q should be considered as 2D vectors with components (nx, ny), (mx, my) and (qx, qy). In the 3D case, they are 3D vectors (nx, ny, nz), (mx, my, mz) and (qx, qy, qz). In these cases, the functions Ψ_nm (TE) appearing in Eq. [16] are

$$\begin{array}{cc}\hfill {\mathrm{\Psi }}_{\mathit{\text{nm}}}(\mathit{\text{TE}})=\text{exp}(i{\phi }_{0,m}+i\gamma b_m\mathit{\text{TE}})·{\eta }_x·{\eta }_y·{\eta }_z^{(0)}\hfill & \hfill (2D)\hfill \\ \hfill {\mathrm{\Psi }}_{\mathit{\text{nm}}}(\mathit{\text{TE}})=\text{exp}(i{\phi }_{0,m}+i\gamma b_m\mathit{\text{TE}})·{\eta }_x·{\eta }_y·{\eta }_z\hfill & \hfill (3D)\hfill \\ \hfill {\eta }_j=\eta (\mathit{\text{nj}},\mathit{\text{mj}};q_{\mathit{\text{mj}}}(\mathit{\text{TE}}))\hfill \end{array}$$ [19]

where the voxel spread functions η_j = η(nj, mj; q_mj (TE)) are defined by Eq. [18] and q_mj (TE) by Eq. [15] with corresponding j = x, y, z. For the 2D case, the function ${\eta }_z^{(0)}=\text{sinc}(q_{\mathit{\text{mz}}}(\mathit{\text{TE}}))$ takes into account the phase gradients across the imaging slices.

Note that if the linear approximation in Eq. [6] is not sufficient for calculating effects of field inhomogeneities across the voxel, then the sinc-function in Eq. [18] and the product of sinc-functions in Eqs. [19] should be substituted by a more general integral:

$${\displaystyle \int }d\mathbf{r}\text{ exp }[-i{\phi }_0(\mathbf{r})-i\gamma \left(b_m(\mathbf{r})-\langle b_m\rangle )·\mathit{\text{TE}}-2\pi i\mathbf{\text{kr}})\right]/V$$ [20]

where integration is taken across a single voxel with volume V. In this case the distribution of magnetic field and phases can be found from an independent experiment, such as high resolution GE MRI with multiple gradient echoes (8).

Similarity Approximation

While solution of Eq. [16] (or its generalization for 2D and 3D cases, Eqs. [19]) involves inversion of large matrices and maybe computationally labor intensive, we can explore an approximation that, as will be demonstrated, provides a rather accurate result. In this approximation we single out the influence of magnetic field inhomogeneities by assuming that the signals from neighboring voxels behave similarly (in the absence of magnetic field inhomogeneities) and make the following substitution in Eq. [16]:

$${\sigma }_m(\mathit{\text{TE}})\Rightarrow {\sigma }_n(\mathit{\text{TE}})\frac{|S_m(0)|}{|S_n(0)|}$$ [21]

In this case Eq. [16] reduces to

$$S_n(\mathit{\text{TE}})={\sigma }_n(\mathit{\text{TE}})·F_n(\mathit{\text{TE}})$$ [22]

where the F-function describing the influence of magnetic field inhomogeneities on image is

$$F_n(\mathit{\text{TE}})=\frac{1}{|S_n(0)|}·{\displaystyle \sum _m}{\mathrm{\Psi }}_{\mathit{\text{nm}}}(\mathit{\text{TE}})·|S_m(0)|$$ [23]

The matrix Ψ_nm (TE) is defined by Eqs. [17] or [19] for the 1D, 2D and 3D cases respectively.

Filtered Images

The above equations can be used either directly or in a conjunction with different filters. An efficient way to reduce Gibbs ringing artifacts and improve SNR is a well-known Hanning filter, however other filters can be employed in a similar manner.

The Hanning filter corresponds to the following transformation of the signal (for 1D case):

$$S_n\to S_n^{(H)}==\frac{1}{2}{S}_n+\frac{1}{4}{S}_{n+1}+\frac{1}{4}{S}_{n-1},$$ [24]

or

$$S_n^{(H)}(\mathit{\text{TE}})=\frac{1}{N}{\displaystyle \sum _k}\mathrm{S̃}(k;\mathit{\text{TE}})\text{ exp}\left[2\pi {\mathit{\text{ikx}}}_n\right]·{\text{cos}}^2(\pi \mathit{\text{ka}})$$ [25]

Equations [16]–[18] should be transformed accordingly:

$$\begin{array}{c}{S}_n^{(H)}(\mathit{\text{TE}})={\displaystyle \sum _m}{\mathrm{\Psi }}_{\mathit{\text{nm}}}^{(H)}(\mathit{\text{TE}})·{\sigma }_m\hfill \\ {\mathrm{\Psi }}_{\mathit{\text{nm}}}^{(H)}(\mathit{\text{TE}})={\eta }^{(H)}(n,m;q_m(\mathit{\text{TE}}))·\text{exp}(i{\phi }_{0,m}+i\gamma b_m\mathit{\text{TE}})\hfill \\ {\eta }^{(H)}(n,m;q_m(\mathit{\text{TE}}))={\displaystyle \sum _q}\text{sinc}(q-q_m(\mathit{\text{TE}}))·{\text{cos}}^2(\pi q)·\text{exp}(2\pi \mathit{\text{iq}}(n-m))\hfill \end{array}$$ [26]

Note that the function η^(H) decays with distance much faster than the function η, Eq. [18] (see Fig. 1).

Figure 1.

The voxel spread function as a function of the distance from a given voxel to the neighboring voxel (0, 1, 2, 3, 4) and the parameter q (range from 0 to 2). Left panel – the original voxel spread function, Eq. [18]; right panel – the voxel spread function for Hanning filtered data, Eq. [26].

A similarity approximation in Eqs. [22]–[23] can also be applied for Hanning filtered data resulting in the following approximation for image data and F-function:

$$\begin{array}{c}{S}_n^{(H)}(\mathit{\text{TE}})={\sigma }_n(\mathit{\text{TE}})·F_n^{(H)}(\mathit{\text{TE}})\hfill \\ F_n^{(H)}(\mathit{\text{TE}})=\frac{1}{|S_n^{(H)}(0)|}·{\displaystyle \sum _m}{\mathrm{\Psi }}_{\mathit{\text{nm}}}^{(H)}(\mathit{\text{TE}})·|S_m^{(H)}(0)|\hfill \end{array}$$ [27]

Multi-channel data

An additional consideration is required when MR data is collected with multi-channel receiver. In this case, all the above results are valid for MR signal originated from each RF channel ν (ν = 1, …, N_ch; N_ch is a number of channels). In particular,

$$S_n^{\nu }(\mathit{\text{TE}})={\displaystyle \sum _m}{\mathrm{\Psi }}_{\mathit{\text{nm}}}^{\nu }(\mathit{\text{TE}})·{\sigma }_m^{\nu }(\mathit{\text{TE}})$$ [28]

where the function ${\Psi }_{\mathit{\text{nm}}}^{\nu }(\mathit{\text{TE}})$ is similar to that in Eq. [17] with the substitution ${\phi }_{0,m}\to {\phi }_{0,m}^{\nu }$.

The signals from the individual channels can then be processed by means of the joint analysis or data from the channels can be combined together. In the latter case, the combined signal can be presented in the form:

$${\mathrm{S̅}}_n(\mathit{\text{TE}})={\displaystyle \sum _{\nu }}{\lambda }_n^{\nu }·S_n^{\nu }(\mathit{\text{TE}})$$ [29]

where ${\lambda }_n^{\nu }$ are weighting coefficients that should be selected to maximize SNR in the data. As demonstrated in (27), in the case of real signals, the optimal weighting factors are ${\lambda }_n^{\nu }=S_n^{\nu }(0)/{\mathrm{\Delta }}_{\nu }^2$, where ${\mathrm{\Delta }}_{\nu }^2$ is the noise variance in the channel ν. In the general case of complex data, this result can be generalized as follows (28):

$${\lambda }_n^{\nu }=\frac{1}{{\mathrm{\Delta }}_{\nu }^2}·|S_n^{\nu }(0)|·\text{ exp }(-i{\phi }_{0n}^{\nu })$$ [30]

An additional factor $\text{exp}(-i{\phi }_{0,n}^{\nu })$ is required to avoid incoherence in the signals from different channels. After that, the F-functions calculated for individual channels can be combined using the same weighting coefficients as in Eq. [30]. Another approximation could employ calculation of the F-function simply based on the combined data.

MATERIALS AND METHODS

Phantom experiment: A spherical flask (3 cm diameter) was attached on the side of a cylindrical bucket (40cm long, 15 cm cross-section diameter). Both were filled with water doped with 3.75 g of NiSO₄·6H₂O and 5 g of NaCl per 1L of water.

Human experiments were conducted on a healthy volunteer with the approval of Washington University IRB after obtaining written informed consent.

MRI acquisition: all data are collected on Siemens 3T Trio MRI scanner (Siemens, Erlangen, Germany) using 3D multi-gradient echo sequence. For brain studies we use a 12-Channel phased-array head coil and a sequence with the following parameters: FOV 256 mm × 192 mm × 120 mm; 11 echoes; TR = 50 ms; min TE = 4 ms; delta-TE = 4 ms; bandwidth = 510 Hz/Pixel; FA = 30; resolution 1 × 1 × 3 mm³. For phantom studies we use a single channel knee coil and the following sequence parameters: FOV 256 mm × 256 mm × 96 mm; 11 echoes; TR = 50 ms; min TE = 4 ms; delta-TE = 4 ms; bandwidth = 510 Hz/Pixel; FA = 60; resolution 2 × 2 × 2 mm³. Cylindrical part of the phantom was placed parallel to the field B₀. The gradient echo time dependence of signal phase in Eq. [12] was determined after unwrapping in a TE domain as described in (28).

Simulations: For computer simulations we use 20-cm long 1D object. The computer imaging experiment mimicked real imaging with parameters that are typical for our in vivo experiments on 3.0 T scanner: FOV 256 mm, number of phase-encoding steps = 256, and spectral bandwidth = 510 Hz/px with gradient echo spacing of 4 ms. Nevertheless, herein we chose echo spacing = 1 ms and number of echoes = 500, to examine the behavior of the F-function in more details and in a broader range than experimental conditions. To further mimic our in vivo study, the inhomogeneous background magnetic field was superimposed on our 1D object with a profile measured from a real human data. A 6^th order polynomial fit was used to present detail field inhomogeneity profile used in calculations. The direction of the background magnetic field gradient was chosen to be the same as the phase-encoding gradient.

Data processing: All programs for data analysis and computer simulations were written in Matlab® (MathWorks, Natick, Massachusetts, U.S.A) according to descriptions in the "Theory" section.

RESULTS: Application to measurement of R2^* relaxation rate constant

Equations [22]–[23] or [27] along with the definition of the matrix Ψ_nm, provide a tool for accounting for the macroscopic magnetic field inhomogeneities. This approach can be applied for analyzing MR signals describing by different models. The most frequently used model is the one, in which the MR signal decay is characterized by a mono-exponential function,

$${\sigma }_n(\mathit{\text{TE}})={\sigma }_n(0)\text{exp}(-R{2}^{*}·\mathit{\text{TE}})$$ [31]

where R2^* is the transverse relaxation rate constant. Since our approach allows elimination of the effects of macroscopic field inhomogeneities, the parameter R2^* becomes tissue specific and can provide important information on tissue properties. For example, it can be used for quantitative assessment of tissue damage in multiple sclerosis (29) and other diseases. Substituting Eq. [31] into Eq. [1] we get for each voxel:

$$S(\mathit{\text{TE}})=S(0)·\text{exp}(-R{2}^{*}·\mathit{\text{TE}})·F(\mathit{\text{TE}})$$ [32]

Fitting this expression to experimental data on the voxel-by-voxel basis, we can create a map of parameter R2^*.

First we applied our technique to computer-simulated data. Simulation was done using a 1D object with homogeneous spin density and background field that resembles the in vivo data. In the simulations, the internal T2^* was set to be infinite (R2^* = 0), so that the signal decay would only be due to the presence of field inhomogeneities. Consequently, the signal time profile should coincide with the F-function. Simulation details are explained in the Methods section. Figure 2 shows frequency profile used in simulations and corresponding frequency gradient across the pixel. An example of F-function at the point where field gradient is rather big (square in Fig. 2) is shown in Fig. 3. Data are plotted against the dimensionless parameter q introduced in Eq. [14]. Figure 4 shows resulting R2^* map obtained with and without correction for field inhomogeneities.

Figure 2.

Frequency profile through the 1D object used in simulation (left panel) and a corresponding frequency gradient (right panel). The frequency profile was adopted from human data and measured going anterior → posterior through the center of an imaging slice shown in Fig. 6.The square is a representative point used for illustrations in Fig. 3.

Figure 3.

Example of the data (circles) corresponding to the point indicated by the square in Fig. 2, where the field gradient is very steep. Left panels (A and C) represent magnitude data and corresponding F-functions; right panels (B and D) represent phase data and their linear fits to initial portion. A and B - no filter, C and D - Hanning filtered data. Red lines in A and C show the F-function calculated with m ranging from −2 to 2, per Eq. [23]; Blue line in A corresponds to the F-function calculated with m ranging from −8 to 8. Data are plotted against the parameter q introduced in Eq. [14].

Figure 4.

R2^* profiles of the simulated data. Dotted line shows R2* profile resulting from mono-exponential fitting without F-function, solid line - with F-function, Eq. [32]. Since signal in simulated data decays only due to field inhomogeneities, the true R2^* should be zero. Application of the F-function gives very good results except for points at the edges, where the background field gradient is maximal.

In Fig. 5, we demonstrate the maps obtained in the phantom study without (left panel) and with (right panel) the F-function accounting for the macroscopic magnetic field inhomogeneities. Figure 6 represents example of the data obtained on a healthy human subject. The middle image represents the R2^* map obtained without correction for field inhomogeneities and right image – the R2^* map with field inhomogeneity correction.

Figure 5.

Example of the images obtained from the phantom study without (left panel) and with (right panel) correction for the macroscopic magnetic field inhomogeneities. The upper row represents R2^* maps and the lower row represents signal intensity maps S(0). Data were analyzed after applying Hanning filter.

Figure 6.

Example of the data obtained from a human subject. Left image – signal intensity map, middle and right images – R2^* map obtained without and with correction for field inhomogeneities, respectively. Data were analyzed after applying Hanning filter.

DISCUSSION

Simulated data presented in Figs. 2–4 demonstrate the main features of the proposed VSF method. Usually one would expect that the presence of magnetic field inhomogeneities should lead to decrease of FID signal with time after RF excitation. However, as was first demonstrated in (8), the signal leakage (Gibbs ringing artifacts, point spread function effects) dramatically increases in the presence of inhomogeneous field. As a result, signal decay in a given voxel is compensated by a “signal inflow” from the neighboring voxels. This effect is seen in Fig. 3A. When no filter is applied during data processing, the signal does not decay at low values of the parameter q (short gradient echo times TE), then shows increased fluctuations around q = 0.3, followed by a steep signal drop. However, applying Hanning filter reduces the signal inflow effects, leading to a rather “normal” decaying signal – Fig. 3C. This is also consistent with the behavior of a voxel spread function shown in Fig. 1. For the unfiltered data, the corresponding F-function calculated with a small number of neighbors (two on both sides of point of interest) is able to capture the signal behavior only for small q (red line in Fig. 3A). The F-function calculated with 8 neighbors on both sides of point of interest captures practically all peculiar details, (blue line in Fig. 3A). When Hanning filter is applied to data, the signal behavior changes: it monotonically decays and displays much smoother curve, Fig. 3C. Moreover, the corresponding F-function agrees very well with the signal even though only two neighbors on both sides of point of interest were used. Also note that the signal phase deviates from the linear fit for q > 0.4. The R2^* map of simulated data, Fig. 4, demonstrates that our approach provides almost perfect correction in calculating R2^* for practically all points except of the voxels at the very edge of our 1D object where magnetic field gradients are the biggest.

Results of the phantom studies are presented in Fig. 5. As described in the Methods, the phantom consisted of a cylinder oriented parallel to the magnetic field B₀ and a sphere positioned next to the cylinder; both filled with doped water. The presence of the sphere creates strong magnetic field inhomogeneities within the cylinder (and vice versa). This inhomogeneities lead to substantial increase in evaluated R2^* values in the vicinity of the interface between the cylinder and sphere if correction for magnetic field inhomogeneities is not taken into account. As we see, accounting for the macroscopic field inhomogeneities by means of our approach (with F-function) significantly decreases the effect of the field inhomogeneities (though not completely at the very edge of the phantom in the area of high field gradients, similar to simulated data): the area of deviation of the parameter R2^* from its undistorted value (achieved in the central area of the map where field inhomogeneities are minimal) is much smaller. Besides, this approach diminishes the artifacts in the amplitude maps (lower panel in Fig. 5).

A limitation of the method to correct data near the edge of an object might be related to the edge effect (insufficient data for calculating frequency gradients at the object boundary) and the strength of magnetic field gradient. The former can be addressed by calculating field gradients at the edge voxels by using one-sided-neighbor equations instead of Eqs. [13]. For example, for a left-sided neighbor, the equation is g_nj = (b_nj − b_nj−1) / a_j; φ_nj = (φ_0,nj − φ_0,nj−1) / a_j. The latter can be addressed by using general Eq. [20] for detail accounting for magnetic field inhomogeneities. However, results obtained on a phantom and shown in Fig. 5 demonstrate that VSF algorithm provides adequate correction for most of the edge voxels except of the ones with very strong gradients.

As in the phantom study, in the human brain example, accounting for the macroscopic field inhomogeneities by means of our VSF approach, also results in a significantly improved R2* map as compared with the map obtained without such a correction as shown in the example in Fig. 6. One can see that the bright spot artifact in the R2* map (central image) obtained without F-function is removed in the right image where magnetic field inhomogeneities are accounted for by means of F-function in Eq. [32].

With increasing parameter q, the deviation of calculated F-function from signal increases. For Hanning filtered data this deviation reaches up to 1% at q about 0.3. Figure 7 represents histograms of MR signal frequency gradients in human brain measured with our GEPCI approach for three directions. Dash lines show boundaries where in our human brain experiments with maximum TE of 44 ms parameter q hits 0.3. By examining this distribution in the whole brain (axial slices coverage 120 mm from top of the brain), we found that field gradient in z direction is the main limiting component; this is because the voxel size is the biggest in z direction (3 mm vs. 1 mm in plane); still 94% of the voxels can be described in terms of our F-function correction approach.

Figure 7.

Distribution of the frequency gradients in x (A), y (B), and z (C) directions measured in human brain. The vertical axes of the histograms are # of pixels in the brain, and horizontal axes are gradients of frequencies in the brain in ppm/cm. Dash lines enclose a range of the gradients where the error in estimating F-function is less than 1% (q < 0.3) under our experimental condition with maximal TE = 44 ms and resolution of 1×1×3 mm³.

When inhomogeneous magnetic field is superimposed on the imaged object, it creates a number of artifacts in a regular “weighted” imaging but it is especially important in a quantitative imaging. Many solutions to different aspects of these artifacts have been discussed in the MRI literature from the very inception of MRI, e.g. (1–20,23–26). Herein we analyze a very specific question – how to correct for the effects of magnetic field inhomogeneities in a quantitative MRI based on different versions of multi-gradient-echo sequences.

One of the applications is based on mapping tissue T2* relaxation time constant (T2* = 1/R2*) exemplified in our paper. Using F-function correction, allows obtaining T2* relaxation time constant that is free from contamination by background field gradients, thus representing internal tissue properties, therefore having a potential to serve as an MRI surrogate for tissue biological properties similar to T1 and T2 relaxation time constants widely used in clinical and research MRI. This T2* surrogate has been used, for example, in the GEPCI technique to quantify tissue damage in multiple sclerosis (28–30). However, utilizing it without correction for field inhomogeneities can lead to substantial quantitative errors.

The difference between T2 and T2* can usually be attributed to the presence of tissue-specific mesoscopic field inhomogeneities (11,31) that are present due to magnetic susceptibility differences between tissue components at the scale that is much smaller than the voxel size. These mesoscopic magnetic field inhomogeneities, being tissue specific, can provide valuable information on tissue structure and functioning. An example is a well-known BOLD effect (32,33) where mesoscopic magnetic field inhomogeneities are generated due to the magnetic susceptibility differences between deoxygenated blood and surrounding tissue. A quantitative BOLD method (qBOLD) has been developed for quantification of blood volume fraction and blood oxygenation level (34–36). In this case the signal model in Eq. [32] should be further modified:

$$S(\mathit{\text{TE}})=S(0)·\text{exp}(-R_2·\mathit{\text{TE}}-\mathit{\text{dCBV}}·f(\delta \omega ·\mathit{\text{TE}})·F(\mathit{\text{TE}})$$ [33]

where the parameters dCBV and δω are proportional to volume and magnetic susceptibility of deoxygenated blood (31). The presence of multiple water components (34–36) can also be incorporated in the model in Eq. [33]. Accounting for background field inhomogeneities is crucial for this quantitative technique. In another application, tissue can be characterized by several compartments/components (e.g., fat, water, metabolites, etc.). In this case, the theoretical model is

$${\sigma }_n(\mathit{\text{TE}})={\displaystyle \sum _p}{\sigma }_n^p(0)·\text{exp}(i{\omega }_p\mathit{\text{TE}}-R{2}_p^{*}\mathit{\text{TE}})$$ [34]

where index p enumerates water and lipid compartments with their frequencies ω_p and relaxation rates R2_p*. This problem has attracted substantial interest in the recent literature especially with respect to imaging fat and water components in the liver (17,18).

CONCLUSION

In this paper we have proposed a solution to a rather old problem – how to account for the presence of background field inhomogeneities in a gradient echo-based quantitative MRI. While a number of papers have been devoted to this problem in the past (see discussion in the Introduction), complexity of the problem still leaves a room for the improvement. The VSF approach developed in this manuscript provides another step in this direction. We have re-analyzed a basic theory of gradient echo MRI signal formation in the presence of background field inhomogeneities, based on this analysis we developed a new method (VSF method) that allows correction for the effects of background field inhomogeneities, provided computer simulations demonstrating a validity of our VSF approach and demonstrated preliminary results in applying the VSF method for quantitative mapping R2* relaxation rate constant in a phantom and the human brain. The obtained R2* relaxation rate constant is free from contamination by background field gradients and represents internal tissue properties. Our proposed VSF method of accounting for background field inhomogeneities in GE sequences offers a promising potential for R2* to serve as an MRI surrogate for tissue biological properties similar to R1 and R2 relaxation rate constants widely used in clinical and research MRI.